Quantum Mechanics

Table of Contents

Reading

  • Quantum Physics - Stephen Gasiorowicz
  • "Quantum Physics", Messiah, p. 463 Appendix A: Distributions & Fourier Transform

Progress

  • Note taken on [2017-09-20 Wed 21:40]
    p. 115

Definitions

Words

plane wave
multi-dimensional wave
wave packet
superposition of plane waves
hilbert space
Banach space withe addition of an inner product
banach space
Space with metric, and is complete wrt. to the metric in a sense that each Cauchy sequence converges to a limit within the space.
isotropic
Independent of orientation.

Bound / unbound wave equations

Bound energy

Restricts us to positive energies quantum_mechanics_0a4abce2547ae964d29a0df05ba6397a4af6d9af.png

Unbound energy

Allows both positive and negative energies quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png

Schrödinger equation

quantum_mechanics_c43a03a35cedaab7b04b7cb206f17fee1b59a195.png

Operators

Position operator

quantum_mechanics_8d9db976d6ea68b03d62b764a0a3bc46acac4931.png

Russel-Sanders notation

States that arise in coupling orbital angular momentum quantum_mechanics_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png and spin quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png to give total angular momentum quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png are denoted:

quantum_mechanics_4bf389c2ec088d1c4a1168a45e461c9c946ae313.png

where the quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png is "optional" (not always included).

Further, remember that we often use the following notation to denote the different angular momentum quantum_mechanics_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png:

quantum_mechanics_0e76355b88ebab98242133c701062efc79ff36a3.png

Stuff

Zero point energy

Heisenbergs uncertainty principle tells us that a particle cannot sit motionless at the bottom of it's potential well, as this would imply that the uncertainty in momentum quantum_mechanics_31a03410ee4f6b70b6c478716d0f20a1726b3a12.png would be infinite. Therefore, the lowest-energy state of the system (called ground state ) must be a distribution for the position and momentum which at all times satisfy the uncertainty principle.

Near the bottom of a potential well, the Hamiltonian of a general system (the quantum-mechanical operator giving a system its energy) can be approximated as a quantum harmonic oscillator:

quantum_mechanics_00cb75b50723a00d0497acc1ff895319bb04336b.png

Quantum Harmonic Oscillator

quantum_mechanics_68fe13aa98eddf941f0582e789ed0840f8415f33.png

Thus, the eigenvalue function of the Hamiltonian becomes

quantum_mechanics_3cd2816b943175f0b691ca29cd52efda73abec9e.png

As it turns out, this differential equation has the eigenfunctions:

quantum_mechanics_b520d2892dadf2405f094688dc9ec8b3e99266ca.png

where quantum_mechanics_3ef4dc0ce45df8fefebf33c12471a2c6cbc776de.png denote the n-th Hermite polynomial, with quantum_mechanics_60900da62e4da296177ff13ae78ebe3f10effcb9.png.

Further, using the raising and lowering operators discussed in Angular momentum - reloaded we can rewrite the solutions as

quantum_mechanics_1efe1207fa0c70940c7dc01c78ec96eb03f5468c.png

Annihiliation / lowering operator:

quantum_mechanics_66e6e422a9782195eeb106195fd0fee953099c58.png

Creation / Raising operator:

quantum_mechanics_175ceab886da44ed4f03de33bce86c59f400201d.png

Solution

There's two ways of going about this:

  1. Solving using the ladder operators, but you somehow need to obtain the ground levels
  2. Solving using straight up good ole' method of variation of parameters and Sturm-Liouville theory
Solving using ladder-operators
  • Notation
    • quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png is the eigenfunction corresponding to the n-th eigenvalue quantum_mechanics_049620bed349a8deabe58aa25ed270a33ddb7c9d.png
    • quantum_mechanics_649e2de6ebe71378c7febad203b2a1fcf0b0d5b8.png, which is dimensionless
    • quantum_mechanics_d6028768e74443d672385020735d0afd45107b34.png, which is dimensionless
    • quantum_mechanics_583c360dccf26d00274abbdc4f4b5963cf5bafab.png
    • quantum_mechanics_af55fe005c48857b07670eba68dfe03769752444.png
  • Stuff

    The Hamiltonian in this case is given by

    quantum_mechanics_3fdb82c25017844e36fac1a2e025d017850ececf.png

    with the TISE

    quantum_mechanics_0847b65914bee86d1018b067083470351c459e99.png

    For convenience we rewrite the Hamiltonian

    quantum_mechanics_6044345343ea1d6769d6de20b66092e0d1ad3ed2.png

    But quantum_mechanics_760f2c4861fe1bf60ee9a0ad12bef3767bb18066.png and quantum_mechanics_5db125ed7043d8d9e39e83f2759122218eef741b.png do not commute:

    quantum_mechanics_e03e2050b68d6a6da443ab912501a1279d19d1a9.png

    So we instead use the notation of quantum_mechanics_14fadeb9e9afd48e814c32e223ee9f6bb95f3110.png and quantum_mechanics_e648c7652091ef6887299d9ad704d221fc55acaf.png which have the property

    quantum_mechanics_283ed7ab9713cbeebcb6071c2c66312175dcff6d.png

    Once again rewriting the Hamiltonian

    quantum_mechanics_d69b74423fe9bc6fbb992ef0cbed2975834735e9.png

    Observe that

    quantum_mechanics_f95c401b38cc4204a309b873142f30c9a4e94b61.png

    Conider the following operation:

    quantum_mechanics_ec8cf203e9708b1672f43676d194b3baf12f48b4.png

    where we in the last line used the fact that for a simple Harmonic Oscillator quantum_mechanics_268b9be0e8ff1d1cf563f1a7b1fa4e77e662f3ed.png.

    Thus,

    quantum_mechanics_0600e4b3fb8938b7141c3381a2be009105e6da63.png

    I.e. applying quantum_mechanics_14fadeb9e9afd48e814c32e223ee9f6bb95f3110.png to an eigenfunction quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png gives us the eigenfunction of the quantum_mechanics_c5deadecb4ffcc24de3a8292e89ef1f015321338.png eigenstate.

    We call this operator the annhiliation or lowering operator.

    We can do exactly the same of quantum_mechanics_e648c7652091ef6887299d9ad704d221fc55acaf.png to observe that

    quantum_mechanics_32351fd41de3c50ccd45b37200f46fad94dac57c.png

    which is why we call quantum_mechanics_e648c7652091ef6887299d9ad704d221fc55acaf.png the creation or rising operator.

    Buuut there's a problem: we can potential "annihilate" below to a state with energy below zero! To fix this we simply define the ground state such that

    quantum_mechanics_616f069f48f997d785c47c72605627fe781af64d.png

    Thus,

    quantum_mechanics_0c76a441c58e27a1af142d4a6a032ac5a97d0e06.png

    And all other higher energy states can then be constructed from this, by successsive application of quantum_mechanics_e648c7652091ef6887299d9ad704d221fc55acaf.png

    • Normalizing the lowering- and raising-operator

      quantum_mechanics_8beb24b730d262609fe4e14312ccd9affe322f4d.png

      Thus,

      quantum_mechanics_44a122e8a5ce90a84b7eb4cf11cdde02c8b87639.png

      Doing the same for quantum_mechanics_e648c7652091ef6887299d9ad704d221fc55acaf.png, we get

      quantum_mechanics_d778a9d5c7d1cc45f4bbdca7016ba73978561f52.png

    • Solving for the ground-state

      We can find the proper solution for the ground-state by solving the differential equation

      quantum_mechanics_d9322a6024db52cad3b745a9a3499993df43f82d.png

      And substituting in for quantum_mechanics_14fadeb9e9afd48e814c32e223ee9f6bb95f3110.png both the quantum_mechanics_760f2c4861fe1bf60ee9a0ad12bef3767bb18066.png and quantum_mechanics_f4e7effdc961ada1b5deefc004f1989f0a584f24.png, so that we get the differential equation. Solving this, we get

      quantum_mechanics_d8575478603041a4715036df1ccdd44c8ceb0a72.png

Solving the differential equation

We start of by rewriting the Schrödinger equation for the harmonic oscillator:

quantum_mechanics_7fdc7328617ba5ae2567752350a77ea510db99c5.png

Letting quantum_mechanics_60900da62e4da296177ff13ae78ebe3f10effcb9.png, we have

quantum_mechanics_e68979dc863b33b8173677a95d37d7164578f94e.png

We'll require our solution to be normalizable, i.e. square-integrable, therefore we need the differential equation to be satisfied as quantum_mechanics_407caa29cd42f9b2436f8953e3a8766db638e038.png, in which case we can drop the constant term:

Letting quantum_mechanics_ecdbb897bafd4fe17a0303dacbfc603cd3f43c08.png which is just a constant (apparently…), we have

quantum_mechanics_ecaa9678b605207b055091a68c6e800e4d6a4aa0.png

I'm not sure I'm "cool" with this. There's clearly something weird going on here, since quantum_mechanics_e29af05574032ace665d996d46b3280fc49866ef.png is a parameter of the potential, NOT a variable which we can just set.

Letting quantum_mechanics_74f198f5a425dbd1afeab44cb7580a8bf5ca9681.png, we then have

quantum_mechanics_608a4be53332497e5ab67e56a04b60b08fbae39e.png

Therefore we instead consider the simpler problem:

quantum_mechanics_44f98d9aee2fd1b5ac82b07c44f5d9670768efbb.png

where we've dropped the quantum_mechanics_81c5f489718d4ff0ca6962103920c3133c0daea4.png, which we will include later on through the us of method of variation of parameters. This clearly has the solution

quantum_mechanics_35063df313253dbb307ee1a807bce10bb30663f6.png

Now, using method of variation of parameters, we suppose there exists some particular solution of the form:

quantum_mechanics_4fbfc162eba5ca5a919cf1c6f92cd902d46bc3f9.png

For which we have

quantum_mechanics_90a4fd1c3f8bd3863ec067f1bc10968f02cbc101.png

Substituting into the diff. eqn. involving quantum_mechanics_81c5f489718d4ff0ca6962103920c3133c0daea4.png,

quantum_mechanics_2e4b081698111f07edfb9f6398af341b996558de.png

Since quantum_mechanics_2449140a739de934f4521413e504b4232c9c351c.png satisfies the original equation, we have

quantum_mechanics_a86521f67dd1d7c507e22b2251051495b967a8b0.png

Seeing this, we substitute in our solution for quantum_mechanics_dbfa60f11e44cc1adad1cd424f4e53321a33807a.png,

quantum_mechanics_10a879b5e31ace712ecd26739948d4681a17bbc3.png

Thus,

quantum_mechanics_75ea3b9387862a27b2e37a24314da5a79505859d.png

Which, if we let

quantum_mechanics_c53d694fa992e7ddd11dd389354657ca681270f4.png

gives us

quantum_mechanics_dea61fac0c9601e5015193cc592706d550bbac31.png

which we recognize as Hermite differential equation! Hence, we have the solutions

quantum_mechanics_bbcb766a130d7679675b6131175a5481343c96cd.png

Why the quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png though? Well, if you attempt to solve the Hermite equation above using a series solution, i.e. assuming

quantum_mechanics_c4b9308b1516eaa3d40d8a3b8f796125511e1f76.png

you end up seeing that you'll have a recurrence relation which depends on the integer quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, and further that for this to be square-integrable we require the series to terminate for some quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png, hence you get that dough above.

Also, it shows us that we do indeed have a (countably) infinite number of solutions, as we'd expect from a Sturm-Liouville problem such as the Schrödinger equation :)

Correspondance principle

The correspondence principle states that the behavior of systems described by quantum mechanics reproduces the classical physics in the limit of large numbers.

Operators

Momentum

quantum_mechanics_629f92d0ff5c99505af3a9412c4fa5a52b7910f9.png

"Derivation"

Considering the wave equation of a free particle, we have

quantum_mechanics_11e55b830a774424662cf0717c5220ac90f47ce6.png

Taking the derivative of the above, and rearranging, we obtain:

quantum_mechanics_ab08723ec863c9264a98b720f770ea551342b5a8.png

and thus we have the momentum operator

quantum_mechanics_976b0da66d69ff62ca6448ee783efe91363852ad.png

Hamiltonian

quantum_mechanics_b533063ce723c88047ed7e73fb873e1927c88847.png

"Derivation"

Where the Hamiltonian comes from the fact that for TISE we have the following:

quantum_mechanics_938d77b42a24fd0ce88978824f8e771c4644ba0d.png

which we can then write as:

quantum_mechanics_c9ded3423699be2363b63f60c0a5ade166ee366e.png

and we have our operator quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png !

In the above "derivation" of the Hamiltonian, we assume TISE for convenience. It works equally well with the time-dependence, which is why we can write the TIDE expression using the Hamiltonian.

In fact, one can deduce it from writing the wave-function as a function of quantum_mechanics_7225b076f6e6326f1636b11d1aad8de58bcc4761.png and quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, and then note the operator defined for the momentum quantum_mechanics_7225b076f6e6326f1636b11d1aad8de58bcc4761.png. This operator can then be substituted into the classical formula for the energy to provide use with a definition of the Hamiltonian operator.

Theorems

Conserved operators

A time-independent operator is conserved if and only if quantum_mechanics_1abaa23f330a3eb43a386d74f0d5484ae65ec86a.png, i.e. it commutes with the Hermitian operator.

Coherence

Stack excange answer

This guy provides a very nice explanation of quantum coherence and entanglement.

Basically what he's saying is that coherence refers to the case where the wave-functions which are superimposed to create the wave-function of the particle under consideration have a definite phase difference , i.e. the phase-difference between the "eigenstates" is does not change wrt. time and is not random .

Another answer to the same question says:

"It is better to think of there being one and only one wavefunction that describes all the particles in the universe."

Which is trying to make sense of entanglement, saying that we can look at the "joint probability" of e.g. two particles and based on measurement taken from one particle we can make a better judgement of the probability-distribution of the measurement of the other particle. At least this is how I interpret it.

Observables

Each dynamic variable is associated with a linear operator, say quantum_mechanics_6c73fafd4edf4f083cdbb9c3c541bf5d36db17e8.png, and it's expectation can be computed:

quantum_mechanics_30602f99d06536e8f29114d365c318db2625ff13.png

when there is no ambiguity about the state in which the average is computed, we shall write

quantum_mechanics_9036218477efbe7e2823006d5015758bb30f66d0.png

The possible outcomes of an observable is given by the eigenvalue of the corresponding operator quantum_mechanics_6c73fafd4edf4f083cdbb9c3c541bf5d36db17e8.png, i.e. the solutions of the eigenvalue equation:

quantum_mechanics_7dcdccdee45fa58f410a7bdc4384ee7e1f2759dd.png

where quantum_mechanics_7704c33b2fbd3b93ee8f13ea48747b917fe49053.png is the eigenstate corresponding to the eigenvalue quantum_mechanics_1fa1bbec54e39f67ec4d93867ec6c5172a4c8e6d.png.

Observables as Hermitian operators

Observables are required to be Hermition operators, i.e. operators such that for the operator quantum_mechanics_6c73fafd4edf4f083cdbb9c3c541bf5d36db17e8.png

quantum_mechanics_a6057f822c33b1ba2ed6c94c48f10d85e47ce1b5.png

where quantum_mechanics_b39bd886e3950dfb0857cae7437127f2d1f08fe1.png is called the Hermitian conjugate of quantum_mechanics_6c73fafd4edf4f083cdbb9c3c541bf5d36db17e8.png.

We an operator corresponding to an Observable to be a Hermitian operator as this is the only way to ensure that all the eigenvalues of the corresponding operator are real, and we want our measurements to be real, of course.

Compatilibity Theorem

Given two observables quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, represented by the Herimitian operators quantum_mechanics_56481186dda922dd45a6f6258565b0441a620e2f.png and quantum_mechanics_9dde26348ed436405c8d6c43924594f7450c884c.png, then the following statements are equivalent:

  1. quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png are compatible
  2. quantum_mechanics_56481186dda922dd45a6f6258565b0441a620e2f.png and quantum_mechanics_9dde26348ed436405c8d6c43924594f7450c884c.png have a common eigenbasis
  3. quantum_mechanics_56481186dda922dd45a6f6258565b0441a620e2f.png and quantum_mechanics_9dde26348ed436405c8d6c43924594f7450c884c.png commute, i.e.: quantum_mechanics_44cfcefd86ca51faded3ffcfaf11e77fcc9e44e4.png

Generalised Uncertainty Relation

If quantum_mechanics_adf98041a80fd57093fbc6b697e7b980d258ec34.png and quantum_mechanics_bc0325de1c38081776bd6af8b7e8c87d7973ef16.png denote the uncertainties in observables quantum_mechanics_37d3d2a40172b2cf7b0e8ea91ec033c3e1f9ecbd.png and quantum_mechanics_57374af10bfdcfe66f9ad361e98cbc9c879f5ec7.png respectively in the state quantum_mechanics_bbbc2b48f1c9c46f7261a2165a1a72524c683ff2.png, then the generalised uncertainty relation states that

quantum_mechanics_daf7a6c58a541e77fee76c0162bb91f187891ebc.png

Momentum

quantum_mechanics_a259d7c2f4d0199850c236dc2ed7083140a24e95.png

Then

quantum_mechanics_de9aaecee04dc00180201a50eb588027bb5fbb74.png

And we have commutability, and thus the we can interchange the order of derivation

quantum_mechanics_060b450021331355690a7feb24d623a88cebbe82.png

and

quantum_mechanics_d1c2d20cfa5b549d94d9eb31fef1a50c0d99a100.png

Hence,

quantum_mechanics_47db7daf783d8915113d21830db1c52788167d75.png

where

quantum_mechanics_11641331c3935ccdbd723c9a1417a42c3ada7c67.png

Separation of variables

Here we consider the TISE for a isotropic harmonic oscillator in 3 dimensions, for which the potential is

quantum_mechanics_5b30ba3bfe0f38d60631974312c64e6e4816bc50.png

We can then factorize the Hamiltonian in the following way

quantum_mechanics_75faba9a2eb6daf8fd9cb1c910a089dc7c650cd3.png

Using this in the Schrödinger equation, we get

quantum_mechanics_5167760aaf7912ddce743121d8cb4a5f3b2336eb.png

And we then assume that we can express quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png as

quantum_mechanics_0f7fb30218025cbc5bffb6b41fdbe4900dfd932c.png

which gives us

quantum_mechanics_2579767b9577f1f2ed1ffabf4851b4743f6abc09.png

Which gives us

quantum_mechanics_50849f3dfebcb3bfe9d67e2405db81608cd9c412.png

And since quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is a constant, we can write quantum_mechanics_59766d46f40cc258f720dadb155a432e5a346c30.png, which gives us a system of equations

quantum_mechanics_9cf777670ac4f1c9f82193af1afa50ba6098b6b3.png

where quantum_mechanics_62214da7392238d382528c8df6deff9881a773ff.png denotes the eigenfunction in the k-th dimension for the n-th quantised energy-level (remember we're working with a harmonic osicallator). Which gives

quantum_mechanics_f87d0f8e5aa106b55bae9e56bec4b0e0c35037bc.png

where quantum_mechanics_deeba3b389441a28534718f062061117e1075398.png denotes the factors of the Hermite polynomials.

Angular momentum

quantum_mechanics_79a64f77262bc3c9c7b78deeed0c09c83a2661b9.png

quantum_mechanics_9193966143c0f3725c0dc7ce51cfce83e261e05f.png

Commutation

The interesting thing about the angular momentum operators, is that unlike the position- and momentum-operators in different dimensions does not commute!

quantum_mechanics_23d645f41bd77514d98d8135f5abb46e610d4dce.png

This implies that the angular momentum in different dimensions are not compatible observables, i.e. we cannot observe one without affecting the distribution of measurements for the other!

Square of the angular momentum

quantum_mechanics_73d75db4f183c93185d366357b6c0d5631384a98.png

or in spherical coordinates,

quantum_mechanics_099f97f0853147e343faf63b9465adfe1408e6c8.png

We then observe that quantum_mechanics_0ef7976ae2bbe1274e36283db47447bb7fb7bef5.png is compatible with any of the Cartesian components of the angular momentum:

quantum_mechanics_437b3baf7d082ba491266e53b37163e51a32ec36.png

which also tells us that they have a common eigenbasis / simultaneous eigenfunctions .

Angular momentum operators in spherical coordinates

quantum_mechanics_b2ca8c14460d7b13fb0310ddd502f818c2bffeb4.png

Eigenfunctions

quantum_mechanics_763e8deb437e5cd979407a09611bbb5156c5e404.png

where quantum_mechanics_695fb17db894ad690095a0a326ef4aabbcecad51.png and

quantum_mechanics_73922a4953522a11782e4ea8249e83c85b7348f1.png

with quantum_mechanics_eda4e3006a3513107b82158dc054efb10f00b02a.png known as the associated Legendre polynomials.

quantum_mechanics_73922a4953522a11782e4ea8249e83c85b7348f1.png

with quantum_mechanics_eda4e3006a3513107b82158dc054efb10f00b02a.png known as the associated Legendre polynomials.

Quantisation of angular momentum

The eigenvalue function for the magnitude of the angular momentum tells us that the angular momentum is quantised. The square of the magnitude of the angular momentum can only assume one of the discrete set of values

quantum_mechanics_e53507e5571c591d648af486b32d398349fc66cf.png

and the z-component of the angular momentum can only assume one of the discrete set of values

quantum_mechanics_1d4bcaada80e235089e6fce50c0f26f0bb99bfd9.png

for a given value of quantum_mechanics_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png.

Laplacian operator

quantum_mechanics_0f55aaedd744008957d4335ab2adbe44ef7efd8f.png

Angular momentum - reloaded

We move away from the notation of using quantum_mechanics_b988cf3bbd4403905104b684ff08b602f65b05d3.png for angular momentum because what follows is true for any operator satisfying these properties.

To generalize what the raising and lowering operators we found for the angular momentum we let quantum_mechanics_62e3944393741f06ec4b19dcb7c0a35f0e405ba6.png denote an (Hermitian) operator which satisfies the following commutation relations:

quantum_mechanics_f1623cb36d06ba7c52eab57b1736e4dfb47af678.png

And since quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png and quantum_mechanics_2202a6283f12a2df9e2cf261219a3923c7799978.png are compatible, we know that these have a common eigenbasis, which we denote quantum_mechanics_8e1dc0c19aaffd1f136318928808d8894b85a09f.png, and we write the eigenvalues in the following form:

quantum_mechanics_80277a2a4ba8e380ea1d223330d60b7d2ebb6b8e.png

Further, we introduce the raising and lowering operators defined by:

quantum_mechanics_7c0831181f43739f91cdb18634a0e49825c48e02.png

for which we can compute the following properties:

quantum_mechanics_705a41b364d487465841934617964df8294d6de3.png

quantum_mechanics_acc6e11a8a41a92627301be6eb2f4ddab0b285a7.png

I.e. we're working with a raising operator for quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png.

Considering the action of the commutator quantum_mechanics_b34fafc7dd343b61fb346eeaee9f849b6e2574d5.png on an eigenstate quantum_mechanics_8e1dc0c19aaffd1f136318928808d8894b85a09f.png, we obtain the following:

quantum_mechanics_3a64e8f246791526d9fae7a09a3ec3b7e5fb4f1a.png

quantum_mechanics_da3818920bbf33e97f5bdac47515a4e8e3569d0b.png

which tells us that both quantum_mechanics_115a8b50c708e40b2e74e652879bc4e9ae115f58.png are also eigenstates of the quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png but with eigenvalues quantum_mechanics_58d192d8de6350c11c72d12adfd43d4ea5150485.png, unless quantum_mechanics_98eabcb34df4c50831a34ed47294ba8004c0d550.png.

Thus quantum_mechanics_2000625e913fc053880bee8edd0a1de38d01c1ef.png and quantum_mechanics_ec197eb0fba8f2f180ef7792d25ee002e61ded1d.png act as raising and lowering operators for the z-component of quantum_mechanics_b1225c094a1add3fb673c962559b8ddd5dc64b09.png.

Further, notice that since quantum_mechanics_ba95201d72fea8a5984deecee4158830b55182ca.png we have

quantum_mechanics_4d84d411e8bae286cb24e24ca56642f9dd1f5885.png

Hence, the eigenstates generated by the action of quantum_mechanics_f2cdb2b5cda8daa3faf0d45ce0a7bee25f9dff27.png are still eigenstates of quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png belonging to the same eigenvalue quantum_mechanics_6911a426313f14d0fc099f5a3fe0cade890c3c07.png. Thus, we can write

quantum_mechanics_95f9c50c961f1fac669f2f60f0bb9c9f7a26810e.png

where quantum_mechanics_074d22945985715c9d38079994ae1aeae2d21e40.png are proportionality constants.

The notation used for the eigenstates quantum_mechanics_5474203aed09a24994d738b88e0ccab727e6383d.png is simply saying that we know that the eigenvalue for quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png does not change under the operator quantum_mechanics_f2cdb2b5cda8daa3faf0d45ce0a7bee25f9dff27.png, and thus only the eigenvalue corresponding to quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png changes with the factor quantum_mechanics_25928aaf7d31a956e84e6b84bbdb7b9f97fcaf19.png (or rather another term of quantum_mechanics_4b98ef863210f65fdc274bc66f9efc84b5cc0937.png) and so we denote this new eigenstate as quantum_mechanics_89487bc1ab71dee38398077b62561d249355bb73.png.

To generalize the raising and lowering operators we found for the angular momentum we let quantum_mechanics_62e3944393741f06ec4b19dcb7c0a35f0e405ba6.png denote an (Hermitian) operator which satisfies the following commutation relations:

quantum_mechanics_f1623cb36d06ba7c52eab57b1736e4dfb47af678.png

And since quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png and quantum_mechanics_2202a6283f12a2df9e2cf261219a3923c7799978.png are compatible, we know that these have a common eigenbasis, which we denote quantum_mechanics_c172ffc09b0a84b6afb70f26a318d5934accb7d9.png, and we write the eigenvalues in the following form:

quantum_mechanics_3643cd6758de0e37b43b8f55b4c72261324dcd0b.png

with quantum_mechanics_5ad6e77146bbc601f880b912a041266864459b9e.png

The set of quantum_mechanics_eb78f994c896ec26756b7198634b5a2d400a75a4.png states quantum_mechanics_f582580bcb2209ad7a8c4c0564fcfe1064e05844.png is called a multiplet.

We introduce the raising and lowering operators defined by:

quantum_mechanics_7c0831181f43739f91cdb18634a0e49825c48e02.png

for which we can compute the following properties:

quantum_mechanics_705a41b364d487465841934617964df8294d6de3.png

quantum_mechanics_acc6e11a8a41a92627301be6eb2f4ddab0b285a7.png

And we have the relations:

quantum_mechanics_830863a9c032489d0cacb6ecde04652a967486d0.png

with

quantum_mechanics_85a2d7d5ad273c3a66ca2c2ae3e3135308aa5163.png

or, if we're assuming quantum_mechanics_bf06fd4916d8daa4581d2db5be7150af0a5649b2.png, we have

quantum_mechanics_a6e4b598ea0d61129b37e49c805cba1b8dbaafaf.png

There was originally also the relation

quantum_mechanics_f36a52c16952306b2b3b8a43966e576e77b9cbb2.png

which I somehow picked up from the lecture. But I'm not entirely sure what this actually means…

Some proofs

Lowering operator on an eigenstate is another eigenstate

quantum_mechanics_da3818920bbf33e97f5bdac47515a4e8e3569d0b.png

quantum_mechanics_f67ad4c124351129e0edf4a293e83f3897a1230f.png

Where the RHS is due to the commutation relation we deduced earlier. This gives us the equation

quantum_mechanics_d9e634b931e4896d7a2e0308af02101dd01f7d19.png

which tells us that quantum_mechanics_ec197eb0fba8f2f180ef7792d25ee002e61ded1d.png is also an eigenstate of quantum_mechanics_8e1dc0c19aaffd1f136318928808d8894b85a09f.png but with the eigenvalue quantum_mechanics_354abc628a2c1a9437524f015bb99417399a3cdd.png.

Just to make it completely obvious what we're saying, we can write the relation above as:

quantum_mechanics_3b4e5f1b6a28b355ff65815a82190434220c7dc6.png

Eigenvalues are bounded

The eigenvalues of quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png are bounded above and below, or more specifically

quantum_mechanics_aa740b7a4b521401ab908d523aaa245dc568f662.png

Further, we have the following properties:

  1. Eigenvalues of quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png are quantum_mechanics_ec6106e8cd70a99da0ee33f4c9078f89e1e15a29.png, where quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png is one of the allowed values

    quantum_mechanics_dfe61b2868ac70b9aa6077306cbc132d70d45200.png

  2. quantum_mechanics_37d098d0b5a0a95f80f26d4af98560940e4c2c8d.png thus we label the eigenstates of quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png and quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png by quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png rather than by quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, so that

    quantum_mechanics_c15003b099bbc403bfb89c093b9cbd4d9f526f81.png

  3. For an eigenstate quantum_mechanics_2a4fc527603bf6fdd9697c76d92714df1a7065ff.png, there are quantum_mechanics_eb78f994c896ec26756b7198634b5a2d400a75a4.png possible eigenvalues of quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png

The set of quantum_mechanics_eb78f994c896ec26756b7198634b5a2d400a75a4.png states quantum_mechanics_f582580bcb2209ad7a8c4c0564fcfe1064e05844.png is called a multiplet

We start by observing the following:

quantum_mechanics_a9b01f7991344502bd4f5bdffd5cf59cf30b320f.png

Taking the scalar product with quantum_mechanics_07b16111740b93cf82612257c2b374eaf0e940a4.png yields

quantum_mechanics_565f35fb9b5ce7ae14b8595df3d88f3e29ee7137.png

so that

quantum_mechanics_aa740b7a4b521401ab908d523aaa245dc568f662.png

Hence the spectrum of quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png is bounded above and below, for a given quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png. We can deduce that

quantum_mechanics_93c4f822c708f12c8d3a46f8c621b250d9a231e2.png

Using the following relations:

quantum_mechanics_08c30e102a1b8cf280eca44c9af0a0f9e19e0a19.png

and applying the quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png to quantum_mechanics_edff9b65dd8910a56330f1ae7cff6b8a62a314a3.png and quantum_mechanics_34ab8ae877f5fcdc9d483be7c782ec1c351212a0.png in turn, we can obtain the following relations:

quantum_mechanics_072eba64aaf5af365c05ce75c423435a6deac0f3.png

Using the notation quantum_mechanics_92d0e62066f7a593c2ef0d0f77ec2932025b05c5.png and using the equality above, we get the equation

quantum_mechanics_b16a59dc92884bac19aebfa186c5562480f6ff08.png

and we see that, since quantum_mechanics_0a7873d9e32bf277217c6921854d41517bc8f55c.png by definition, the only acceptable root to the above quadratic is

quantum_mechanics_276698e5d83047d45e29ac9b5fda4984d5c183af.png

Now, since quantum_mechanics_78fa99fb127f7b5f3c8f05d119a52d08e9eaf2db.png and quantum_mechanics_f908bf786e84001413297648785a64608d8c3d7f.png differ by some integer, quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png, we can write

quantum_mechanics_9d76b2a7b7bbadccdef66b8b6633474c44499dac.png

Or equivalently,

quantum_mechanics_588a7e849bb4b5fcf5ed7410eab71d985c8ea4cd.png

Hence, the allowed values are

quantum_mechanics_cac853c3bd3cc028a66c6c93600d7ac59f9c3cfe.png

for any given value of quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png, we see that quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png ranges over the values

quantum_mechanics_88f1031bd7119b69b216cee2a39595303e9d1f8e.png

which is a total of quantum_mechanics_eb78f994c896ec26756b7198634b5a2d400a75a4.png values.

Concluding the following:

  1. Eigenvalues of quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png are quantum_mechanics_ec6106e8cd70a99da0ee33f4c9078f89e1e15a29.png, where quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png is one of the allowed values

    quantum_mechanics_dfe61b2868ac70b9aa6077306cbc132d70d45200.png

  2. Since quantum_mechanics_37d098d0b5a0a95f80f26d4af98560940e4c2c8d.png we can equally well label the simultaneous eigenstates of quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png and quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png by quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png rather than by quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, so that

    quantum_mechanics_757c2bc1e6ad99c5801c13ae87bf0ce38a4a227d.png

  3. For a eigenvalue of quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png, there are quantum_mechanics_eb78f994c896ec26756b7198634b5a2d400a75a4.png possible eigenvalues of quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png, denoted by quantum_mechanics_68fe957370a2d3496c29aa4da9ac8e5f1612642f.png, where quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png runs from quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png to quantum_mechanics_e81807b912ab07c13988615bdddb378707e96fae.png
  4. The set of quantum_mechanics_eb78f994c896ec26756b7198634b5a2d400a75a4.png states quantum_mechanics_f582580bcb2209ad7a8c4c0564fcfe1064e05844.png is called a multiplet
  • Which step makes it so that the j in the case of "normal" angular momentum only takes on integer values?

    In the case where we're working with "normal" angular momentum, the spherical harmonics are also eigenfunctions quantum_mechanics_f1d3e7750adaa76c093dfe5966d2d0f7a3512d23.png. Since we require that the wave function must be single-valued, the spherical coordinates must be periodic in quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png with period quantum_mechanics_40ce122da09675999df03f3fff4b24d3df1132fb.png:

    quantum_mechanics_408df21e87d355126043e6f41ccdf9c15d104463.png

    this implies that quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is an integer, hence quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png must also be an integer.

Properties of J

  1. quantum_mechanics_9abefa766e9470a4c542cdfdc93b621aa035aad9.png implies

    quantum_mechanics_ab31c719c9de1604e94f2a2d2de9dd256f7aa391.png

    I.e. quantum_mechanics_015470e61d290a19122b54baa240e6f6445fd161.png, thus quantum_mechanics_249a2761b81b2a224c94b4c7eb2e46dd46e34bc0.png is positive definite

  2. quantum_mechanics_3cf76a2a49de4b4249b22d35468d295569e7f873.png such that

    quantum_mechanics_9d2441116d881c9996c4c67553bd71040a89266d.png

  3. This is :

    quantum_mechanics_66bf1fd28e5023e6f591cf74d3e61dbafb4e7415.png

    and

    quantum_mechanics_2c85425f944ea85558bfa7987f35194d726e49bf.png

  4. :

    quantum_mechanics_e258618782989d0ff098e4e02fdbce19c7b793ea.png

  5. quantum_mechanics_37ba76336bac725ad1407cf93048f2e9407ea090.png, quantum_mechanics_524bb287a0a6fe90faffe5904a82f9d7b778f2e7.png implies that quantum_mechanics_860631252ec3e276d885150be94d1613bd66b971.png

Normalization constant

quantum_mechanics_8b6ece6a7df43056c4dd9a1b8b77e4f5bf51f5e2.png

Using the Property 3 in Properties of J, we get

quantum_mechanics_95c9848b236bbadc551e28bf36b303d0b3b21d35.png

which gives

quantum_mechanics_32fc7b3ce17357da7eb471b23c94345707a3a2d8.png

since quantum_mechanics_035a25038509ea3831dad828626166652075f1b6.png

Doing the same for quantum_mechanics_c4138aac94c29181e870e6cdbbc831f87801e8b9.png we get

quantum_mechanics_ee845857aa162ab7d0c50442e65a4e55dcc2fa57.png

Computing the matrix elements

quantum_mechanics_87b04b508111d3eaaa1977cdacafd7f3fe8094e0.png

The matrix is quantum_mechanics_a8176b67efdfb58b5b653a68fcb4f1d6d84f0f24.png

And for the raising and lowering operators

quantum_mechanics_3d78922b4109bfa5ab9f226025c131305773c807.png

Q & A

DONE Which step makes it so that the j in the case of "normal" angular momentum only takes on integer values?

In the case where we're working with "normal" angular momentum, the spherical harmonics are also eigenfunctions quantum_mechanics_f1d3e7750adaa76c093dfe5966d2d0f7a3512d23.png. Since we require that the wave function must be single-valued(?), the spherical coordinates must be periodic in quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png with period quantum_mechanics_40ce122da09675999df03f3fff4b24d3df1132fb.png:

quantum_mechanics_408df21e87d355126043e6f41ccdf9c15d104463.png

this implies that quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is an integer, hence quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png must also be an integer.

I'm not sure what they mean by "single-valued". From the notes they do this:

quantum_mechanics_6739647b0184b81d3a78b1af2121a1a4807cdc23.png

of which I'm not entirely sure what they mean.

From looking at looking at the spherical harmonics from a perpespective of a Sturm-Liouville problem the eigenvalue quantum_mechanics_c0e7e486574eda3063c12f9f7e267aa761dc1ee4.png for some non-negative integer quantum_mechanics_2d34cebe6c4de4f298757abf5037eeaa2db8a3cf.png, is due to regularity on the poles quantum_mechanics_0d62324a3f0c2ca80233459330db1c98d00ba8bc.png.

Central potential

Notation

  • quantum_mechanics_7bec91615d02a5342b34270ae4a669947491ecaf.png is the angle vertical to the xy-plane
  • quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is the angle in the xy-plane
  • quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png is the mass (avoids confusion with the magnetic quantum number quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png)
  • quantum_mechanics_d86190240fc08d8dfdbf734ca628129ce9bb1e60.png is the position vector
  • quantum_mechanics_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png denotes the magnitude of the position vector

Stuff

quantum_mechanics_79e4018fdd104984da8741ae3a47b6f2dc00f779.png

In spherical coordinates we write the Laplacian as

quantum_mechanics_f29198d77be442fba987bc8245a8b9936fd9615a.png

we can then rewrite this in terms of the angular momentum operator:

quantum_mechanics_2770d78cfd60d54ebf1f3997862b740736990868.png

and thus the TISE becomes

quantum_mechanics_4449df13459f0747b07b8c3e54b6ee4a2c230d58.png

quantum_mechanics_36ebaeafedac40775f7122da426e8653aacb9861.png

where quantum_mechanics_dc071ba24ddc996e067991d2178ce5d0a3c1083a.png, then any pair of quantum_mechanics_ee37a25593d64c40a8cebd719444b1f9db2d7c9f.png commutes, so there exists a set of simultaneous eigenstates quantum_mechanics_7f8f0ce62585b72cb2b19a3331196b288669d02b.png.

Hydrogen atom

Notation

  • quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png effective mass of system
  • quantum_mechanics_7eb1b1b47675ba92f3a26b3b63b9b0a4bd4917d6.png
  • quantum_mechanics_487b64633468ccc1575485a7ea01c73d53bc1181.png
  • quantum_mechanics_c4b5c9db7f463a72d863bd5d4c5bd03dda157bbf.png
  • quantum_mechanics_c1b17c2b98255e2a994325dc94b462844c179faf.png
  • quantum_mechanics_bb7f55b4405bfeaaf103cbfb92d297461e1cfc54.png
  • quantum_mechanics_48d8d654bbbe9069c6f78c9827858b80939f1c04.png
  • quantum_mechanics_11cf9eec6c4eca7ca0dbf5906e3e943afaf4e7c0.png is the Bohr radius
  • quantum_mechanics_2cb9247520011b6e7a870750820c39d650c8973f.png
  • quantum_mechanics_8a461218e742a9331c339eca5fceeac102312acc.png

Stuff

Superposition of eigenstates of proton and electron

(Coloumb) potential:

quantum_mechanics_621024becbc653184d1b73e77eeca728f7ea68d1.png

quantum_mechanics_560ae1234bb169f7800a7db801f96d47972551e3.png

where the effective mass is

quantum_mechanics_1e82c99ef48dc9a2725c805f27a11d279f1d5eaa.png

The TISE is:

quantum_mechanics_a58aa1f93f643b18f0639ab4c37d9e1559381ac1.png

We make the ansatz: quantum_mechanics_cc69b2c4735231e3a9935e75c64ad06c3f7a1bce.png

Thus,

quantum_mechanics_ba81ea7992309e75a0387f9c29100a4655d88acc.png

Thus, we end up with

quantum_mechanics_2ea7d6a00bdd4f35211c01d20eb4ba240089a116.png

and we let

quantum_mechanics_c0c81dae4629ed39a5915a88a701bfdc64e34ada.png

We now introduce some "scales" for length and energy:

quantum_mechanics_9e6161cd88a1f09567558f7b8d82b647e01ee9bf.png

where quantum_mechanics_ecfe5c9e2e17245a0d3e468e7b5f1d92f5aae3fb.png is the Bohr radius. Further, we let

quantum_mechanics_fc1a8b15ed0f7c7f3360cf30dae3647e89582cdd.png

Substituting these scales into the TISE deduced earlier:

quantum_mechanics_4a39e1b363aea16d233bf4e3f79836e27c65d7b3.png

quantum_mechanics_e1cbef21ddc776bc3c29493f78f1c30d97610bd7.png

We tehn introduce another function of quantum_mechanics_1a4a18e739bc08439a04efa4dc6013a508ff1163.png instead of quantum_mechanics_92c19b87231ba07022797e54178e93233f72cd8d.png :

quantum_mechanics_519bd0824986b11fb64ce7b26d041dee8bee13ac.png

which gives us

quantum_mechanics_85581e2786e1f51ffebca66ba9eb87ab0604f5b5.png

with boundary conditions: quantum_mechanics_91fb777f7a0dedd96393a8916a96b774fe057c48.png.

We then consider the boundary condition where quantum_mechanics_ef5badfea7ff552a3410b7813a8767fa75fbc9fb.png :

quantum_mechanics_5b4ee623fc8cd1d38fc77df41dae870f2bfb42f7.png

But since we need it to be bounded:

quantum_mechanics_f7c2b5bd7b2708216e3387bdc2831bc7d201a3e2.png

Another ansatz: quantum_mechanics_88064f2057f3cf911acd92091404404c390c925c.png, which is simply the familiar method of variation of parameters, where we tack on another function of quantum_mechanics_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png, the independent variable, to obtain another solution quantum_mechanics_943a454508f23fbcd22c28ab08560a6c82166f4d.png which is independent of quantum_mechanics_f6dd4a98a1ff408295e3159ce8c0e15f7cd1d373.png.

If we then plug our quantum_mechanics_0b910a933242a1c2e06ac066632229c61143c6e3.png into the [tise-wrt-u-rho], which ends up being

quantum_mechanics_d146557f0d3b54b9225e0cbbd564ef1cca6d3173.png

where due to the behaviour of quantum_mechanics_99e968550ec7110a8118911118e7820c127061ca.png we have

quantum_mechanics_974315cf2dc465530363b23fd2fe68ff7dd50fb2.png

Ansatz:

quantum_mechanics_1d6950ff5bbe1ee765f68dbf2fc40cc201c1cb4e.png

Spin - intrinsic angular momentum

Notation

  • quantum_mechanics_286e27847008e130779346dd377ab762501933b2.png denotes the eigenstate of a system with spin (eigenvalue of quantum_mechanics_eab1858cf96b1ed3dcebeed8406e98845e73ef5c.png) quantum_mechanics_9e114dbc2f5503741d2ef70303cfbf796adbb892.png in the z-direction, and quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png being the eigenvalue of quantum_mechanics_10ba3f41db5704694bd2f9910297c3dc4f9f22df.png
  • quantum_mechanics_0546eedebb42b09ee61e6f064ea7877c0a1207ac.png since the first number (quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png) is provided in the second anyways
  • quantum_mechanics_a9b3b9e0168ef0d180de722430a6ea189420bf55.png and quantum_mechanics_de01bf8c179267bbb0bb734904918b20c7e39182.png
  • quantum_mechanics_1e726ae0dc1f9e7d173f884bd25d6df36e43c7f9.png etc. we use to represent the coefficient of the function quantum_mechanics_916d5fa611eb6b9fa234a8738215fa32fe3ade68.png in some basis
  • when we say a system has spin quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png, we're talking about a system with eigenvalue of quantum_mechanics_10ba3f41db5704694bd2f9910297c3dc4f9f22df.png to be quantum_mechanics_776c0048a5fc513347b77b4946c1fac7ac206253.png

Stuff

Now that we know

quantum_mechanics_f92e5ea7567b64f04cd08112ba5cc5c988abd46b.png

  • quantum_mechanics_f5f18b430e8d0e2bdb6494e43ed41deca9bf97fb.png eigenvalues: quantum_mechanics_21adc31900a55c3239d4a254a9f7ac27effcbbb7.png
  • quantum_mechanics_c40b11f569ff2c712a6acaa454d6cacb193ee078.png eigenvalues: quantum_mechanics_9c5e37ba83d6466fa269f49d781df639d434bfde.png

From the orbital angular momentum, we have

quantum_mechanics_f29a3d993307a8b81f775dac9d39e4bebe6aa878.png

quantum_mechanics_7abe9bbedb846c9873bfab831fc8b40d9df5bc9a.png

In the case of orbital angular momentum , we have the requirement that quantum_mechanics_f56b556b5aef28b769c52c4c0bf2d3213e67eb2e.png, which implies quantum_mechanics_3f3e117bfc5417544533930abdd79f160bdc7ec4.png, rather than quantum_mechanics_524bb287a0a6fe90faffe5904a82f9d7b778f2e7.png which we found in the Angular momentum - Reloaded.

quantum_mechanics_6b2ade9e68f043283b6508ebcb1a18b1302b3f56.png are diff. operators, which implies quantum_mechanics_590fb11474cd873d5330a73acfd6352d8d93ac3d.png diff. operators

For a given value of quantum_mechanics_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png, quantum_mechanics_157c77490049378ed03cb2126a3d791fb8577d45.png since quantum_mechanics_ef9dab4b0c85cb57230141f0c253c62619c1b85b.png is the maximum for a spherical harmonic.

quantum_mechanics_47cd7f40c8de86a9ab517b18836a2ae5d80201bb.png

Now using the properties derived for the more general quantum_mechanics_6c106e4f397fe213b8a3c36a17d7dc758a031184.png operator we can deduce the all the other spherical harmonics by the usage of raising and lowering operators.

quantum_mechanics_49f0a98c7453dc60f2a98252168c5ded8d4db6cf.png

And it satisfies the commutation relation

quantum_mechanics_d2c7fe3a88762e65ebc214f80bfc110564a74170.png

and

quantum_mechanics_c7c8b369911a043b484a41fe21e6cae0f3e24934.png

Thus, we can construct a basis using the eigenstates of quantum_mechanics_9c2066df451700c442f1b492da9f64cc38a20295.png, i.e. they form a C.S.O.C..

And as we've seen earlier, if an operator satisfies the above properties, we have:

  • eigenvalues of quantum_mechanics_10ba3f41db5704694bd2f9910297c3dc4f9f22df.png: quantum_mechanics_bb4003a2aad00c86e97bbce3db5036f1b97e5e31.png, quantum_mechanics_70284b822ac11216f6d167751db7f0da5bd249b1.png
  • eigenvalues of quantum_mechanics_eab1858cf96b1ed3dcebeed8406e98845e73ef5c.png : quantum_mechanics_58d42511bf5b4c16adf8d8e2d24ddb21ddcef802.png

For a given value of quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png, quantum_mechanics_e57a7a9ec66fa97af5c71fb59094b870cd2c0804.png has quantum_mechanics_795c9466f13eb2ee4dd61d7710515ab8705a0fc8.png elements

quantum_mechanics_73d244e8fd3adf9def9d2ea4a52706176f8420cd.png

And the eigenfunctions have the relations:

quantum_mechanics_d07d60a89e9447517d823f85c6eeba3e6677f758.png

Example: system of spin 1 / 2

Since quantum_mechanics_32d47689ed8eb2703ed36d793867242ad1e48b50.png, we have:

  • eigenvalue of quantum_mechanics_10ba3f41db5704694bd2f9910297c3dc4f9f22df.png is quantum_mechanics_f38e173cd936f35cac89d1c1b2a68261117ca117.png
  • eigenvalue of quantum_mechanics_eab1858cf96b1ed3dcebeed8406e98845e73ef5c.png are quantum_mechanics_9b2f7cd706dbd9c0d9cb89e647402c2db2424101.png

The basis of the space of the physical states is then

quantum_mechanics_bbe0c5a6728b00a8cfe59209288a9a196ba6a03d.png

where we use the notation

quantum_mechanics_e59e74010c6dee5aeeb6d1909bcfc12792bc0150.png

Thus, any eigenstate we can expand in this basis:

quantum_mechanics_fcc98534f26f90079b6a7fd65bc9b5a36705dd41.png

and then

  • quantum_mechanics_6d22af1e5faf9673233f09139fcdee6baacc2cd6.png is the prob. of finding the system in quantum_mechanics_11d1163e616bdb47023d88e7908aede14367f199.png measuring quantum_mechanics_7b41e2bb44ac34bfe071a9c09eb1667878fe55f9.png
  • quantum_mechanics_1d8674e56cfb8934400f36cc116cc12424a0523c.png is the prob. of finding the system in quantum_mechanics_eb1997d82d80553a5bfc87e3fef0041e2c5bcb59.png measuring quantum_mechanics_7b41e2bb44ac34bfe071a9c09eb1667878fe55f9.png

TODO Pauli matrices

Since we can write

quantum_mechanics_f5d57b98befd42af1b0c1197f01e201cee23e325.png

Addition of angular momenta

Notation

  • quantum_mechanics_76ce06c7c1d11ea426c224cdabadaa6c874db65d.png, without integer subscripts refers to the total angular momentum , and thus quantum_mechanics_fa0421cd3289f387792835bda476e81e71c79846.png refers to the z-component of this total angular momentum
  • quantum_mechanics_23d8af094b3b874015ee7eda7464fe92dbf5cf95.png refers to the z-component of the angular momentum indexed by, well, 1
  • quantum_mechanics_02c63728fe2c2f5cdca61ee31fa33eeb87c7c9a3.png refers to the square of the angular momentum indexed by 1

Stuff

This excerpt from some book might be a really good way of getting a better feel of all of this

In this section we deal with the case of having two angular momenta!

For example, we might wish to consider an electron which has both an intrinsic spin and some orbital angular momentum, as in a real hydrogen atom.

Total angular momentum operator

quantum_mechanics_1b48bd8f3cdf7d039dd975f9c0184282813cf0bb.png

where we assume quantum_mechanics_dafe0a5d90fec1e013a16914ddc9bd2c030461bd.png and quantum_mechanics_7dfe47d8af6abcbc575f413b9a8f840ba377f043.png are independent angular momenta, meaning each satisfies the usual angular momentum commutation relations

quantum_mechanics_c8263518f81615dafacd3741c604ae7aaebc6ee6.png

where

  • quantum_mechanics_0b6a6df5744a8bdb696acaf5f611240bba4257d0.png labels the individual angular momenta
  • quantum_mechanics_48240e3abdb96e9e8392205dd073401bf6939856.png stands for cyclic permutations.

Furthermore, any component of quantum_mechanics_d8e5d143ec66883eb959afe3daedbab6126dc95f.png commutes with any component of quantum_mechanics_ce2ef55afdaf265e59de8a9a4bcb55a7f2b07aad.png:

quantum_mechanics_a5ba7d80af594ebe8f9fd4f3945e22804916c89e.png

so that the two angular momenta are compatible.

It follows that the four operators quantum_mechanics_030c7b0584a5f170cc5aff52e67b2484a1e5b494.png are mutually commuting and thus must posses a common eigenbasis!

This common eigenbasis is known as the uncoupled basis and is denoted quantum_mechanics_5c86e384f698c83c74bb6cef3df3c3f019f0009b.png.

It has the following properties:

quantum_mechanics_e7bc3e4fc41a237bb7fb92a320795b13a536b9e4.png

The allowed values of the total angular momentum quantum number quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png, given two angular momenta corresponding to the quantum numbers quantum_mechanics_9858160f78e2dbdd8819b92e5da57aeb15a36d1b.png and quantum_mechanics_b2f45f76c457079c699cda336ba3abf0b0645e17.png are:

quantum_mechanics_edbcc5f48af77d521eab6cf9c5476de1148a2ff4.png

and for each of these values of quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png, we have quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png take on the quantum_mechanics_eb78f994c896ec26756b7198634b5a2d400a75a4.png values

quantum_mechanics_0cdc3953547db969415118b4f34066e12de141de.png

The z-component of the total angular momentum operator quantum_mechanics_fa0421cd3289f387792835bda476e81e71c79846.png commutes with quantum_mechanics_02c63728fe2c2f5cdca61ee31fa33eeb87c7c9a3.png and quantum_mechanics_6c8628b093c7d4eb4d276cb48c219047ccae0bef.png, thus the set of four operators quantum_mechanics_5da07ca48b3a8603843a8aace8672552e99d12da.png are also a mutually commuting set of operators with a common basis known as the coupled basis, denoted quantum_mechanics_a5df283351a96f5d049770f75e47373b32720dd6.png and satisfying

quantum_mechanics_0cc16b91de4bea46526a380355efe0899991c742.png

these are states of definite total angular momentum and definite z-component of total angular momentum but not in general states with definite quantum_mechanics_23d8af094b3b874015ee7eda7464fe92dbf5cf95.png or quantum_mechanics_d625f51f771ba2d12960b91e3acc320e6beb72af.png.

In fact, these states are expressible as linear combinations of the states of the uncoupled basis, with coefficients known as Clebsch-Gordan coefficients.

I think what they mean by "states of definite angular momentum…" are the states which are coupled in a way that one eigenvalue decides the other?

TODO Example: two half-spin particles

Suppose we have two half-spin particles (e.g. electrons) with spin quantum numbers quantum_mechanics_f9e2220eaae9941eb03993f89938be7bf7183780.png and quantum_mechanics_d5822ec0976e00260a9a6b012ebaf78c3e3b1ead.png.

According to the Addition theorem, the total spin quantum number quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png takes on the values quantum_mechanics_74d58a58dddfe3978862ff0ae869e2f8f901f529.png and we require quantum_mechanics_b6d7cb54df6d7f73cfc17dee5df8fdd6cb95cb48.png.

Thus, two electroncs can only have a total spin:

  • quantum_mechanics_526aebf00d04b1bf139a69e10980dcdc1c6e0747.png called the triplet states, for which there are three possible values of the spin magnetic quantum number quantum_mechanics_f4411ec2c8c81ce3610626e374585e0a65ed02f9.png
  • quantum_mechanics_6895ac723a44d1e88cb199e38db32983bcf92d72.png called the singlet states, for which there are only a single possible value quantum_mechanics_89f377ec98053f954d7f489b26b7e17cf1e59a49.png

Let's denote the elements of the uncoupled basis as

quantum_mechanics_d820d0bdd2be4a3a693e2dde3894b9b647e9649b.png

where the subscripts 1 and 2 refer to electron 1 and 2, respectively. The operators quantum_mechanics_2acf04cde247d3cbe04b7a5734623453cb1ff462.png and quantum_mechanics_9f545e1d88c2543825d0f359db736f15edfe0d6e.png act only on the parts labelled 1, and so on.

If we let quantum_mechanics_559531395c623532e3d35b433a54823dbb7f83e1.png be the state which has quantum_mechanics_5fd7351b278f79ea0829b24ab09d27b21397231e.png and quantum_mechanics_8977d887f2f26eaa0b59f6a19d1537417e284bcb.png then it must have quantum_mechanics_b55dccbca97964e8e96f377140e015474bee0e34.png, i.e. total z-component of spin quantum_mechanics_4b98ef863210f65fdc274bc66f9efc84b5cc0937.png, and can therefore only be quantum_mechanics_526aebf00d04b1bf139a69e10980dcdc1c6e0747.png and not quantum_mechanics_6895ac723a44d1e88cb199e38db32983bcf92d72.png.

The eigenstates ends up being:

quantum_mechanics_67db69ee9357c6df70a26ea7e9cb9e4239da95e3.png

Identical Particles

Stuff

Systems of identical particles with integer spin quantum_mechanics_273961987a27aed5496864eadd3781db8f62e66a.png known as bosons , have wave functions which are symmertic under interchange of any pair of particle labels (i.e. swapping the states of say two particles).

The wave function is said to obey Bose-Einstein statistics .

Systems of identical particles with half-odd-integer spins quantum_mechanics_b8c636ad16f7a0d7597b62d4158b3eba4ec851c1.png known as fermions , have wave functions which are antisymmetric under interchange of any pair of particle labels.

The wave function is said to obey Fermi-Dirac statistics .

In the simplest model of the helium atom, the Hamiltonian is

quantum_mechanics_4ef8407f68ee01b0e956de06c94740ca050fd87f.png

where

quantum_mechanics_c776eac0e0fb5655bd52513bea31b1152a0a0bab.png

This is symmetric under permutation of the indices 1 and 2 which label the two electrons. Thus it must be the case if the two electrons are identical or industinguishable : it cannot matter which particle we label 1 and we which we label 2.

Due to symmerty condition, we have

quantum_mechanics_b74fc75136a63f8c0d2a5a869248672dc5b63cf9.png

Suppose that

quantum_mechanics_9709213cd9df84a6c7651937fdbad731dec209de.png

so we conclude that quantum_mechanics_2e4b17216478bf997e475bf971ca0674b8e337fa.png and quantum_mechanics_34f9741ddc82e614ce9667173c6b19ff6ab99624.png are both eigenfunctions belonging to the same eigenvalue quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png. Further, any linear combination of the two is!

In particular, the normalised symmertic and antisymmertic combinations

quantum_mechanics_cecf7d6e900c3609b5e2954b63158c35da17c1ab.png

are eigenfunctions belonging to the eigenvalue quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

If we introduce the particle interchange operator, quantum_mechanics_88970c9afcdfb3d545cc1adc30c566fc0d841801.png, with the property that

quantum_mechanics_f113dcf99dd4a05068417354cf440a18531e12d3.png

then the symmertic and antisymmertic combinations are eigenfunctions of quantum_mechanics_88970c9afcdfb3d545cc1adc30c566fc0d841801.png with eigenvalues quantum_mechanics_e13087d27f02ec26aa39d3435042df6752635a61.png respectively:

quantum_mechanics_4254411d83a3648fb2568993a2e54b712ca1fbdf.png

Since quantum_mechanics_a6de197dcc88fa9d1523da3e3800e1d7e5537e26.png are simultaneous eigenfunctions of quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png and quantum_mechanics_88970c9afcdfb3d545cc1adc30c566fc0d841801.png it follows that:

quantum_mechanics_20e9ccd599376dc8713e612937d4fb9f64bbb88f.png

Fourier Transforms and Uncertainty Relations

This is heavily inspired by a post from mathpages.

Notation

  • quantum_mechanics_3d9ee3a5d37114aec43cd7179e69d2ad99ade22f.png
  • quantum_mechanics_cc8e7ed71b04f4e5932daf4a4d46846933e064df.png is such that quantum_mechanics_87b6decad495c014487c0a5e1a17e721b84ec6e0.png, i.e. the ket corresponding to the eigenvalue quantum_mechanics_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png

Stuff

Gaussian integral
  • Basic

    Taking the square of the integral quantum_mechanics_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png we get

    quantum_mechanics_2930e89cb0d69c1c215d1fec5f42ce05d75c2a08.png

    Reparametrizing in polar coordinates

    quantum_mechanics_49f0992712d5005a92034f8788fb148beec06e32.png

    which gives us the Jacobian transformation

    quantum_mechanics_9bfcb78145fc4b7f0ee748c14e1dac797566c7bd.png

    so the incremental area element is

    quantum_mechanics_dc7dd0455da37bb39be52d2ea1a5aa113c81ba04.png

    Hence the double-integral above becomes

    quantum_mechanics_3f5f3c6f719aa0b42a7e63343659764e266c07fc.png

    and we since

    quantum_mechanics_9dac27f20b5ae074f4eab7e3c6598f7c822cb211.png

    we get

    quantum_mechanics_12fdee8dd84c1b6796c4d629ea48cb31675ee193.png

    Finally giving us

    quantum_mechanics_c1823abeb31a1595d7899b0245fc2759993d9263.png

  • More general

    If we let quantum_mechanics_330ce23271295a2a533d0ffe9cd7523e3fee78cd.png, i.e. we consider some arbitrary quadratic, then

    quantum_mechanics_2b07bdc73eea2b824a315ddf2fb9889eb21821ef.png

    in terms of which we can write

    quantum_mechanics_59856a0a3ec46b6a7c32508ed8f4767d52b11cdb.png

Fourier transform of a normal probability density

For any function quantum_mechanics_b4625aa4535b6123d93c130d4dc4772a395916cd.png we have the relation

quantum_mechanics_5db22b910d5178cb9d3266db1a87728765a56594.png

Now, if quantum_mechanics_b4625aa4535b6123d93c130d4dc4772a395916cd.png is a normal probability density, we have

quantum_mechanics_21ce5bb9a20a2c98b8354ef3ced4dda7c74feb9c.png

which has the Fourier transform

quantum_mechanics_96df4ff4979bf76355b1e0d7b0acaf0586215fcf.png

Where we can write the exponent in the integral of the form quantum_mechanics_cb8e1d89dd0950a4b422a48eff09a76ca5f84011.png with

  • quantum_mechanics_d122141de0c2145a2d6a7f137754f156b62dfe09.png
  • quantum_mechanics_ceeef89726214cd90637694c37b5517d3cf12fdb.png
  • quantum_mechanics_cf3b4be5f30b8dba224807885943afd0a97f5aac.png

hence the Fourier transform of the normal density function is

quantum_mechanics_9769db9cb9760e156bb469f3414ba37aa04540a6.png

If we then choose our scales so that the mena of quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is zero, i.e. so that quantum_mechanics_543a00fe9e998f934248907d93b0102228aa6894.png the expression reduces to

quantum_mechanics_bd62a96a838b52453443412825701c2fd78325c9.png

which means that the Fourier transform of a normal distribution with mean quantum_mechanics_96f53e8f2667720f54bd85623f46cbf545733989.png and variance quantum_mechanics_68668609fc72c879ba0fabd0c25275964a5e4af8.png is a normal distribution with mean quantum_mechanics_96f53e8f2667720f54bd85623f46cbf545733989.png and variance quantum_mechanics_fefff837ca88a28bd5eb1797d00263d439100797.png.

Which tells us that the variances of the distributions quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png and quantum_mechanics_b2db59aeb94cbc1050079892ff07b21b493513b7.png satisfy the uncertainty relation:

quantum_mechanics_84a2afb0275542c6d3fb5e223dedbee9d112a015.png

This is the limiting case of the general inequality on the product of variances of Fourier transform pairs. In general, if quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is an arbitrary probability density and quantum_mechanics_b2db59aeb94cbc1050079892ff07b21b493513b7.png is its Fourier Transform, then

For any probability density function quantum_mechanics_b4625aa4535b6123d93c130d4dc4772a395916cd.png and it's the Fourier Transform quantum_mechanics_2b236c4da465e6410c6667e470a0883f7c110991.png we have

quantum_mechanics_95e9ef9439c7a0c3107671a5699a5407b55ade04.png

quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png and quantum_mechanics_b2db59aeb94cbc1050079892ff07b21b493513b7.png are two different ways of characterizing the same distribution, one in the amplitude domain and one in the frequency domain. Given either of these, the other is completely determined.

Remember, due to the Central Limit Theorem, i.i.d. the sample mean of any random variable with finite variance follows a Normal distribution

Hence, the equality above is saying that if we have an infinte number of realizations of some random variable quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png then the variance of this will be bounded below by the inverse of the variance of the Fourier Transform.

Hmm, I'm not 100% sure about all this. How can we for sure know that the Fourier Transform of the probability density will have a sample variance which decreases quicker than the actual density function quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png?

Why can we guarantee that the variance of the Fourier transform does not decrease faster than the variance of the density quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png?

We're saying that variance of random variable whos sample average converges to quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png has a variance of quantum_mechanics_35299519cb4c8debab4f4d9b201c1cc8942a9c83.png, which is fiiine. Then we're saying that some the variance of some random variable whos sample average converges to quantum_mechanics_b2db59aeb94cbc1050079892ff07b21b493513b7.png has a variance of quantum_mechanics_0ff7ae414c9daf47b3965ba4bc34f671b5cad6c8.png, which is also fine. Then, we're saying that the variance of this random variable is always going to be greater than the limiting variances for these random variables, whiiiich you really can't say.

Application to Quantum mechanics

The canonical commutation relation is the fundamental relation between canonical conjugate quantities, i.e. quantities which are related by definition such that one is the Fourier transform of the other, which for two operators which are canonical conjugates, we have

quantum_mechanics_99359736d1442bb258f60a01e5f52dd3a9b2dfdb.png

Equivalently, if we have the above commutation relation between two Hermitian operators, then they form a Fourier Transform pair.

Where, if we take the commutation-relation the other way around the the sign of quantum_mechanics_55ec47ca8d1ca79b0c44d3cb3638a908b0a282ce.png changes, i.e. it's "symmetric" in a sense.

Now, if we then take some state quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png and compute the probability amplitudes that the measurements corresponding to the operators quantum_mechanics_56481186dda922dd45a6f6258565b0441a620e2f.png and quantum_mechanics_9dde26348ed436405c8d6c43924594f7450c884c.png will return the eigenvalues quantum_mechanics_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png and quantum_mechanics_f8ba0bcc022c6d477cbfde325f315454468205bb.png respectively, then we find

quantum_mechanics_3e5914f3fa2b6df2df3c6df9922637b686b09b6b.png

where quantum_mechanics_60b335ad0bf65f6eda027284474b286e4006470b.png is the bra corresponding to the eigenstate quantum_mechanics_cc8e7ed71b04f4e5932daf4a4d46846933e064df.png with the eigenvalue quantum_mechanics_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png of the operator quantum_mechanics_56481186dda922dd45a6f6258565b0441a620e2f.png.

That is; the probability amplitude distributions of two conjugate variables are simply the (suitably scaled) Fourier transforms of each other.

We saw earlier that the variances of two density distributions that comprise a Fourier transform pair satisfy the variance inequality:

quantum_mechanics_f548b24b8460f521a4a2decf7feb58bdfdac5d47.png

Heisenberg Uncertainty Principle

How is this interesting? Well, as it turns out, the position operator quantum_mechanics_880013a88afa007890fe6bee9aa74edbce42be1a.png and the momentum operator quantum_mechanics_bd333f10ceba0f45997c593d5014f729e84e0315.png have the commutation relation

quantum_mechanics_b54d370094095a85ba53013974584e7204c1b7b2.png

which then tells us that they are conjugate varianbles, i.e. the probability amplitude distributions of the two are scaled Fourier transforms of each other! Which means they satisfy the following inequality

quantum_mechanics_585ce485c2755e1bf712c8fb838a2cd06a58162b.png

which is just the Heisenberg uncertainty principle!

Bra, kets, and matrices

Time-independent Perturbation Theory

Notation

  • quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png is the Hamiltonian corresponding to the unperturbed / exactly solvable system
  • quantum_mechanics_6e9ab7a59a50c2029174aa8b4e260decbc8f0128.png and quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png represent the n-th eigenvalue and eigenfunction of the unperturbed Hamiltonian
  • quantum_mechanics_d666b0a239e3937f0f76611567d5f6aff5490e20.png and quantum_mechanics_f8ac70980af51a75a9075f6f518f20ead0399744.png is the energy and ket of the i-th perturbation term
  • quantum_mechanics_049620bed349a8deabe58aa25ed270a33ddb7c9d.png and quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png is the corrected (perturbed with all the terms) eigenvalue and eigenfunction, i.e.
  • quantum_mechanics_166ffa148eb6b90032dd84891c3eb0331aea6e3b.png
  • quantum_mechanics_742a6383de791a9c13c0859cb489091b948ad66a.png
  • quantum_mechanics_2b4d71209349ac820c256a2f414c357ee5cfc404.png
  • quantum_mechanics_4083b612ecc54e42ebc22fba924730c1a0ac4ddf.png denote the total angular momentum (rather than a general ang. mom. as we've used it for earlier)

Overview

  • Few problems in Quantum Theory can be solved exactly
    • Helium atom: inter-electron electrostatic repulsion term in quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png changes the problem into one which cannot be solved analytically
  • Perturbation Theory provides a method for finding approx. energy eigenvalues and eigenstates for a system whose Hamiltonian is of the form

    quantum_mechanics_08ad796c544bd95927878933480fc25bfb1a5844.png

    where quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png is the Hamiltonian of an exactly solvable system, for which we know the eigenvalues, quantum_mechanics_6e9ab7a59a50c2029174aa8b4e260decbc8f0128.png, and eigenstates, quantum_mechanics_b945545edddaedb892437be680fbea133dfc31b2.png, and quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png is small, time-independent perturbation.

Does not converge

Perturbation does not actually guarantee convergence, i.e. that each term is smaller than the previous.

But, it is something called Borel summable, which is a method for summing up divergent series.

Short and sweet

Let

quantum_mechanics_a6739a09a0dddbd4ebacfd81892027ef336e7555.png

where:

  • quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png is a real parameter used for convenience
  • quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png is some small perturbation to the Hamiltonian, e.g. quantum_mechanics_ebedc1da7d56eff0c0831a1569d6eac772bc7865.png

Then, the corrected eigenvalues and eigenfunctions are going to be given by:

quantum_mechanics_04c5725e4f9a044bf5302f4d1dbd1e247ca56832.png

quantum_mechanics_e058a87d9016c08809e69dcf9ed510f5fbcfb181.png

"My" version (first order mainly)

It is convenient to consider the related problem of a system with Hamiltonian

quantum_mechanics_a6739a09a0dddbd4ebacfd81892027ef336e7555.png

where:

  • quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png is a real parameter used for convenience
  • quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png is some small perturbation to the Hamiltonian, e.g. quantum_mechanics_ebedc1da7d56eff0c0831a1569d6eac772bc7865.png

Then the eigenvalue problem we're trying to solve is then:

quantum_mechanics_0847b65914bee86d1018b067083470351c459e99.png

Assume quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png and quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png posses discrete, non-degenerate eigenvalues only, and we write:

quantum_mechanics_a03444cfd1fdaf6d35aa4c18fbe1de18b9821169.png

where quantum_mechanics_b945545edddaedb892437be680fbea133dfc31b2.png are orthonormal.

The effect of the perturbation on the eigenstates and eigenvalues is defined by the following maps:

quantum_mechanics_2e46dbd76903721bc3e464cd1e5efce763095069.png

where

quantum_mechanics_0847b65914bee86d1018b067083470351c459e99.png

Thus, we solve the full eigenvalue problem by assuming we can exapnd quantum_mechanics_049620bed349a8deabe58aa25ed270a33ddb7c9d.png and quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png in a series as follows:

quantum_mechanics_985ea1f90770169364946ec8c2816150c8986a40.png

were the correction terms quantum_mechanics_83b9d3ea8a85a1efc83d95c601e946ccc2c4048b.png and quantum_mechanics_f7f4eca9be433271a884ed9606a2a6aaccb0f539.png are of succesively higher order of "smallness": the power of quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png keeps track of this for us.

The correction terms quantum_mechanics_f7f4eca9be433271a884ed9606a2a6aaccb0f539.png are not normalized (by defualt)!

Substitute these into the equation for the full Hamiltonian (can be seen above):

quantum_mechanics_eb3f133c7b8415cf01655513748ade41fed89d7b.png

and equating the terms of the same degree of quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, giving:

quantum_mechanics_4cf5dbeb8621cdcc90e6f0348324576f2c3b0886.png

Now let's, for the heck of it, take the inner product of the coefficients of quantum_mechanics_d921e7480a6232c87a118b0d591262b3129cb91a.png with some arbitrary non-perturbed state quantum_mechanics_79ecf90cabf865ad87cd08f290557084b2a98ba2.png:

quantum_mechanics_74e88c3ca8c4b3182a2749b4f58f7f80d359c7e6.png

which (due to quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png being Hermitian), becomes

quantum_mechanics_764e5561fb8843a1a57bee2c47169f5bf312372b.png

Let's first consider the correction (of first order) to the energy (then we'll to the eigenstate afterwards) by letting quantum_mechanics_6f26cd7d1f07674c429e76b65c912af3b3b40f96.png (since quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png is arbitrary):

quantum_mechanics_6f498d5255c9ca61d93946870e06012c5a2b7e65.png

LHS vanishes, and we're left with

quantum_mechanics_04c5725e4f9a044bf5302f4d1dbd1e247ca56832.png

which is the expression for the energy correction (of first order)!

To obtain the correction to the eigenfunction itself, we explore the following: quantum_mechanics_f87540160fadbe364d62d7a427c9b0d9539e6c99.png,

quantum_mechanics_9e6476d4146c07ef0481c271d93e78db71f61dac.png

where quantum_mechanics_9172e344cf06cd92c0cc29ef3b79c531704c00ab.png when quantum_mechanics_f87540160fadbe364d62d7a427c9b0d9539e6c99.png, thus

quantum_mechanics_ef2f07973972fe1f50f4478b655e1873152e046e.png

Hence, we can simply expand quantum_mechanics_15df30eb2cc9984ad57a899240f933954e34cfea.png in the basis of all quantum_mechanics_f87540160fadbe364d62d7a427c9b0d9539e6c99.png:

quantum_mechanics_9ce92f53e8dedfaf79f6ebf2f816b82d7add2dae.png

Aaaand we have our expression for the eigenstate correction (of first order)!

Preservation of orthogonality

Let's consider the inner product quantum_mechanics_3fccca5425cb9161df984ad5a9d9e8ec4ad166b4.png

quantum_mechanics_50bebea0b67dee36b14e57892043b9567a7a1f73.png

where we've picked out the terms which will be non-zero in quantum_mechanics_3fccca5425cb9161df984ad5a9d9e8ec4ad166b4.png. Observe that the non-zero terms are of different sign, thus

quantum_mechanics_b661af19ddbbdb49cfa7c6a261cd6e67fc859e57.png

Thus orthogonality is preserved (for first order approximation).

Notes

  • Corrected quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png needs to be normalized, thus we better have

    quantum_mechanics_c779ffa241bfbf527012a018059d4e64d09ddb30.png

  • We require that the level shift be small compared to the level spacing in the unperturbed system:

Example: Potential well

quantum_mechanics_efe3eb06a8661e115b7d747f9d7b779270436f71.png

with

quantum_mechanics_b7301fa52987e75a2873c17df2cb2272d4d1648f.png

where quantum_mechanics_d05f7faefbef9b3c10f76b500dbaf12b2ab66df6.png,

quantum_mechanics_ef1f51c305da133b69f1763bb029009ec111a7bb.png

Thus, the first order perturbation correction is

quantum_mechanics_ce3d7cbe61cfefd4e18b6ae65fe0e7c74cf269b4.png

quantum_mechanics_182b7e6ff962c4ccf6a84aefec9b055984822fa1.png

And the second order perturbation correction is

quantum_mechanics_7e8b54d4b218840158b944210ba06fb7dd6259d6.png

Letting quantum_mechanics_6f26cd7d1f07674c429e76b65c912af3b3b40f96.png, we have

quantum_mechanics_ee62782b75b7c714dd21023af6efcbc58de79fda.png

Degeneracy in Perturbation Theory

Notation

  • States are ordered s.t. first quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png states are degenerate
  • quantum_mechanics_ff7a1486c6e7bea2dea361f86424ae7026ac32f4.png
  • quantum_mechanics_abce2fc4e2c39ec369b3711400c6239e32d329c8.png
  • quantum_mechanics_b312e6a07e339f2d01681c6ccc22ff1613d56282.png

Stuff

Remember, this is the case where quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png has degenerate eigenstates, and we're assuming that the "real" Hamiltonian quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png never is degenerate (which is sort of what you'd expect in nature, I guess).

Consider the eigenstate quantum_mechanics_6e9ab7a59a50c2029174aa8b4e260decbc8f0128.png with degeneracy quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png:

quantum_mechanics_ef7f8a4f7ac21c189ef7cfcae3715774ebb0f726.png

Then

quantum_mechanics_67a91f81667c3e74ea9799404a3b23808b634906.png

We're just saying that we've ordered the states in such a way that we have state of quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png degeneracy occuring in the first quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png indices.

Since we have a g-dimensional subspace to work with for the degenerate eigenstate, now, instead of just finding the coefficients as in non-degenerate perturbation theory, we're looking for the "best" linear combination of the eigenbasis for the degenerate case!

quantum_mechanics_27bcdb6b37c17962108e8002f2c91c3458578d1e.png

and want to find the "optimal" coefficients / projection.

Proceeding as for the non-degenerate case, we assume that

quantum_mechanics_61bb9c9dfa5a6b78f399abe2dabfd6264fbf2695.png

where we assume we can write the following

  • quantum_mechanics_ff7a1486c6e7bea2dea361f86424ae7026ac32f4.png
  • quantum_mechanics_abce2fc4e2c39ec369b3711400c6239e32d329c8.png

Substituting the expansions into the eigenvalue equation and equating terms of the same degree quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png, we find:

quantum_mechanics_3e30ecb7ee75ec0396a126a4c21ff9382236632b.png

Once again, taking the scalar product with some arbitrary k-th unperturbed eigenstate quantum_mechanics_5bf49549aafd4f7adada58d671cd02ab412a54fb.png, we get

quantum_mechanics_290772f2b0be27ab00cfe0da8df3416d8bae933a.png

substituting in

quantum_mechanics_be381b00c74f45d5bc038861b442b37c195051a0.png

we get

quantum_mechanics_7ff1efce88aa9f8e826a2eea29e6ba950653c88b.png

Now we consider the different cases of quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png.

quantum_mechanics_7f8b4de20949c2937b7f3da04602ac930bf1d902.png - degenerate states

In this case we simply have quantum_mechanics_074328e20ac3eb46417647fe76e2458723f87eb3.png and thus the first term vanishes

quantum_mechanics_69c0460bdf018ffffbd214222523f333ea2d15de.png

which is simply a "linear algebra eigenvalue problem" (when including for all quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png)!

We'll denote the "linear algebra eigenvalues" as roots, to distinguish from the eigenvalues from the Schrödinger equation.

quantum_mechanics_4aae30740eeea5dfdf8adc4a2ff7dbcc1e98e9d2.png

where

quantum_mechanics_a6f8d7151cfa7e03bb26cc83c700d61ee67339d4.png

We get the roots quantum_mechanics_9abf87953dd6622eeada039461bf23ff1842ee72.png then have the different cases:

  • quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png distinct roots => we've broken the degeneracy of quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png, yey!
  • one or more repeating roots => there's still some degeneracy left from quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png, aaargh!
  • all equal roots => move on to 2nd order expansion, cuz 1st order didn't help mate!

If we have quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png distinct roots, we end up with

quantum_mechanics_c383c8a7dee780c35f1baca1002c45d2e73abe08.png

Which is great; it's the same as the non-degenerate case! Therefore, we'd like to arrange for this to always be the case.

Suppose we can find some observable represented by an operator quantum_mechanics_56481186dda922dd45a6f6258565b0441a620e2f.png such that

quantum_mechanics_54ac85e638b617c6f40b895da05aac64b28dbb68.png

Then there are simultaneous eigenstates of quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png and quantum_mechanics_56481186dda922dd45a6f6258565b0441a620e2f.png:

quantum_mechanics_0acd59d8e117002432b9e6c75a477c7c105224bb.png

If and only if the eigenvalues quantum_mechanics_9a647135fd5a5f9cd536b762592275f9be89ed32.png are distinct, then we have a C.S.O.C., and we can write

quantum_mechanics_3a3e5f3b3782291f857badf203085de203c38923.png

Thus,

quantum_mechanics_388013381b278b46babd97bf1bab92e8b07fe367.png

which is

quantum_mechanics_014ecfa3e0b8676948d6cb032ea9e20355eab038.png

Therefore,

quantum_mechanics_bc5f01156392bebd12ea547416f30768684c5c32.png

Hence, if quantum_mechanics_4a016dd74e9708e858a6c7b64c3036013b9f16f1.png for all quantum_mechanics_6f51fce2570d7ad3b80f479fd51470ee5a78d81f.png, then quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png is diagonal, as wanted.

Now, what if quantum_mechanics_0cfbaf8099829dc22926c29f06d382850bbea3bf.png are not distinct?! We then look for another operator quantum_mechanics_9dde26348ed436405c8d6c43924594f7450c884c.png, such that

quantum_mechanics_5127da40edb282b8ad5d1dbf3bf3af7907f52d84.png

i.e. quantum_mechanics_69f0017613d71ad3d1fbe7c5a3bd818a339025cc.png is a C.S.O.C., and letting quantum_mechanics_6ce4e3d3bed07e32c4d13126e5fc788214b9c344.png we can repeat the about argument.

We're saying that if we can find some operator which commutes with both quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png and quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png, we can make quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png diagonalizable, i.e. making our lives super-easy above.

quantum_mechanics_e5a05bb82ecb4a0436b23c7ff599636fe7bac06d.png - non-degenerate states

We simply take our expansion for quantum_mechanics_f1930a81077bc69a4f0de3fd91205667487cf457.png and do exactly what we did for the non-degenerate case!

Example

Central potential, spin-(1/2), spin-orbit interaction:

quantum_mechanics_2b3e685d7fcab2379f5243e12d2f3bcbf6822559.png

We know that

quantum_mechanics_cba45c9fb4da399684e21e12cb97cc00cb9cbbed.png

so if we use the coupled basis quantum_mechanics_7d9492a78eca737e57b9712c5c729d71dbc777ac.png, we can use the non-degeneracy theory to compute the 1st order energy shifts!

quantum_mechanics_0d54fd8875f11d61867d995f4b0e3874568c75f6.png

but

quantum_mechanics_76a4fc592ac5c493c39728d0931dd527c995e333.png

because of the quantum_mechanics_5b05c8ea129b3aefda7e125c3c88744a924823e0.png term in quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png.

Example: Hydrogen fine structure

Notation

  • quantum_mechanics_e6b6e706af6982c85d6933081c101c54aad451cb.png denotes the spin-orbit term, whose physical origin is the interaction between the intrinstic magnetic dipole moment of the electron and the magnetic field due to the electron's orbital motion in the electric field of the nuclues

Stuff

quantum_mechanics_53772e1345ff34bf594259ed3501721746ed688b.png

with

quantum_mechanics_018337a8b6a3a94d9b93ec8d7b72b87fccc35ea6.png

quantum_mechanics_56f0b832e4ee6484cd8c52a58db8492702a44c0c.png

We start with the kinetic energy

quantum_mechanics_1028ea281751d5e3567359dee7cbf6dcde2e87a9.png

Where

quantum_mechanics_29da9fe71a48307a3bceeb20954b9bae5f79deb3.png

Therefore

quantum_mechanics_e45ac8143b7d265af4c4e08bc995fe51dbaf1058.png

Which appearantly give us

quantum_mechanics_cf95e73ec1f9c300b124c77ee92044536a0e3e26.png

quantum_mechanics_a1e4a8b21f71682125ac765ccd065aaea7aec4c0.png

And for the spin-orbit interaction we have

quantum_mechanics_a7ad4241fb92ce331191ea5f39056c1d1c13666a.png

quantum_mechanics_17992bece3f9c1ff8dd6fe2079589e0f41c80940.png

which gives us

quantum_mechanics_1868d69828edfa69401536d7855d4c7596240ad0.png

Applying this to some ket

quantum_mechanics_f959e5a47588b5455d9a6bce15f36b91ca018c26.png

Thus,

quantum_mechanics_7f1f75f34d9326e6c57a5b36abaedf57228834bc.png

where

quantum_mechanics_d8b3afa14b33970efb0dafa163bf01c023229a95.png

thus, if

quantum_mechanics_865f48c443758c2e48cb44e7d6ebfc40858d5480.png

quantum_mechanics_ed15493e7e7e456bcf01794442eff8429d4767ce.png

Which gives us

quantum_mechanics_4919dbb82fe80e1b02f5781bc2603ea8a13d392a.png

implying

quantum_mechanics_6375bd3dff80eef91bfbd9f19c505ef0b12d73dd.png

And finally for the Darwin-correction

quantum_mechanics_272307b8cffdb1fcc8c7d76ae652b59f93d3e674.png

Observe that quantum_mechanics_91277b199ae8a8bbb44984e259740edc8d0071ed.png as quantum_mechanics_657834bf5bcf0058db65610e43c5ed8370b128d4.png for quantum_mechanics_dfdaa474eae3b083aa9c880bba55b2f5e97f16f0.png, then

quantum_mechanics_234c109039e1af5be370f2a717a2a75fda345654.png

where

quantum_mechanics_5dc45a12f12de17f5c75364c45d74fb9126c7a45.png

Which gives us the final Darwin correction

quantum_mechanics_a894005f128c3ce4d35136cd80c84163009a85bc.png

Giving the total correction

quantum_mechanics_e5106fda6c556f6a0b81162cdbf2f78c0f35dd7f.png

Example: Helium atom

Notation

  • quantum_mechanics_74a16435e6d92b548b364cc146cd5d7661dece1f.png represents the spin-down state
  • quantum_mechanics_7a9581d5e3aebf337502fabe0528d56cb5fa0e9a.png represents the spin-up state
  • quantum_mechanics_a57464e88fd6b19da991a0fae78c036afc070846.png is the states for the coupled basis, where quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png is the total spin quantum number and quantum_mechanics_cb8c94861c7efdc83228ad9793f6f730786c1c08.png is the total magnetic quantum number

Neglecting mutual Coulomb repulsion between the electrons

Neglecting the mututal Coulomb repulsion between the electrons we end up with the Hamiltonian

quantum_mechanics_e18266b1e35d007b0d48b8f8764774b6b101cb7f.png

It turns out that this yields the ground-state energy just

quantum_mechanics_611a1298c6c1850c4f88a0c56c64687f31fd023d.png

but experimentally we find

quantum_mechanics_66268ad80c16e40ab2d5dbe28b090f5347be4fe2.png

For the Helium atom we use the convention of denoting the ground state with quantum_mechanics_3e67b91c275ca363f9ef19a10395e1bb74f0cccc.png rather than quantum_mechanics_fe2c53a21d8a6ea03f4395abf662960ba035999b.png as for the Hydrogen atom.

Stuff

In the simplest model of the helium atom, the Hamiltonian is

quantum_mechanics_eefa94e25ce8df7bc2fa5f45ef09aa6c473eeb79.png

where

quantum_mechanics_d67e3b9eaf91049c962d8fc28384804408719704.png

The term added to quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png represents the repulsion of the two electrons, and in the quantum_mechanics_9f993d43cf8116684e49c5137ccac2c570769de5.png we have the negative term which represents the attraction.

Also, observe that this is symmertic under permutation of the indices which label the two electrons. Thus, the electrons are indistinguishable, as one would want.

The states of the coupled representation for the two spin-half electrons are the three triplet states:

quantum_mechanics_f37328ff84f6591b9877faf00e35dd21b5d13fab.png

and the singlet state:

quantum_mechanics_80eabe96512b47ee26b65a5d46933072d9b09c67.png

  • triplet states are symmetric under permutation
  • singlet state is anti-symmetric under permutation

The overall 2-electron wavefunction is a product of a spatial wavefunction and a spin function:

quantum_mechanics_1a4994b8a4b3b32da3755019df1e58f26c8fa364.png

The tensor product between the spatial wavefunction, quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png, and the spin function, quantum_mechanics_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png, represents some function quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png such that

quantum_mechanics_dc29e0841636eeb33a4b402541aed040fedcbf7a.png

where quantum_mechanics_23be504fdd2187f45c6fb66dcaa6585cf16baadd.png is as defined above.

Thus, the total 2-electron wavefunction has the following symmetries:

  symmetry of quantum_mechanics_f99568718c1ff190b2d972cfb589249f91e3ae3a.png symmetry of quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png symmerty of quantum_mechanics_6dc1061c5364c3b1728a113bfcdea121ee21a980.png
quantum_mechanics_7cdeceb00609d7e8d3a80c0a41442c6f037eb2ed.png (singlet) anti sym anti
quantum_mechanics_276aa9e3035aee69e38dfe5f2727ba46387d4a1c.png (triplet) sym anti anti

Consider the case of the Helium atom, where (if we're neglecting the inter-electron interactions) we would have the spatial solutions

quantum_mechanics_e3955b9aa081834035bfe32fe4c03739db1dcd9c.png

since we can just use separation of variables and solve the Schrödinger eqn. for each electron separately, both having identical solutions.

Now, suppose further that quantum_mechanics_fe2c53a21d8a6ea03f4395abf662960ba035999b.png, and thus quantum_mechanics_624777a530cb3bdee407f60650d92b1c79ca9b21.png and quantum_mechanics_328f3c5701bcfc3d92a6f55308c54ff4b1adb4f8.png, then

quantum_mechanics_14002eaebf4321d1ad16b59352d53d967642cc39.png

Now consider the case where they would swap places, i.e. we were to interchange them (we're assuming they're completely opposite to each other, since i 2-electron orbit, they're almost always opposite of each other), then

quantum_mechanics_eb99d9fe96dbc126839f823d176b61248bc165ac.png

Now, we know that

quantum_mechanics_c89720de95f5e075e8ab60f74ccd30a3c22c309b.png

Therefore, quantum_mechanics_4838e9e7ebed1e2f4d6ed90fafe902ee234da1dd.png, as we have above, is symmetric:

quantum_mechanics_b94c166bd5babc8f734d452761f3c49fb4036124.png

Thus,

quantum_mechanics_07f01c466042efa25d270a23a5463264a1ee9138.png

BUT we know that two fermions cannot occupy the same state at the same time, hence the above is not sufficient! We need the wave-functions to be different, further, for the wave-function of the entire Helium to stay the same we require the wave-functions to be anti-symmetric under interchanging of electrons (for quantum_mechanics_5fc4bb06ed804d5c094d56a6947eb34a4e7b1983.png, that is). And this is why we introduce the quantum_mechanics_23be504fdd2187f45c6fb66dcaa6585cf16baadd.png described earlier

Ground state

quantum_mechanics_335ac8a3b42b5f14af8b466ee6a1909170deadeb.png

Compute the first order correction to quantum_mechanics_4070caa01fede33fe80e39178121e1df2177be4e.png, we compute the expectation of the perturbation quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png, wrt. wavefunction:

quantum_mechanics_43a4cdf7981db47dc9c48add0fb13fd4ee5d91ce.png

which gives us a correction of

quantum_mechanics_c516255269f0eb4e2f709cad74dcc027fab37993.png

giving us the first order estimate of the ground-state energy

quantum_mechanics_edeaa23bc3c66f515ce555ec2187926407998613.png

which is pretty close to the experimentally observed value of quantum_mechanics_0ff747caf1d20161ed282acf399a11b120257c65.png.

Stuff

Define a new operator

quantum_mechanics_f7bb0a10896a5b755e955a309b4fc85253f97230.png

With

quantum_mechanics_c59100d2a103687884024876662e9acf04917a8e.png

Then we want to expand wrt. quantum_mechanics_224ae917d74dc2133d4403064c971bf562d4db50.png

quantum_mechanics_9090f8549251ee91f6806e0953279c6f48a8bbb9.png

Setting up the Schrödinger equation, we have

quantum_mechanics_e47eb63de8cc1b8ae557a0f4e29fd1ef7a1c05c1.png

Now, collecting terms involving the different factors of quantum_mechanics_224ae917d74dc2133d4403064c971bf562d4db50.png, we have:

  • For quantum_mechanics_0eb4c77097cbe8d379840ef3ea3cb6f8cf3ea8c5.png we find:

    quantum_mechanics_49afd4d30f5ac078c582cca3e59aae0be787fb38.png

    and therefore quantum_mechanics_89cd00ae56215e01e459fa9498c733f6d843858e.png has to be one of the unperturbed eigenfunctions quantum_mechanics_05818fa6392f4aa68ded22132bc3fc1d14553294.png and quantum_mechanics_56c0febd9e3d033181b28ce430bcace978b9ab21.png, i.e. the corresponding unperturbed eigenvalue.

  • For quantum_mechanics_36618a8478f59fd497ff1c3ae5edfb9a30bda857.png we have:

    quantum_mechanics_b7848282ac5e5f3aae4ec95f53f99703d3dd8605.png

    Which we can take the scalar product of with quantum_mechanics_89cd00ae56215e01e459fa9498c733f6d843858e.png:

    quantum_mechanics_de028da7703caf82eb28417f279cb0fa52110744.png

    Since quantum_mechanics_e4f7745a0be0968167c6050e260388054101fec9.png, we have

    quantum_mechanics_55a2846cc1e2f861dc7a200c52649ac5fbf931ea.png

    Hence, the eigenvalue of the Hamitonian in this case becomes:

    quantum_mechanics_dd9056ef9e3ca29c52c7f1a645549d1fc9525e7d.png

This is the correction to the n-th energy level! So to obtain the complete correction due to the perturbation, we have to do this for each of the eigenfunctions which make up the wave-function of the entire system we're looking at.

He atom

quantum_mechanics_1abc449f62c9db281cd60d6bf83152df1bacbd0c.png

quantum_mechanics_b05cf18550ac592d3c1b35c3aee4dcbdcf01d1ee.png

quantum_mechanics_40048e767f6a0fbaef67e211ae2fdc9340c40dc1.png

quantum_mechanics_14c9ee632f21095363304fc42ca39ac1ec51a439.png

Time-dependent Perturbation Theory

Notation

  • quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png is the Hamiltonian corresponding to the unperturbed / exactly solvable system
  • quantum_mechanics_6e9ab7a59a50c2029174aa8b4e260decbc8f0128.png and quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png represent the n-th eigenvalue and eigenfunction of the unperturbed Hamiltonian
  • quantum_mechanics_d666b0a239e3937f0f76611567d5f6aff5490e20.png and quantum_mechanics_f8ac70980af51a75a9075f6f518f20ead0399744.png is the energy and ket of the i-th perturbation term
  • quantum_mechanics_049620bed349a8deabe58aa25ed270a33ddb7c9d.png and quantum_mechanics_73feeecca073cdaae2c97718592184f4e4a20543.png is the corrected (perturbed with all the terms) eigenvalue and eigenfunction, i.e.
  • quantum_mechanics_166ffa148eb6b90032dd84891c3eb0331aea6e3b.png
  • quantum_mechanics_742a6383de791a9c13c0859cb489091b948ad66a.png
  • quantum_mechanics_773b1a757c18b3360058e3ab6fdf745a22a06109.png
  • quantum_mechanics_3133b23187c025b410040a6326ff1c324d5b9b8c.png
  • quantum_mechanics_967bc4f01b4e87cb25a4ed37e5424ce892ffb71a.png
  • quantum_mechanics_03800e61ee2ab1f7ef2dde2b1ba71986ee92c402.png
  • quantum_mechanics_b06103d1e1b2ec7f2b04b228c84ed6cbc4d7ba3e.png is the density of final states

Expression for quantum_mechanics_44c1a05102abb7d15654cf17a95f5240e43caf6f.png

  • Solution to the TIDE can be written

    quantum_mechanics_475e338c0d0f73f142509d1b1c88323e394f1fee.png

  • Generalize to the perturbed Hamiltonian

    quantum_mechanics_e1447fe7ea994dc60a31f8726183771ca002611d.png

    the coefficients quantum_mechanics_3645f2afbf496874d7816be2f7f82d11cb9b737b.png become time-dependent:

    quantum_mechanics_0fa043b9904fc564d890988cf58da2e75c774d3b.png

Observe that lack of quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png here!!!

We're not yet using the perturbation here, which would be

quantum_mechanics_a6739a09a0dddbd4ebacfd81892027ef336e7555.png

This is comes in the next section.

  • Probability of finding the system in the state quantum_mechanics_3a4400366eeeda1f166635890351cdc33d873d4e.png at time quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png is then

    quantum_mechanics_b81da9c23a1a1d9bd9a4b8c60deeb97eca38cd5e.png

    where we have used the orthonormality of quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png.

  • Substituting into TDSE:

    quantum_mechanics_b3af0511eae525da84b490d72609fad29d41fcbe.png

    which gives:

    quantum_mechanics_44e8a874483a7be7cabe5acb25ae9506f12d0ca3.png

  • Taking scalar product with arbitrary unperturbed state quantum_mechanics_3a4400366eeeda1f166635890351cdc33d873d4e.png:

    quantum_mechanics_41a70a77f66f0fec8b2da99de740a3a69c31240b.png

    which gives:

    quantum_mechanics_85a3bfc76f7abf88eebbbf66ca1d2dc92c9cbad5.png

Perturbation

  • Now consider the Hamiltonian related to the perturbed one above:

    quantum_mechanics_c88a3f13bc74875f68bc493c47c3675c98fa17c0.png

  • Assume we can expand quantum_mechanics_f7ae591990d4b793df00916b56f53bdcf775dcc2.png in power series:

    quantum_mechanics_8d6f13b39404fea048fd5b326ec060fbc370c45c.png

  • Substitute in the equation for quantum_mechanics_76d944cf77eaf8d7d8264f9918674c0495833606.png derived above (factor of quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png on RHS is from quantum_mechanics_2d4ef05c0a80db81d27b78e1f4d6c89e87107d7e.png now instead of quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png):

    quantum_mechanics_d5f588c1eaaf423dcc7272dff477530cf3e5b076.png

  • Equate terms of same degree in quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png:

    quantum_mechanics_ea0bdf19b4dc35968fcf2049cb16936379288f11.png

    • Zeroth order is time-independent; since to this order the Hamiltonian is time-independent => recover unperturbed result
  • Integrating first-order correction gives:

    quantum_mechanics_3a6cdc7bf45075645b1fd0d53c8df08e8261a007.png

  • Suppose initially system is eigenstate of quantum_mechanics_9ebf4ddec95719ac2541a12dea3510005cfb18a7.png, say quantum_mechanics_79ecf90cabf865ad87cd08f290557084b2a98ba2.png, then for quantum_mechanics_9914d027e836093efa89c1b85cf9a04490b86b2b.png, /probability of finding system in different state quantum_mechanics_3a4400366eeeda1f166635890351cdc33d873d4e.png at time quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png

    quantum_mechanics_94976f845423a7b406d368c78c4eab234d64e12c.png

  • Thus, transition probability of finding the system at a later time, quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, in the state quantum_mechanics_3a4400366eeeda1f166635890351cdc33d873d4e.png where quantum_mechanics_29303534c3909cda48ee0ef9cdb8b61292562c1c.png, is given by

    quantum_mechanics_0844f16a28b9792dd6149215338255ccf2e6fa21.png

TODO Time-independent pertubations for time-dependent wave equation

  • Suppose quantum_mechanics_ddfef0cd793e5a5ae41498a1194d0e91960b7abb.png is actually independent of time

Fermi's Golden Rule

  • Interested in transitions not to single state, but to a group of final states, quantum_mechanics_8b46b89ea0fbe16704b2d053ca55ad40d48c8b33.png, in some range:

    quantum_mechanics_888dd778b7140cca2d54f13e4dadc053104d059b.png

    e.g. transitions to continuous part of energy-spectrum.

  • Total transition probability:

    quantum_mechanics_2f3027afeb70c13d1a33227f6c3e84a14c0db554.png

  • Assume quantum_mechanics_a7203594e682f1ec03795996fa77a9db19db59c8.png to be small, s.t. can treat quantum_mechanics_40104a2cfa7f9d2916b2077423c91617552a49f2.png constant wrt. quantum_mechanics_f6199d49f470cca13d1db6e3accff88bd33dcb1c.png on this interval:

    quantum_mechanics_0d90b4a9b6436ffe82cc1ba55456e7138ad89125.png

  • Change of variables quantum_mechanics_c0a70d4a1ce6d20dbb9fd3a510de27e3080ff779.png, we get

    quantum_mechanics_cb561c15d1499f65d16c7e56d317e4c938e3c7b1.png

    where we've used

    quantum_mechanics_837eb6432ef6010380616d215430379e7e01f3da.png

Number of transitions per time, the transition rate, quantum_mechanics_3d136f0fc4860468633907421c098b9feb0eef24.png, is then just

quantum_mechanics_9c508dba53e56ae7f4adcd53a4aefe94976d197b.png

quantum_mechanics_829471a782f0f50e2ec0446098ea5da0b2b54ac4.png

Harmonic Perturbations

Notation

  • quantum_mechanics_8183706118a3adc8bcc20e33d4b448e0caa1541f.png is the time-independent Hermitian operator

Stuff

  • Transition probability amplitude is a sinusoidal function of itme which is turned on at time quantum_mechanics_9b50c3c5095e3def3ea0a38afe0602245b66e068.png
  • Suppose that

    quantum_mechanics_2a0004c11a46770f9d87e62690dd96d8e9ada174.png

    where quantum_mechanics_8183706118a3adc8bcc20e33d4b448e0caa1541f.png is the time-independent Hermitian operator, which we write

    quantum_mechanics_e27e1de63bf59e87c48187c44d875b3d39f5a378.png

Radiation

Notation

  • quantum_mechanics_fca38f90babf39231c9ef1172d9d6b1375b49614.png is the dipole operator

Electromagnetic radiation

quantum_mechanics_fca38f90babf39231c9ef1172d9d6b1375b49614.png is the dipole operator where

  • quantum_mechanics_1a4a18e739bc08439a04efa4dc6013a508ff1163.png is unit charge

Einstein A-B coefficients

Notation

Spontanoues emission (thermodynamic argument)

We have

quantum_mechanics_e83a001a5917a4a72885eb136a3debffeb8d5c8f.png

For absorption we have

quantum_mechanics_9046bcb447bb94d0b84b7b506bb711e2d3ecf5d6.png

Now, suppose we have emission , with transition quantum_mechanics_c3f4e953d474def9d0c0e1175f684c75edafdfc0.png:

quantum_mechanics_9dabe49d46dfc5365e1b92c0e7b1cd54425fec32.png

Note that this is quantum_mechanics_ba9c875ec1c1e18e67f37a0199fe174bab938b3c.png, NOT quantum_mechanics_2882062ef214a70c6657355ed6900c7006552cd0.png as introduced earlier.

The distribution between the number of particles in the different states is then given by the Boltzmann distribution

quantum_mechanics_3b856b605b4ca549e7cf39ca804befc2501b4ebc.png

Plank's law tells us that Black body radiation follows

quantum_mechanics_af5e4dd665c02963a63a4a2bf347ffa9cb73d0b5.png

Which we can rewrite as

quantum_mechanics_a6afc989f470838a5109c1ffa6098fb9cce7f652.png

Substituting back into the epxression for spontaneous emission

quantum_mechanics_9525d381e3ee7bbc0d9beac13d1cb2590584b759.png

One obtains the same result in QED.

Selection rules

Goal is to evaluate quantum_mechanics_34e067f2abab4f8407e99635be2afe969f12ac10.png.

Hydrogenic atom
  • electric dipole operator is spin-independent
  • work in uncoupled basis and ignore spin

quantum_mechanics_765c36f3507c8ac5cd525d6e68275693dae573c0.png

Which implies

quantum_mechanics_c2990bd91290c186193b6c6d86fd51a7c87fa91e.png

Which implies either of the following is true

  • quantum_mechanics_4052e6c5d92ffe166265c56c3b6725987a9be74c.png
  • quantum_mechanics_8f7becb268ac7f3911f3beebe9f096a47a8edb71.png

This is called a selection rule.

Doing the same for quantum_mechanics_03f2a398abdc13ca82cac1636553b0b16ff43c2b.png, we get

quantum_mechanics_c3403095c586e9bc65292d11bacb92dc2b2b1525.png

Considering the matrix elements, we get

quantum_mechanics_3f710937c2fb579f2cdb3334393fe88494588c5e.png

which becomes

quantum_mechanics_7a98771b7f8b0f8fbb96c31a71b3481cb352a687.png

so we deduce that

quantum_mechanics_7daf985741cffadc2d666153af20e8e0b53787ef.png

giving the selection rule quantum_mechanics_5eedcc6049c7a91500d3be722854b480402287fe.png.

We then conclude that the electric dipole transitions are only possible if

quantum_mechanics_335d01904db719e935a0d799f96a62c9364d54e3.png

Which is due to

quantum_mechanics_2a3514d89de8b8b211e5d9cf30374480b7a79614.png

hence quantum_mechanics_9d887b6f787c165e22e49e266c18a6e61be961c0.png would be zero unless quantum_mechanics_1b85090945afe4beccc8f1d5fc7432e1bffb05b6.png.

Parity Selection Rule
  • Under parity operator quantum_mechanics_8e9a46e9e26709eda4de2b2568b0e694d5c73936.png, electric dipole operator is odd multi-electron atoms
  • See notes :)

Quantum Scattering Theory

Notation

  • quantum_mechanics_7bec91615d02a5342b34270ae4a669947491ecaf.png is the scattering angle
  • quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png is the incident angle
  • quantum_mechanics_71e3e78e19ee2e47dfebe55e0ae6c7b74493ac36.png is the incident momentum
  • quantum_mechanics_de796225943d4f14c2b642a6aa6752c8bd5bc1b7.png is the scattering momentum (i.e. momentum after being scattered)
  • quantum_mechanics_09646f2be98ff673b1215d2ea539008f339d3110.png, thus when we talk about the direction and so on of quantum_mechanics_ed16129729c0be1ee8ff5ca34ae68982a2ba81cd.png (wave-number), it's equivalent to the angle of the momentum
  • quantum_mechanics_f85a375e8ca610c42bb7d02ca742abb5870ad3f0.png is the solid angle
  • Rate of transitions from initial to final plane-wave states

    quantum_mechanics_48965a4ed7afb268b08f3a62b2f94ba55c06ee3a.png

  • quantum_mechanics_b7b40645f376b67d1dc717f3bb3deeba382e6fbe.png is the density of the final states: quantum_mechanics_05e925069ee60b39b1883867c8692ec1762fd1cb.png is the number of final states with energy in the range quantum_mechanics_147c6467615b5bcfb5704499814c608178a2a803.png
  • quantum_mechanics_5d6e1371783195b7a912fd64e5228163d7941537.png denotes fixed potential (i.e. we say the "potential source" is fixed at some point and quantum_mechanics_d86190240fc08d8dfdbf734ca628129ce9bb1e60.png is then the position of the incident particle, and we treat the potential from the fixed scattering-particle as a regular "external" field)

Setup

  • Beam of particles
  • Each of momentum quantum_mechanics_04cdf724b0e79b2f71355ae5cea52983d55564c0.png

quantum_scattering.png

The incident flux is the number of incident particles crossing unit area perpendicular to the beam direcion per unit time.

The scattered flux is the number of scattered particles scattered into the element of solid angle quantum_mechanics_f85a375e8ca610c42bb7d02ca742abb5870ad3f0.png about the direction quantum_mechanics_7bec91615d02a5342b34270ae4a669947491ecaf.png, quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png perunit time per unit solid angle.

The differential cross-section is usually denoted quantum_mechanics_f236dcfdbd7eafa080943bdd2c3054d128e284be.png and is defined to be the ratio of the scattered flux to the incident flux:

quantum_mechanics_641b2c1b2e903602166d9a00fa365c32b30571e3.png

The differential cross-section thus has dimensions of area: quantum_mechanics_204ca00c9cf0556b521ac15129e8de5b02c3a716.png

Total cross-section is then

quantum_mechanics_51d00fb88a29ae65eb6fa751f8e0d36544c5b52c.png

Born approximation

  • Can use time-dependent perturbation teory theto approximate the cross-section
  • Assume interaction between particle and scattering centre is localized to the region around quantum_mechanics_08836347d69b2e1f3f960129a55da2f494bbbdd0.png
  • Hamiltonian

    quantum_mechanics_d7d3ccbde5875b024cf5b50f38358e182cc04857.png

    and treat quantum_mechanics_a44f433a3f3c177a2ea85107c84bb82c41731368.png as perturbation

  • Wave-functions are non-normalizable, therefore we restrict to "potential-well" scenario, as we can take the width of the well as large as we "want"
    • Since we're working in 3D: box-normalization

Density of final states

  • Final-state vector quantum_mechanics_e0802bd1ba78994012f6f57c857e186e60242f7b.png is a point in k-space
  • All quantum_mechanics_e0802bd1ba78994012f6f57c857e186e60242f7b.png form a cubic lattice with lattice spacing quantum_mechanics_91226c56c3f32a032c18bb8d8e4f55fad47ba8d6.png (because of the potential well approx. discretizing the energy)
  • Volume of k'-sphere per lattice point is quantum_mechanics_354aacb0f79d57a2426ddf0f2d7ac9e1b71336fd.png
  • # of states in volume element quantum_mechanics_627cc0dc3d4fc3614aab38c44e1c4b97773c3651.png is

    quantum_mechanics_fb123e8b95b70d6c922bf2003d772501a8b2b481.png

    using spherical coordinates

  • Energy is

    quantum_mechanics_bea7bf7b286171e33e8c96fc4962a9afaa73aa87.png

    thus, the quantum_mechanics_b7b40645f376b67d1dc717f3bb3deeba382e6fbe.png, the density of states per unit energy, is the enrgy corresponding to wave-vector quantum_mechanics_e0802bd1ba78994012f6f57c857e186e60242f7b.png. Therefore,

    quantum_mechanics_07ecdb6032edd0d7a4604dfd291ecf2eaae5a568.png

    is the # of states with energy in the desried interval and with quantum_mechanics_e0802bd1ba78994012f6f57c857e186e60242f7b.png pointing into the solid angle quantum_mechanics_f85a375e8ca610c42bb7d02ca742abb5870ad3f0.png about the direction quantum_mechanics_d355a07585ee38f0061020001bbcabed0f0fc0d7.png.

  • Final result for density of states

    quantum_mechanics_62ad10f5b60a20759b7e9629ce1e56aa54203cb3.png

Incident flux

  • Box normalization corresponds to one particle per volume quantum_mechanics_f84894a300289dc5c0ce37fbce0bd1df633310cd.png

    quantum_mechanics_548d8b74c94e511c71d588cfb3869c2449ca64da.png

Scattered flux

  • By Fermi's Golden Rule

    quantum_mechanics_5995df0eb1e29d6ffddb0614e4e93c1b85215932.png

  • Is # of particles scattered into quantum_mechanics_f85a375e8ca610c42bb7d02ca742abb5870ad3f0.png per unit time
  • Scattered flux therefore obtained by dividing by quantum_mechanics_f85a375e8ca610c42bb7d02ca742abb5870ad3f0.png to get # of unit time per unit solid angle

Differential Cross-section for Elastic Scattering

  • Can compute the cross-section using scattered flux and incident flux found earlier

    quantum_mechanics_2fdafa860f461813f4712afbef80bec26aba229d.png

  • If potential is real, energy conservation implies elastic scattering, i.e.

    quantum_mechanics_a3283da6d272bdd622e124ffda5051be33c7087a.png

The Born approximation to the differential cross-section then becomes

quantum_mechanics_b5d16a95be2fc8ca47eba90537c5b15ecc008a99.png

with

quantum_mechanics_64a40fb00adbea866e1ffb5df7f65c36adc929d7.png

where quantum_mechanics_ea30f852e1aaece64f87022beb3d0a4bb565b115.png is called the wave-vector transfer.

(this is just the 3-dimensional Fourier transform of the potential energy function.)

Scattering by Central potentials

  • Can simplify further
  • Work in polar coordinates quantum_mechanics_c80dc7fdd9fc8b4b976ce49ed5776efad866b358.png, which refer to the wave-vector transfer quantum_mechanics_ea30f852e1aaece64f87022beb3d0a4bb565b115.png so that

    quantum_mechanics_d8e7e5eb48313530c9356b12733500692a5ec59d.png

  • Then

    quantum_mechanics_b4ed2cdad5200b0f3b44d98f709d6ba2ff781291.png

  • Born approximation then becomes

    quantum_mechanics_3031fd806f36f89b2eebd6b8f04e64bfcb197a59.png

    which is independent of quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png but depends on the scattering angle, quantum_mechanics_7bec91615d02a5342b34270ae4a669947491ecaf.png, through quantum_mechanics_1641d18cc980f8db14cdff95d7417a8526eef446.png.

  • Trigonometry gives

    quantum_mechanics_02f127e8efe68e7400487e8907b4f0589b10a75b.png

Quantum Rotator

  • Two particles of mass quantum_mechanics_937ac307652c479877aab0f9607fea79ea25f7d0.png and quantum_mechanics_921df09a52c87777e1ac388adf65f0cce2ab0929.png separated by fixed distance quantum_mechanics_0f4e8d0395a52a6a0e30bffd378754b4d7cea9fb.png
  • Effective description of the rotational degrees of freedom of a diatomic molecule
  • Choose centre-of-mass as origin: system is completely specified by angles quantum_mechanics_7bec91615d02a5342b34270ae4a669947491ecaf.png and quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png (since quantum_mechanics_0f4e8d0395a52a6a0e30bffd378754b4d7cea9fb.png is fixed)
  • Neglecting vibrational degrees of freedom, both quantum_mechanics_5926a2dcef2ea762e4e3fb75bcf57e6712557db8.png and quantum_mechanics_226aa776fce313197cd05377735658272e641364.png are constant, and quantum_mechanics_d67207fc51a7371e22c60d26210647ae6ffbdbf9.png
  • Since origin is frame of CM:

    quantum_mechanics_0a4277f2d8a93b8c368f7ec15e84ad6b64f4bd8a.png

  • (Classically) Moment of intertia of the system is:

    quantum_mechanics_180529e3abd2f99c4ef6bb2601be8a3eff123cc8.png

    where

    quantum_mechanics_5af42c2fea5fbcac83ceb2fd0491ac8ee88c30f3.png

  • Classical mechanics:

    quantum_mechanics_2e0eba9c8075a290020c89a1101c4be04cd09bbd.png

    where quantum_mechanics_36e6e3116a238091202bae070d7f195c50486509.png is the angular velocity of the system. The energy can be expressed as

    quantum_mechanics_e41159fb0d4beaca09ea5cdfe58d0d8414b029ad.png

  • Correspondance principle:

    quantum_mechanics_89c107d7efbb9c83ab08b1c53f709f7551aee1b6.png

  • Since we're neglecting vibration, quantum_mechanics_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png is fixed, hence the wave function is independent of quantum_mechanics_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png:

    quantum_mechanics_d4d2ffd66fff83fd001dbf2f5f989ce6d5a19fdc.png

Two-body scattering

  • So far assumed beam of particle scattered by fixed scattering centre; interaction described by quantum_mechanics_a44f433a3f3c177a2ea85107c84bb82c41731368.png
    • Useful for electron-atom scattering due to regarding the atom as infinitively heavy
  • Consider two-particle system; turns out it's the same!
  • Hamiltonian with relative separation

    quantum_mechanics_890c30b82ae500c6d6646b2828e7597cf5504881.png

  • Centre of mass and relative position vectors

    quantum_mechanics_70d670175077b6cb75f74063d04cd8ce1291d58b.png

  • Then

    quantum_mechanics_fdbfa5710a0799711dd8e630d8a87477a0c7e471.png

  • Rewrite gradient operators quantum_mechanics_ebe658ef4f28ce5660e7b25b451dcab2d533410b.png and quantum_mechanics_c82618557e9d74f2119f941996a90d3757286b39.png by

    quantum_mechanics_46972c5e15ae8654122b0e2a6191416ad1b8a0fc.png

    where

    quantum_mechanics_406e01e0242ce6421d7e4e41ed6ab026135b656a.png

    and so on.

To see this just apply the differential operator wrt. quantum_mechanics_ca2a495fb8e64acb1cc942d55440d084d83b588a.png to some function quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png:

quantum_mechanics_b74064ed4823f228b72c675eafa0c328852248e0.png

  • Then

    quantum_mechanics_f130e2201a506d3e429191efd4b788d5a32cab7e.png

    where

    quantum_mechanics_a139d77d88188ed667fc1ecfa5dd5543b3ae9ff3.png

    with quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png called the reduced mass

  • Can be viewed as

    quantum_mechanics_766d64000633755c9ea0cfbcbf545a9f03d2c977.png

  • quantum_mechanics_f258b047eac821a37005e0a4dc062d3c70d27d73.png describes free motation (kinetic energy) of centre of mass
  • In CM frame the CM is at rest, hence

    quantum_mechanics_7afc2a47cbeff10529338e90289db649e67ba9bb.png

    which is identical to the form of Hamiltonian of a single particle moving in the fixed potential

quantum_mechanics_5d6e1371783195b7a912fd64e5228163d7941537.png

  • Implies CM cross-section for two-body scattering obtained from solution to single particle of mass quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png scattering from fixed potential

quantum_mechanics_029db7f5fe173347ed121aca758e366fbfeffd95.png Ion and Bonding

Notation

  • quantum_mechanics_83bc72bfcb75deba5f65643ddbce6fa2cf599fa4.png is the mass of a proton
  • quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is the mass of an electron
  • quantum_mechanics_ea3e91227b6e0e64366687b8608dd0bcef81741d.png reduced mass of the electron/two-proton system

    quantum_mechanics_11a362d411ebd6c36ad54d2ac0932a24ef459882.png

  • quantum_mechanics_92ef1f690b8025412462ee0289bf3884448fe698.png is the reduced mass of the two-proton system

    quantum_mechanics_772e89f0621067aca15d50044a5be9e146c50c46.png

  • eigenfunction is called gerade if parity is even, and ungerade if the parity is odd

Stuff

  • Schrödinger equation is

    quantum_mechanics_bc7c28099e59e9a93451183ff6dd5291d4d9a720.png

  • Nuclei massive compared to electrons, thus motion of nuclei much slower than electron
  • Therefore, treat nuclear and electronic motions independently, further treat the nuclei as fixed, i.e. we only care about quantum_mechanics_294e8a599428eb321e4b495827fcd6096bd0cb6a.png

Central Field Approximation

Notation

  • quantum_mechanics_4cd082f9db7ea80260b7e962dd0e38070da0d1a2.png is angular momentum quantum number
  • quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is magnetic quantum number
  • quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png is mass

Stuff

Starting with the TISE, we have:

quantum_mechanics_79e4018fdd104984da8741ae3a47b6f2dc00f779.png

Rewriting in spherical coordinates, using the Laplacian in spherical coordinates and assuming

quantum_mechanics_bac53416daf81a584fedcf33f8f0c7af2506cb82.png

we have

quantum_mechanics_5387dc77764a273067a3c049af19a42a832bb641.png

which becomes:

quantum_mechanics_2fabb4e7d59808d7418cbe9722067febc6ae611a.png

Suppose

quantum_mechanics_4afda4f970f3c8330337b2367175daa3ff8f8285.png

then the above equation becomes

quantum_mechanics_c1dcac179506c25ecefece7a1323be57e83accaf.png

Thus,

quantum_mechanics_27e8a5b484adbfca4f7fdef5a9fb4296ddf29991.png

Substituting back into the equation we just arrived at, we obtain

quantum_mechanics_831af6a9911fe1956b0a9e0292677be147783299.png

Multiplying by quantum_mechanics_17bffe02d589bbbc96f6efbdbbdd12efb5d9607b.png

quantum_mechanics_ee830ae45f84362679a01d4d65354827f2dda6b5.png

Which we can rewrite as

quantum_mechanics_49fd8e9e9a1eb528b04ae3f2a90c86e1d6b776c6.png

Therefore we consider this is a effective potential:

quantum_mechanics_a8b2c5bd0f70a6c0fe7e4dd51c56e41dafd5f884.png

Example: charged nucleus

Consider the problem of an atom or ion containing a nucleus withcharge quantum_mechanics_9c920cb9c335f79cf404425d19b535aa5c305878.png and quantum_mechanics_b3845b676dc3083a0ac31a0a29810621f9c09bcc.png elecrons.

We assume nucleus to be infinitively heavy and we assume only the following interactions are present:

  • Coulomb interaction between each electron and the nucleus
  • Mutual electronic repulsion

Then the Hamiltonian is

quantum_mechanics_64aa2dea5f912ccec86083592f531af6885ae6cd.png

Presence of electron-electron interaction terms (2nd sum) means that the Schödinger equation is not separable.

Therefore we introduce the Central Field Approximation, which is an example of a independent particle model , in which we assume that each electron moves in an effective potential quantum_mechanics_a80c4ae68e89ab3487713a3f39113104da4d3ae4.png, which incorporates the nuclear attraction and the average effect of the repuslive interaction with the other quantum_mechanics_1e66cdd84daa6689871191626a353bce08a60764.png electrons.

We expect this effective potential to have the following properties:

quantum_mechanics_a1d9207f950296d6a76caad20c51e1f73e8b9953.png

  • quantum_mechanics_ba48fd4683c45042a623a66b1a5d683801f9e464.png means the nucleus dominates
  • quantum_mechanics_cfb50a303f48a17f4db8888ecff355acd38dba71.png means that we can treat (nuclues) + (the rest of the electrons) as a point-charge

Variational Method

  • Especially useful for estimating the ground-state energy of a system
  • This is closesly (if not "exactly") related to the Rayleigh quotient from Linear algebra

Main problem is guessing a trial function quantum_mechanics_ed53b66dec28fff87e99da125bbb5dd5d7e4ba5e.png:

  • Recognize properties of the system and observe that quantum_mechanics_ed53b66dec28fff87e99da125bbb5dd5d7e4ba5e.png ought to have the same properties (e.g. symmetry, etc.)

Useful identity:

quantum_mechanics_5fb0beffd9c5cc818a6c7d343059e29ec6a079bb.png

Consider a system described by a Hamiltonian quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png, with a complete orthonormal set of eigenstates quantum_mechanics_39d535c4677de27599c36b5ef9c8a79d94515317.png and corresponding energy eigenvalues quantum_mechanics_5668fdeb0d8b650e911170a69dc042f528c816c5.png ordered in increasing value

quantum_mechanics_06a1b88e5b0cd40e53b99f7936399b1836fcb319.png

We observe that

quantum_mechanics_8c3dccc56a87924213c95be3e359c43f4b4baf17.png

since quantum_mechanics_e7ce7019798cbbeadc757ad2f7724b34d23791c0.png for all quantum_mechanics_32df88ec08d3c40d5bae099216a3a2e1142599fe.png. Thus we have the inequality

quantum_mechanics_12dc791f6248a55b792df5ab17afbfd62f910edf.png

Thus, we can estimate quantum_mechanics_4070caa01fede33fe80e39178121e1df2177be4e.png as follows:

  1. We chose a trial state quantum_mechanics_022ef254bd8e3eb0e2ed30aefabf5269694e77f6.png which depends on one or more paramters quantum_mechanics_07e474680afb45dc9cd09009076e1333641ea974.png.
  2. Calculate

    quantum_mechanics_4c204ab9c29ae1cf7b5d855918da4d9dd5060073.png

  3. Minimize quantum_mechanics_84792c2d61a2924b75e3c627b7fea861031aa55b.png wrt. the variational parameters quantum_mechanics_07e474680afb45dc9cd09009076e1333641ea974.png, by solving

    quantum_mechanics_13555512be52336b6eeed0e0992612c970697be8.png

The resulting minimum value of quantum_mechanics_add09316f6765383a87050335c92fe3c1effc116.png is then our best estimate for the ground-state energy of the given trial function.

Hidden Variables, EPR Paradox, Bell's Inequality

  • Hidden variable theories suppose that there exist parameters which fully determine the values of the observables, and that QM is an incomplete theory where the wavefunction simply reflects the probability distribution wrt. to these parameters

EPR though experiment

  • Two measuring devices (Alice and Bob)
  • Electron sent from the middle where due to conservation of momentum (total spin quantum_mechanics_6895ac723a44d1e88cb199e38db32983bcf92d72.png needs to be conserved) we need:
    • Alice to receive quantum_mechanics_85c616b710937f8b9c71b4f400174742eff9e4ab.png or quantum_mechanics_f938e594168c2e335602f9f5006f2cf3e842ca95.png
    • Bob receives the state which Alice does not
  • Distance between measuring devices is too large to exchange any information between the the two measuring devices (even at speed of light)

Consider the following cases:

  • Both measures spin-state in z-direction: measuring Alice's completely determines what we measures at Bob, due to spin being conserved
  • Alice measures along z-direction, Bob measures along x-direction: measuring Alice's does not completely determine Bob's measure, rather, both spin-half states will be equally likely for Bob

The state of the "electron" measured by Bob will be correlated to the direction in which Alice makes the measurement!

Bell's Inequality

Notation

  • quantum_mechanics_23a41686415bf15002851802933b520d67febc2d.png spin component in direction quantum_mechanics_493eec6a2fdfbfd0dcc20ff18d41045c90c7e002.png of the particle travelling to quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, e.g. quantum_mechanics_9b3f539101443c17cd097804ea05ecd3cd879b3c.png refers to taking on a specific configuration (remember we can only measure one of these directions)
  • quantum_mechanics_b9fe35504be72b76e5babd54961c09e4a4c0d444.png spin component in direction quantum_mechanics_493eec6a2fdfbfd0dcc20ff18d41045c90c7e002.png of the particle travelling to quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png
  • quantum_mechanics_c388a4e4d61a3d4dfdfb8107e2b53503de7eb329.png is the fraction of particle-pairs belong to the group quantum_mechanics_6d997cc48e8b713d187449fe1cf8d9e91dec6c17.png
  • quantum_mechanics_e8989324acda4ea72727dd13cfa6302c460399b7.png denotes the probability of measuring quantum_mechanics_1105616c4cf4d32f024e46c034660f9a9289dee9.png and quantum_mechanics_85c004bebf4d2e4ac6d646265677aba3e3492a28.png together

Stuff

Since we're assuming that the spin-half pair is produced in a singlet state, this tells us immediately that

quantum_mechanics_b633eec5443aab2f831cb7be70a82397ecea31e1.png

Suppose then that we're measuring quantum_mechanics_c5d0497c3147731441af41dc7eaeacbb6f7ebf6b.png and quantum_mechanics_ebb1ca19ca27363c5b474c32fa0ff515f5138911.png, we marginalize out all the other possible states to obtain the distribution:

quantum_mechanics_e63ecb5f9c7bb2dd8082f3a623057e1f63409df3.png

Observe that

quantum_mechanics_2fc3ac17863d859beb6e692dc3ddd3f22b48a81e.png

Thus,

quantum_mechanics_cc1d63c06021194b27d76fe7835352f8c29a25b1.png

Doing the same for the pairs quantum_mechanics_25d6fd8ccf10ca349cd37f67eea723393df7d5b6.png and quantum_mechanics_87dbd8548d12b5fb14f07ee76e26864a36d76179.png we observe that

The Bell's Inequality is for measurements made of two spin-half particles in directions quantum_mechanics_662567f12c16b2f88db990790fdc0f6823bfed11.png:

quantum_mechanics_1548466ce516629d97a042a92f9064939925c3ed.png

which defines a requirement which needs to be satisfied for a theory to be a realistic local theory.

Components possessing objective properties which exist independently of any mesaurements by observers and that the result of a measurement of a property at point quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png cannot depend on an event at point quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png sufficiently far from quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png that information about the event at quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, even travelling at the speed of light, could not reach quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png until after the measurement has taken place.

Theories meeting these requiremenets are called realistic local theories.

Quantum communication

Notation

  • quantum_mechanics_2fc48343c20e7d1810d36ee3a8190d88ae251b97.png, i.e. these all denote the same thing
  • Convention to represent these two states

    quantum_mechanics_69166d12b96b38dc12e201dc716876048236373b.png

    which is often parametrized in polar angles

    quantum_mechanics_560f9613cccea46c6051f5038d323f77ae2f0070.png

    with quantum_mechanics_64eea29bfc7be48e3158021bd3588709f3cb904b.png and quantum_mechanics_5cae5595cca81b04f52178cb276924a433b841d5.png.

  • Bell's states:

    quantum_mechanics_2cf3117d2c7c1def1a67d9970a70fc01118721f0.png

Secure communication

An entangled state between systems quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png implies the wavefunction of the combined quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png systems cannot be written as a tensor-product of independent wavefunctions for each system, i.e.

quantum_mechanics_789b5d219d23ca44afb259a551e456705b6c2ccc.png

The Bloch sphere is just the parametrization of a two-qubit system in polar angles in 3d:

quantum_mechanics_560f9613cccea46c6051f5038d323f77ae2f0070.png

with quantum_mechanics_64eea29bfc7be48e3158021bd3588709f3cb904b.png and quantum_mechanics_5cae5595cca81b04f52178cb276924a433b841d5.png.

A frequently used complete set of two-qubit states in which both qubits are entangled, are the Bell states,

quantum_mechanics_c62bf8c37f6e5f98f8b7b396f35343c985e09c44.png

No-cloning theorem

It is impossible to create an identical copy of an arbitrary unknown quantum state.

Formally, let quantum_mechanics_00a944c26f0a49f01fe53ef567cd6c5ae6819767.png where quantum_mechanics_991879c7fa97073b972673f091ed54ad80ca2ecc.png denotes a Hilbert space.

Suppose we're in some initial state quantum_mechanics_5c8eebb60e0289bba600b311bdef7f0479b11171.png, then initially we have

quantum_mechanics_2dc56d4de5447e3504e5c8b4d4566a156a26f9e5.png

Then the question is: can we turn quantum_mechanics_5c8eebb60e0289bba600b311bdef7f0479b11171.png into a duplicate state of quantum_mechanics_44dcf12668047a91d8f4c8123be8b3d65e7098a8.png?

  1. Observing quantum_mechanics_44dcf12668047a91d8f4c8123be8b3d65e7098a8.png will collapse the state, thus corrupting the state we want to copy
  2. Can control quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png of the system and evolve quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png with unitary time evolution operator quantum_mechanics_2d3a960231083503fab5ed0075a401a7f9b12fa4.png, where by unitary operator we mean that it preserves the norm and coherence.

Answer is no.

Suppose quantum_mechanics_75ef81a6278d7ea44a6421c77b360a52ce7f1ceb.png on quantum_mechanics_5ef57531625becc28e364c3e7e2d8b82241f1616.png such that

quantum_mechanics_fda21faf686eeb4d1f2d3cf44936926d8c619bba.png

where quantum_mechanics_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png and quantum_mechanics_d21892f3bdfae9ec08a78f3061484c467c1030ec.png are two arbitrary states in quantum_mechanics_991879c7fa97073b972673f091ed54ad80ca2ecc.png.

quantum_mechanics_b1ece3b39fae3736224448d6fcbbe8af558999f9.png is the angle which in general can be a function of the states on which quantum_mechanics_5f00944bfa9ea5e5d44ff4410e0ce40f19d5760d.png is acting. Then we have

quantum_mechanics_52da9724f9eb35df0ec119874f2aa94d703439a8.png

This equation leads to the condition

quantum_mechanics_1e3c447721cb363824989bf913994ab63f386d0a.png

I.e. states are either orthogonal or the same.

Quantum teleportation of a qubit

  1. Two spin-half particles are prepared in the Bell state

    quantum_mechanics_7c89c6d9f02279e336bf410eee4b5bbb18dae2be.png

  2. Establish a quantum channel by giving one of the electrons to Alice (quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png) and one to Bob (quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png); due to the EPR-stuffs we know that if Alice makes a measurement, she can use a classicial channel to tell Bob which state she measured, which tells Bob that his electron is in the opposite state.
  3. New particle given to Alice (Bob also knows that quantum_mechanics_3f50e5819e39baedffbe9c030549361ae8c4f2ee.png also takes this form)

    quantum_mechanics_0223dd07fd867f79ddd09d909731fa5d35041c28.png

  4. Combine with the EPR states (referring to the ones Alice and Bob received beforehand)

    quantum_mechanics_fc1f5f528a25b212e42434d4c6f617114506bbed.png

    which we can rewrite in terms of the Bell-basis:

    quantum_mechanics_fcde55dc584721adf78c1848a18a3ce5a4e9211f.png

  5. Alice makes measurement of the two-state system consisting of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_6ecabf46b855c7cbc7b0c57a13b6f1559d122040.png (this can in fact be performed physically)
    • Collapses quantum_mechanics_071fd226f65c474cb3fcc75f04f962d1909d0b0a.png to one of the quantum_mechanics_a56a45f1199099e02b36e8cf2ca207ede846e1e7.png or quantum_mechanics_b0f248e010ccbb05a87bd59b5282e5995f170e08.png
    • Alice now knows which of the states quantum_mechanics_a56a45f1199099e02b36e8cf2ca207ede846e1e7.png or quantum_mechanics_b0f248e010ccbb05a87bd59b5282e5995f170e08.png her system is in
    • Breaks entanglement with quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png and forces quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png to take on the "remainder" of the system, i.e. one of the

      quantum_mechanics_f39965056e7b0ee4d370f04a3fe2399d1d4159a8.png

  6. Alice sends back to Bob which state quantum_mechanics_621a04a94bc9d6d3d927f74c68f743bad92435c7.png collapsed to
    • Since Bob knew the expression for quantum_mechanics_fdfe843b1ff43668a82fd58d668e566da309c51c.png, by learning which state quantum_mechanics_11294cacbadfea8983ca62b8d7f232f5a5035f96.png is in, he can fully determine what state his own particle is in, i.e. the coefficient of quantum_mechanics_7d80e2faa5a53f36220f87b5345e0e66476ed9b3.png and quantum_mechanics_676c5721ccf75f54000bc06aec0681d19d563bab.png
    • This tells Bob which transformation he needs to perform to get his own particle into the state which quantum_mechanics_fdfe843b1ff43668a82fd58d668e566da309c51c.png was originally in!

Superdense coding

Superdense coding is a procedure that utilises an entangled state between two participants in a way that one participant can transmit two bits of information through the act of sending over just one qubit.

Q & A

DONE Is there a correspondance between "unitary transformation / operator" and unitary matrices (when representing the operator as a matrix)?

Check out the definition of a

TODO Not quite sure about how representations used in the Superdense coding step-by-step procedure in the notes

Quantum computing

A quantum register is simply the entire state vector.

Usually we write it in the following form:

quantum_mechanics_7fde166a94f4c918a0ef5beaa70fcea972e921c1.png

where:

  • quantum_mechanics_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png are called control or input register (does not necessarily make sense, it's just a term)
  • quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png are called the target or output register (does not necessarily make sense, it's just a term)
  • quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png and quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png are states labelled by quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png and quantum_mechanics_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png bits

Example: quantum_mechanics_fa81f26e14c14fc309276fcc3d8274e434ac008f.png is a control ket specified by a binary number of length 2, i.e. a linear combination of the states quantum_mechanics_91951249c12c42b8e4903ed4dc7e2d51b29796ef.png, quantum_mechanics_034ab299f67f673fa18219bca066127700de7ad6.png, quantum_mechanics_12bc07508d98703d3b8b9981bf85a3efa8d5796c.png, and quantum_mechanics_7546dec5c965e2a148db1c3a9d75d7b6b5066c3c.png.

Algorithms

Notation

  • quantum_mechanics_692b0a5535b8c711e46eddcef758b3e06871fe1c.png is called the oracle
  • Quantum register :

    quantum_mechanics_7fde166a94f4c918a0ef5beaa70fcea972e921c1.png

Deutch's algorithm

quantum_mechanics_8953e412fad7244b7534972e49a6e559cdb8f656.png

And we're wondering

quantum_mechanics_0dd29438406aaef8108349cf43a376d3e0510835.png

quantum_mechanics_17a4b65964419bf3b5d0e4fb3d0a86e2739754c0.png

quantum_mechanics_3b65f4031397ff5af9fb6f90a1402e5f101671bf.png is working in quantum_mechanics_e7832931ac33d553a5f34521db0fecba78af1b81.png, thus

quantum_mechanics_c08a72d8b57e10cc9fe290d8468081e15e655cbe.png

quantum_mechanics_03000be31dc7c6d72fd1ac194621b77fd9661ae3.png

where

quantum_mechanics_24fca3b64cb72582f05430b266559dee1cf78f46.png

and

quantum_mechanics_42a74adf6de13d7aaaa50bd0b337d67ebbddbfc5.png

Applying the operator above, we get

quantum_mechanics_4b876f0c3af4ab5ed0cdce904c4a75c18e8dacfc.png

Thus, for quantum_mechanics_8fbcc4865dbc27c3bebbe96ade4dcfa86ef14074.png this leads to

quantum_mechanics_b769397cb6478d1638b4a58b0adce7148a3f6775.png

and for quantum_mechanics_21aee773f15ecf2ef1fe50d78952d6e8deb24edd.png, we have

quantum_mechanics_2b385b915bac76b109afdac9dc2a022e1ecc4172.png

i.e. just by looking at the control gate quantum_mechanics_cc6b5c35669c3b0901ef37cf78827ca69227d7b2.png, we get the answer!

Grover's algorithm

  • Search algorithm
  • Reduces time-complexity to quantum_mechanics_bb35fbbc879425f91b287c6d56a0404f0a57f142.png wrt. number of items quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png

Cohen-Tannoudji - Quantum Mechanics

1. Waves and particles

Equations

Planck-Einstein relations

quantum_mechanics_79877b618089ea0c097bc6cb28bf358ac3aae42b.png

where quantum_mechanics_ed16129729c0be1ee8ff5ca34ae68982a2ba81cd.png is the wave vector s.t. quantum_mechanics_16277ae44714b47e622d390db15d95893840f295.png.

de Broglie

quantum_mechanics_19d887db2795ee9613317cef658449f07f799478.png

Eigenstates

With each eigenvalu quantum_mechanics_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png is associated an eigenstate, i.e. an eigenfunction quantum_mechanics_d3da0bc3d2536b674f17158f23c488c52b61fc7d.png s.t. quantum_mechanics_5d2e581e1c4731032be9801f89e01cb03dfad184.png where quantum_mechanics_0eca193b6f6cd17cb61e1065c28719e1d5237614.png is the time the measurement is performed, i.e. the measurement will always yield the same quantum_mechanics_3778a663528a3e4db5ce811ed0c6cd9f5831579f.png.

Scröding equation

quantum_mechanics_2bd4ecd006fd952d376f769635c0481e496d0f2f.png

where quantum_mechanics_9524608936f1ac579d357fdfb91d46c838e0b209.png is the Laplacian operator quantum_mechanics_fb6d8e24c6903139facfe4739f5c3181b6a960c4.png.

  • Free particle

    No force acting on the particle → quantum_mechanics_888ab3bf49ae8f9f65124473f69cae26d8650a92.png

    quantum_mechanics_04ce776eed2dee9a4b5e9532f0a5fab09ec23e0b.png

    which has the solution

    quantum_mechanics_70ea1e11658374de887ac5c07fc297400143d547.png

    on the condition that quantum_mechanics_ed16129729c0be1ee8ff5ca34ae68982a2ba81cd.png and quantum_mechanics_e29af05574032ace665d996d46b3280fc49866ef.png satisty the relation:

    quantum_mechanics_72d8a26c2e22ea527883c11559db226b451e0502.png

    According to de Broglie we have:

    quantum_mechanics_40fb5f0e71a6b578fb06314937f503b9f48484f8.png

    which is the same as in the classical case (no potential → only kinetic → get the above).

    A plane wave of this type represents a particle whose probability of presence is uniform throughout all space :

    quantum_mechanics_68723682f765eeccf68202099ae3708c9784991a.png

    • Form of the superposition

      Principle of superposition tells us that every lin. comb. of plane waves satistfying the relation for quantum_mechanics_e29af05574032ace665d996d46b3280fc49866ef.png specified above, will also be a solution of the free-particle Schrödinger equation.

      This super-position can be written:

      quantum_mechanics_d132655a839df7393a3c4c49971d29af981ddc10.png

      where quantum_mechanics_6d917f40a8601bf75e90114282bd9bf1e70ccc9a.png represents the coefficients, and we integrate over all possible quantum_mechanics_ed16129729c0be1ee8ff5ca34ae68982a2ba81cd.png (quantum_mechanics_cfaaa50aaf6e4126036affb94e1d398e08e0c654.png).

  • Time-independent potential

    In the case of a time-independent potential, quantum_mechanics_3af4fca489d6c670df66ef3ebfe95869fbe162fd.png we have:

    quantum_mechanics_3e99363152b75ea115c5bb77b02572ea7c214af1.png

    where quantum_mechanics_714fa2046a941c515a379c619260a35169c81cb3.png is the Laplacian operator .

    • Separation of variables. Stationary states
      • Separation of variables

        By making the assumption that quantum_mechanics_9ef11a48f80d3f4bccbfcc3a26ad74d1574223f5.png can be separated as follows:

        quantum_mechanics_3a5c8f4908ac8d25f3b38e0c64b15123bad81a0a.png

        We can rearrange the Schrödinger equation as follows:

        quantum_mechanics_1440319e390a43fde78533097f63e032a50e312f.png

        Where LHS only depends on quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png while RHS only depends on quantum_mechanics_d86190240fc08d8dfdbf734ca628129ce9bb1e60.png, which is only true if both sides are equal to a constant.

        Setting LHS to quantum_mechanics_6fc3b15d8d1ef648e35ddac98d170699594cf31f.png (just treating quantum_mechanics_e29af05574032ace665d996d46b3280fc49866ef.png as some arbitrary constant at this point), we obtain a solution for LHS:

        quantum_mechanics_8a0a467256d52c578d7d54ff54b94ebf986d6dda.png

        Figure why we can set it to quantum_mechanics_6fc3b15d8d1ef648e35ddac98d170699594cf31f.png.

        Now do the same for RHS we get the final solution for the full plane wave function:

        quantum_mechanics_8fff35f8788ef79b750f95ba65c0cd0a642ae6d7.png

        where we've let quantum_mechanics_f4b3f0ddaa487c913c23b041a1a1aff86335addc.png (which is fine as long as we incorporate the constant quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png into quantum_mechanics_8b5308072f6400ab2e33da1e7f15f74b310959fe.png ).

        The time and space variables are said to be separated .

      • Stationary States

        The time-independent wave function makes it so that the Schrödinger Equation only involves a single angular frequency quantum_mechanics_e29af05574032ace665d996d46b3280fc49866ef.png. Thus, according to the Planck-Einstein relation, a stationary state is a state with a well-defined energy quantum_mechanics_a65717821dd4b97ed2e8c91ed1f17ccc2a119fa9.png (energy eigenstate).

        We can therefore write the time-independent Schrödinger Equation as:

        quantum_mechanics_7a524df5061c3cdabe2c9b78752788b1947c0206.png

        or:

        quantum_mechanics_a94b08df79d35d36100d8839306b3d1a73470532.png

        where quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png is the differential operator :

        quantum_mechanics_4c032984d91641621938b92d4e05834400a11fb4.png

        Which is a linear operator.

        Thus quantum_mechanics_4568021f4f24ba2c1b51fa137f4aa3bad82d53cc.png tells us that quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png applied to the "eigenfunction" quantum_mechanics_8b5308072f6400ab2e33da1e7f15f74b310959fe.png (analogous to eigenvector) yields the same function, but multiplied by a constant "eigenvalue" quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

        The allowed energies are therefore the eigenvalues of the operator quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png.

        I'm not sure about the remark by about the eigenvalues being the only allowed energies..

      • Superposition of stationary states

        We can then distinguish between the various possible values of the energy quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png (and the corresponding eigenfunctions quantum_mechanics_8b5308072f6400ab2e33da1e7f15f74b310959fe.png ), by labeling them with an index quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png :

        quantum_mechanics_c275ba69a4601783e56bcc1a1af4408dbb0765ff.png

        and the stationary states of the particle have as wave functions:

        quantum_mechanics_0453fc578889677ca6f4021b45d909a0b7b5927c.png

        And since quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png is a linear operator , we have a have series of other solutions of the form:

        quantum_mechanics_c1947ecd42858f28fd1cf0770ba32a3061394dc9.png

  • Requirements

    A system composed of only one particle → total probability of finding the particle anywhere in space, at time quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png, is equal to quantum_mechanics_4468973182b954eeeb1a22bfe0c5b928511fa9f2.png :

    quantum_mechanics_02e56d3acf7b406b13bbd0829456e55c48f91fa4.png

    We therefore require that the wave-function quantum_mechanics_9ef11a48f80d3f4bccbfcc3a26ad74d1574223f5.png must be square-integrable :

    quantum_mechanics_35cb0c6af1dea921b642b714bb1b8cfdc0244a8d.png

    This is NOT true for the simplest solution to the Scröding equation:

    quantum_mechanics_d1fcef059b15a2a195f91fed2077f9083d68b5cf.png

    Where we have

    quantum_mechanics_88b1bbe896598003611e15e2608cfff3c694eb0b.png

    And thus,

    quantum_mechanics_041bb48bfbe223499f8c5f89af7f96ef1c04a6fe.png

    Hence it's NOT square-integrable.

    They then say: "Therefore, rigorously, it cannot represent a physical state of the particle. On the other hand, a superposition of plane waves like [this one] can be square-integrable."

Heisenberg
  • Deduction

    Here we only consider the 1D case, allowing us to write the wave function as as superposition of over all possible quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png :

    quantum_mechanics_8480e1fa14259f9befdc67d2db15d6cf54408963.png

    Suppose that quantum_mechanics_bf8f77673f3bd5832ce6d6775f8ec9041e728bff.png has the following shape:

    modulus_fourier_transform_of_wave_function.png

    Now let quantum_mechanics_922d669a7e878ff8675ec746501b970ea68e9370.png be a function of quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png s.t. :

    quantum_mechanics_d30697ef5b56ca084e2da58ccce31575ef98d993.png

    Further, we assume that quantum_mechanics_922d669a7e878ff8675ec746501b970ea68e9370.png varies sufficently smoothly within the interval quantum_mechanics_c82c7d66eec5b90aea7bb779f7ddd23834dd9a51.png. Then, when quantum_mechanics_0c1eea3d70e50936fcd8c86714d9d6b02b283d5b.png is sufficiently small, we can approx. quantum_mechanics_922d669a7e878ff8675ec746501b970ea68e9370.png using it's tangent / linear approximation about the point quantum_mechanics_e286da926383bafdc9300893ae629db22c9c65b8.png :

    quantum_mechanics_7f21bd17b2598ca771f25094a22bd054385339f0.png

    which enables us to write the wave-function as:

    quantum_mechanics_a88d2168b6c043ced5f0a0fc279ad1510add17b7.png

    where

    quantum_mechanics_fb40e655efe37a5510b1571d7fe241778670cea9.png

    I belive this note turned out to be that it was in fact integrating from quantum_mechanics_edf526239d7ca12a8437916d53d23fda71d4fc3e.png to quantum_mechanics_f6f02b10995dc1353548c5ff27838dfc549df734.png.

    I presume that when they write quantum_mechanics_7a064dd7e1dd3682b4329f6ff2da3db05f27cea3.png they mean "integrate from quantum_mechanics_b9d52f86261c715ef5f0242ec108a57de95cfc95.png to quantum_mechanics_cbfa53921af5496c56986a42ee5f2cb72316b875.png..right? Since the "width" quantum_mechanics_0c1eea3d70e50936fcd8c86714d9d6b02b283d5b.png "contains" most of the integral.

    This gives us a useful way of studying the variations of quantum_mechanics_b4e7ac7a36e5524c7a8eff2ea4c6942e67903428.png in terms of quantum_mechanics_7a84c9a383f9772338016d101ccc096be06af784.png.

    • If quantum_mechanics_39318d71c491324725887c9c4e70abc2373e54b2.png is large, i.e. quantum_mechanics_7a84c9a383f9772338016d101ccc096be06af784.png is far from quantum_mechanics_c462c980a45481116745a8647094b2b1d245df0f.png, quantum_mechanics_0934436a6718b90bd72f8f059bfa7d8fe045294e.png will oscillate a very large number of times within the interval quantum_mechanics_0c1eea3d70e50936fcd8c86714d9d6b02b283d5b.png. Due to the high frequency the osciallations will cancel eachother out, thus quantum_mechanics_6f965a67c8062378fcae7c1dcc42d2998a91e075.png
    • If quantum_mechanics_39318d71c491324725887c9c4e70abc2373e54b2.png is small, i.e. quantum_mechanics_7a84c9a383f9772338016d101ccc096be06af784.png is close to quantum_mechanics_c462c980a45481116745a8647094b2b1d245df0f.png, quantum_mechanics_0934436a6718b90bd72f8f059bfa7d8fe045294e.png will barely oscillate, and so we will end up with quantum_mechanics_b4e7ac7a36e5524c7a8eff2ea4c6942e67903428.png being a maximum.
    • Relating to momentum

      As seen previously, quantum_mechanics_71f0dfe3c17263d701b5b12be2acafcc0eefe04c.png appears as a linear superposition of the momentum eigenfunctions in which the coefficient of quantum_mechanics_11477427fad660c556f56c6bf1ea3ab6a7dca275.png is quantum_mechanics_b41020a22d2840f46451777a79d3d3772902751c.png. We are thus led to interpret quantum_mechanics_0dc60a69b8202c8698a63a6c482f1d4ac0fecece.png (to within a constant factor) as the probability of finding quantum_mechanics_52aac10b42bb121efe4dcb562d951cfe3ee918f2.png if one measures, at quantum_mechanics_c404bbd46048dc77b64f1dd1e1f62f3ad32ef3c5.png, the momentum of a particle whose state is described by quantum_mechanics_0b2dbda6c6b2e03029d35ac061998e011bd6b702.png.

      The possible values of quantum_mechanics_7225b076f6e6326f1636b11d1aad8de58bcc4761.png, like those of quantum_mechanics_7a84c9a383f9772338016d101ccc096be06af784.png, form a continuous set, and quantum_mechanics_0dc60a69b8202c8698a63a6c482f1d4ac0fecece.png is proportional to a probability density : the probability quantum_mechanics_ebb4ac5303358a5e9a9aafd8ed0cb0b5cd9cb50f.png of obtaining a value of between quantum_mechanics_bffdacaf90809aaa7d0b379ceaa07fceafc8f07f.png and quantum_mechanics_a52bb7c7e97df5f5bd1806d7d11a87ac9d670a33.png is, to within a constant factor quantum_mechanics_93b446f562082c90fe92f15abd52b75b81871e92.png. More precisely, we can rewrite the formula as:

      quantum_mechanics_061d21e762081d0097416ab8722c24078faf3e23.png

      we know that quantum_mechanics_43e43b104d63833a4a14b30c784829aa7d106122.png and quantum_mechanics_71f0dfe3c17263d701b5b12be2acafcc0eefe04c.png satisfy the Bessel-Parseval relation:

      quantum_mechanics_73df127d04e066526dd8a9e6e1072a51089b91e9.png

      We then have

      quantum_mechanics_0e94a78adebb99f57b6657d66b1eb2063546d5fd.png

      which is the probability that the measurement of the momentum will yield a result included between quantum_mechanics_7225b076f6e6326f1636b11d1aad8de58bcc4761.png and quantum_mechanics_f2ed492ab041f7c0bb4aad91c7937d9379994911.png.

      Then, writing the relation

      Honestly, I'm not seeing this.

      This is based on the argument that quantum_mechanics_52b1f52d29884a21fb9e9af515c52fee791227a3.png for the quantum_mechanics_0a67499a4bffbb10cb63e93e3e97932598d28117.png to be significant, which I don't get.

  • Q & A
    • DONE We're assuming a certain "family" of distribution for k; does this affect our deduction?

      This quantum_mechanics_b41020a22d2840f46451777a79d3d3772902751c.png represents the coefficients for each of the different wave-functions ("different" meaning parametrized with different quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png, and thus different wavelength / frequency), allowing us to write some arbitrary wave-function as a superposition of an infinite number of plane waves.

      Why should it follow the "distribution" shown above? Why can't quantum_mechanics_b41020a22d2840f46451777a79d3d3772902751c.png have an arbitrary "distribution", e.g. multiple peaks?

      • Answer

        Yeah, this is wierd.

        Let's just wait until we learn about how this relates to the Fourier Transform.

    • DONE What's up with the alphas on the boundary of the integrals?
      • Answer

        It's supposed to be quantum_mechanics_f6f02b10995dc1353548c5ff27838dfc549df734.png not quantum_mechanics_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png.

    • DONE Why is it the complex conjugate of the wave equation when writing it as a function of the momentum?
      • Answer

        It's NOT the complex conjugate, it's just a notation to say it's a different function, i.e. quantum_mechanics_a5e6f5a187348e4cb8374cc8202bdd03cb6ea6a3.png.

Wave equation for free particle

Separable solution in as a function of quantum_mechanics_7225b076f6e6326f1636b11d1aad8de58bcc4761.png and quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png

quantum_mechanics_60ef2d6b20179aeb477fdaa3e0648fd165421e1e.png

or

quantum_mechanics_8506bb94f17942a5b30796ead06fcae8b7c1d5b5.png

Complements

H: Stationary States of particle in one-dimensional square potential "well"
  • Setup

    Here we consider the time-independent Schrödinger Equation with a step-wise constant potential .

    Thus, the time-independent Schrödinger Equation becomes:

    quantum_mechanics_f0ae0230fa8c955e3ba7efea9e96224cbe37eb57.png

  • Stationary wave-function in different potentials

    We can solve the TISE with constant potential using integrating factors:

    quantum_mechanics_accfb831def12425e4a66166e7310028994eb9ef.png

    which we can be solved as a regular quadratic:

    quantum_mechanics_f2dc7d5795d55ff081b4c19cdec220111dd182af.png

    quantum_mechanics_982ff8aad2604e162db6c83375beb0bc5a0de0b4.png

    then whether or not we have a complex solution depends on whether or not quantum_mechanics_bad4a8dcfdcf2e44609284243ea5cb60ef1a18bf.png is non-negative or not.

    • E > V - energy of particle is GREATER than potential barrier

      We write (quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png being defined by the the following equation)

      quantum_mechanics_3555a50c8a4b57e188ebd0a62015a7c49fe55129.png

      Then the stationary solution to the TISE with constant potential can be written

      quantum_mechanics_b24d57d25ed7f88e225abeb6dab2695576f750bd.png

      where quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_6ecabf46b855c7cbc7b0c57a13b6f1559d122040.png are complex constants.

    • E < V - energy of particle is LOWER than potential barrier

      Let quantum_mechanics_af84e5c235f63b7bf7fddcb13027387ba4d61db0.png be defined by the following equation

      quantum_mechanics_8f759edd36a725aa573ef9f7405d4098b785a3e6.png

      with solution to TISE with constant potential can be written:

      quantum_mechanics_4865db2e2c0d6d590383b861a1acf16cfb50331e.png

      where quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png and quantum_mechanics_62eee427e4a640894f553b65324621ac0757255b.png are complex constants

      Different order of quantum_mechanics_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png on LHS.

      Also notice the fact that using the "integrating factor" to solve the ODE, our exponentials aren't positive in this case.

    • E = V - energy of particle is equivalent to the potential barrier

      In this case we simply end up with the differential equation

      quantum_mechanics_248bb091eb4f1f1e410a9f39c2cab4b304d99959.png

      Which we can simply integrate twice and obtain a linear function expression for quantum_mechanics_a284dbf7473556cebbba8621e88420c98f5232ad.png.

  • Stationary wave-function at potential energy discontinuity
    • Computation outline
      1. Solve the TISE with constant potential using integrating factors, leaving you with a quadratic.
      2. Let quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png be such that quantum_mechanics_b01f46fbd6d20b1684928bd2ca0718e97c8205ae.png, for convenience
      3. Obtain a general solution for both wave-functions depending on the quantum_mechanics_d1571a0bb997904b0c74560c30628f1a1ba84152.png in the different regions.
      4. Assume the wave-function to be continues across the boundaries (discontinuities of the potential), i.e. we require

      quantum_mechanics_b0e57a06c4aa3c8bdcee688142a16adbb509182a.png

      1. Solve for the coefficients of the general solutions obtained.

      qm_potential_step.png

      In some cases, solving for all coefficients of the different wave-functions is not possible due to not having enough constraints.

      In this case it might make sense to set some constant to zero, and instead solve for the ratios , not the coefficients themselves. Solving for ratios reduces the order of the system of equations by 1, and setting one of them to zero reduces it further by 1. In the case were we have a simple potential barrier in 1D, using the above methods to reduce the order then allows us to compute solve equations for the ratios, providing us with a reflection coefficient (which also implies a transmission coefficient ) and thus the probability of reflecting of the barrier.

    • Potential well

      qm_potential_well.png

      We would follow the same as described above, but now we would have three wave functions, with boundary conditions:

      quantum_mechanics_70233bf4924da0c5a734db1c5c68a049de44cc75.png

      for the left side of the potential-well at quantum_mechanics_ca2a495fb8e64acb1cc942d55440d084d83b588a.png, and for the right side at quantum_mechanics_e7c0a38090ff26392b3492806caff9191c1b4a1d.png:

      quantum_mechanics_2646eb7890ab1bdb6101c52dd8bfef2fd8966ce2.png

      i.e. we simply use the exact same method, but now we have an additional boundary to consider.

      The solution ends up looking as:

      quantum_mechanics_d59315a3539fab1112c48ccaa6735875b70eb0cf.png

      where we note that if we have a particle coming from the left, we set the coefficient of the left-traveling before the barrier to 0:

      quantum_mechanics_e7c0b4bc52e1c1649e4f1c561e51724e81a20244.png

      i.e. we can

      We can "think" of each of these terms in the wave-function to be a super-position of eigenstates. Viewing it in that way, it makes a bit more sense I suppose.

    • Infinite potential well

      Same procedure as regular potential well, but now the wave-functions quantum_mechanics_d7c9083cc3ee95ae7430bfc85b110f4e0ac68ade.png and quantum_mechanics_6de68ff5734d3281fe658bbd26e5f45f7bc65003.png (i.e. wave-functions outside of the potential well) are equal to zero for all quantum_mechanics_7a84c9a383f9772338016d101ccc096be06af784.png ! That is,

      quantum_mechanics_3c86ca983c2a4f6fb82d0a6b5a42ad0bf034300d.png

      This leads us to the quantization of energies , where we have multiple solutions satisfying the boundary conditions specified above. If the potential well is of "width" quantum_mechanics_a48375c7d11c6e5c0584515ab8c07eecb5e78ca8.png, we end up with the solution

      quantum_mechanics_96778eb4095f2cd34beacf6cd9f770d162af82fc.png

      Which falls out of the fact that the boundary conditions above corresponds to the inverse wave-length being a integer multiple of the width of the potential well.

      Each of these energy levels have their own eigen-function quantum_mechanics_6026c03e845e7bf21559f8d9abba4def5de3e62f.png.

2. The mathematical tools of quantum mechanics

Notation

  • quantum_mechanics_d5e808bc081ddc7b1096b6f9c51abd69a0dc64e1.png is the set of all square-integrable functions which are everywhere defined, continuous and infinitively differentiable. Together with the inner product quantum_mechanics_3f1e1f0e7b173b4cbf11c1199d15dc9fe4a1bc6b.png this defines a Hilbert space
  • quantum_mechanics_75debb2b7799d9964f3d3023c9acfe019ed4436b.png denotes a wave function
  • quantum_mechanics_740319c31f392c2a05b5616b2b90523b2efa880c.png is a ket or ket vector, which is an element or vector of quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png space, e.g. quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png
  • quantum_mechanics_d0ade1f0229b9925f1e0fd54889393ad82846a91.png is the state space of a particle, and is a subspace of a Hilbert space. It's defined such that we associate each square-integrable function quantum_mechanics_6693245bc2c82fb03d07415a90697c9c38f2e410.png with a ket vector, i.e. quantum_mechanics_4b29f88f00ea5236ade47fa9581de3dec8148ad7.png
  • quantum_mechanics_209d2b7186dfee7e1db01e0a9439b4a111888963.png is the state space of a (spinless) particle in only one dimension
  • quantum_mechanics_02a77b19a4c56c321496005d799c0fbb159221f6.png is associated with a complex number, which is the scalar product, satisfying the properties of an inner product.
  • quantum_mechanics_41520134b153f2d86b85d3201bfc9a9f0156f9a8.png denotes the set of linear functionals defined on the kets quantum_mechanics_24c129fe5e2d71cb1d60c725fe8fb929a66ae0ef.png, which constitutes a vector space, called the dual space of quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png.
  • quantum_mechanics_5c6b8a91d6630c0a8205dee182a464a2cd83775c.png is a bra or bra vector of the space quantum_mechanics_41520134b153f2d86b85d3201bfc9a9f0156f9a8.png, which represents a linear functional quantum_mechanics_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png
  • quantum_mechanics_3fb0a2ec8eb0db2fa214edde2ab20ce098fb6c52.png, i.e. the linear function quantum_mechanics_44901563cb581b56f700fdb8b1753b1b6e23fcf2.png acting on the ket quantum_mechanics_24c129fe5e2d71cb1d60c725fe8fb929a66ae0ef.png
  • quantum_mechanics_cc1538f6215835fdf8fdbffc6dde7d62fd483b3c.png denotes a discrete basis
  • quantum_mechanics_f792ac100a12dcf9d2396c65936206abd0622223.png denotes a continuous basis
  • quantum_mechanics_fdd148668186f6263d474aee31e3fc100a1e933c.png (quantum_mechanics_39a209439d76789bdc64136603574115dae27559.png) denote a projection operator for a discrete (continuous) basis
  • quantum_mechanics_48a6ba9a3894dc40eccee906fa188a614719497f.png denotes the degeneracy of the eigenvalue quantum_mechanics_9c17aa32c24ff4dbc420cd454dec844aef475372.png
  • quantum_mechanics_89bfa0eb4c1f4910c438bb8f40856993e26a7432.png denotes the i-th (degenerate) eigenvector corresponding to the eigenvalue quantum_mechanics_9c17aa32c24ff4dbc420cd454dec844aef475372.png, if this eigenvalue is non-degenerate then the quantum_mechanics_f78e53d727e82c8e43205bb55f2368f6dc0affa3.png in the super-script can be dropped
  • quantum_mechanics_a5c263e9545d465bd6638ba3846a145713bb4232.png denotes the eigensubspace of the eigenvalue quantum_mechanics_9c17aa32c24ff4dbc420cd454dec844aef475372.png of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png

Dirac notation

Overview

The quantum state of any physical system is characterized by a state vector, belonging to a space quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png which is the state space of the system.

"Ket" vectors and "bra" vectors
  • Dual space

    A linear function quantum_mechanics_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png is a linear operation which associates an complex number with every ket quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png :

    quantum_mechanics_ed4ab28c712079a2c1871ea965f382733f605f57.png

    Linear functional and linear operator must NOT be confused. In both cases we're dealing with linear operations, but the former associates each ket with a complex number, while the latter associates another ket.

  • Generalized kets

    "Generalized kets" are functions which are not necessarily square-integrable, but whose scalar product with every function of quantum_mechanics_92757546978c1aad3468f8a69e8d334af2bfadac.png exists.

    These cannot, strictly speaking, represent physical states. They are merely intermediaries.

    The counter-examples to the "normal" definition of a ket are in the form of limiting cases, where the result of applying the bra to the ket is well-defined, but it's not square-integrable (it diverges when taking the limit) and thus not in quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png.

    See the book at p.115 for more information about this.

    But in short, in general, the dual space quantum_mechanics_41520134b153f2d86b85d3201bfc9a9f0156f9a8.png and the state space quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png are NOT isomorphic, except if quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png is finite-dimensional.1 I.e. the following is true:

    quantum_mechanics_c30a1b53ced36e085f287d3415e55b8e69b1f231.png

    But the otherway around is NOT true.

Linear operators
  • Overview

    Consider the operations defined by:

    quantum_mechanics_de23af3aeb0b964914a82cf1d217a146db676b08.png

    Choose an arbitrary ket quantum_mechanics_44dcf12668047a91d8f4c8123be8b3d65e7098a8.png and consider:

    quantum_mechanics_42a248bc3065013b95d89ea1ceedd87b8a818425.png

    We already know that quantum_mechanics_4a5950c5b181272bc155018875bc1b8061465f36.png ; consequently, the equation above is a ket, obtained by multiplying quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png by the scalar quantum_mechanics_9371b2b331220eab7fab75b26ffce27ce9d74a1e.png. Therefore the quantum_mechanics_84cacfa982a629ea0973a6c621997219c84e0d6f.png applied to some arbitrary ket quantum_mechanics_44dcf12668047a91d8f4c8123be8b3d65e7098a8.png gives another ket, i.e. it's an operator.

    If quantum_mechanics_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png :

    quantum_mechanics_3b682ab56bf221d719b391350bbe33cbb01dfb3b.png

  • Projections

    Let quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png be a ket which is normalized to one:

    quantum_mechanics_1adfc0815f886ffc5a38337fc00103787447a2b1.png

    Consider the operator quantum_mechanics_59b9f8e4113390b367b0e808a271afda4a98bdff.png defined by:

    quantum_mechanics_53cd2fd869aafdf903701ddc075e5b1d66994b80.png

    and apply it to an arbitrary ket quantum_mechanics_943c3700edbf5c9e06ceb1641231eb84e5c7e5c2.png :

    quantum_mechanics_73a74dd51e5150fc2dede01ff81c666f15443ee5.png

    Which is simply the projection of quantum_mechanics_943c3700edbf5c9e06ceb1641231eb84e5c7e5c2.png onto quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png : first take the inner product, not needing to normalize wrt. quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png due quantum_mechanics_d05ba51bc42dd2e1e74f9fdd25776e7bcd41693d.png, and then multiply by quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png to get the component.

    We can also project onto a basis by taking the um over quantum_mechanics_3562cbb63088796937fabb85ee7bfb612bac9ee1.png, and using this as an operator (applying each of them to the target ket as above). Thus we get the linear superposition .

Hermitian conjugation
  • Linear operator on a bra

    quantum_mechanics_ffa489754eb8c458b812ff764bc579b62748b6c6.png

  • Adjoint operator

    "Correspondence" between kets and bras.

    With every linear operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png we associate another linear operator quantum_mechanics_644edafe00ea76c2f90e45cd1b9df7b247b2e40e.png called the adjoint operator (or Hermitian conjugate ) of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

    quantum_mechanics_b5c8c523675e57ee7bfb215ee1f4dec01424de52.png

Representations in state space

  • Choosing a representation amounts to choosing an orthonormal basis, either discrete or continuous, in the state space quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png

For a discrete basis quantum_mechanics_cc1538f6215835fdf8fdbffc6dde7d62fd483b3c.png

quantum_mechanics_6c7c3964a02c2034a88ac6897cfa589de9fea5b5.png

For a continuous basis quantum_mechanics_f792ac100a12dcf9d2396c65936206abd0622223.png

quantum_mechanics_ba49c41ca25e281af69390423118521232df8777.png

Representation of operators

Given a linear operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, we can, in a quantum_mechanics_cc1538f6215835fdf8fdbffc6dde7d62fd483b3c.png or quantum_mechanics_f792ac100a12dcf9d2396c65936206abd0622223.png basis, associate with it a series of numbers defined by

quantum_mechanics_952ea9daaf6cedcb248a3333041639ae8f93bbac.png

or for a continuous basis

quantum_mechanics_67f5830832011acab2d67451715a64c1df92dbe9.png

We can then use the closure relation to compute the matrix which represents the operator quantum_mechanics_703fbe2e1ea8d69d29b497a14d67db427a8197ca.png in the quantum_mechanics_cc1538f6215835fdf8fdbffc6dde7d62fd483b3c.png basis:

quantum_mechanics_ea444c8c83ccf387962d14caaff5139db10f11d5.png

And equivalently for the continuous basis quantum_mechanics_f792ac100a12dcf9d2396c65936206abd0622223.png

Matrix representation of a ket

Problem: we know the components of quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png and the matrix elements of the operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png in some representation. How can we compute the components of quantum_mechanics_9adac8924a724186d0b46158b9a90bebc52c8118.png?

In the quantum_mechanics_cc1538f6215835fdf8fdbffc6dde7d62fd483b3c.png basis, the "coordinate" quantum_mechanics_8724a4216b8924dc19b9e181303237244117ed48.png of quantum_mechanics_4c8f7faf78c2a4d22961ef60cc4983b451ba189b.png are given by:

quantum_mechanics_ad4d943a07ca598dd58d42bf5cef164966a26f37.png

Inserting the closure relation between quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png, we obtain:

quantum_mechanics_43bf6ff736bf82c6bc1a8e002fe75a99b1575d39.png

Or for a continuous basis quantum_mechanics_f792ac100a12dcf9d2396c65936206abd0622223.png we have:

quantum_mechanics_1ea19ee60bf5e125c0825d926f30516650417858.png

Eigenvalue equations for observables

quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png is said to be an eigenvector (or eigenket ) of the linear operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png if :

quantum_mechanics_43e59617dc83e43c27b9d16ce867d520f6524e26.png

where quantum_mechanics_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png. We call the equation above the eigenvalue equation of the linear operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

We say the eigenvalue quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png is nondegenerate if and only if the corresponding eigenvector is unique within a constant factor, i.e. when all associated eigenkets are collinear.

If there exists as least two linearly independent kets which are eigenvectors of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png with the same eigenvalue, this eigenvalue is said to be degenerate.

To be completely rigorous, one should solve the eigenvalue equation in the space quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png, i.e. we ought to only consider those eigenvectors quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png which have a finite norm.

Unfortunately, we will be obliged to use operatores for which eigenkets do not satisfy this condition. Therefore, we shall grant that vectors which are solutions of the eigenvalue equation can be

Two eigenvectors of a Herimition operator corresponding to two different eigenvalues are orthogonal.

Consider two eigenvectors quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png and quantum_mechanics_943c3700edbf5c9e06ceb1641231eb84e5c7e5c2.png of the Hermitian operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

quantum_mechanics_55bfc18f88fdd29faf75be8dc4e880c1b67f15bf.png

Since quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is Herimition, we can write the above using the corresponding bras:

quantum_mechanics_2a79d79784c2645b533a1d160027b6460133ce91.png

Multiplying the above on the left with quantum_mechanics_fed715cf5dfda1186c93ca7bf08e2a8c4be01a55.png on the right:

quantum_mechanics_2b55a57eb1a801cb9693ea4337239c7d78fc4b2a.png

Which implies

quantum_mechanics_f5c3bd4750cf7c7f3a69c2475fde9ad49043dd07.png

Hence, if quantum_mechanics_1dc1e800b69599a7cb7098d80fbe028adb1b67cf.png we have orthogonality, i.e. quantum_mechanics_3026b7f5f18461365394f852f10d4f4303c47318.png.

The Hermitian operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is an observable if the orthonormal system of eigenvectors of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, described by

quantum_mechanics_8c4771d550d8d5b344507bd163f03d2d7f2aed63.png

form a basis in the state space. That is,

quantum_mechanics_dd8513808b9a4ee86a1a00b78109809fc45cc6f8.png

Sets of commuting observables

Consider an observable quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and a basis of quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png composed of eigenvectors quantum_mechanics_89bfa0eb4c1f4910c438bb8f40856993e26a7432.png of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.

If none of the eigenvalues of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is degenerate, then the various basis vectors of quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png can be labelled by the eigenvalue quantum_mechanics_9c17aa32c24ff4dbc420cd454dec844aef475372.png, and all the eigensubspaces quantum_mechanics_a5c263e9545d465bd6638ba3846a145713bb4232.png are then one-dimensional. That is, there exists a unique basis of quantum_mechanics_6419e4346778fa5d3dcc17fa8c3cb50c449a3f4d.png formed by the eigenvectors of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png. We then say that the observable quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png constitutes, by itself, a C.S.C.O.

If, on the other hand, one or several eigenvalues of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png are degenerate, then the basis of eigenvectors of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is not unqiue. We then choose another observable quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png which commutes with quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, and construct an orthonormal basis of eigenvectors common to quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png. By definition, quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png form a C.S.C.O. if this basis is unique (to within a phase factor for each of the basis vectors), that is, if, to each of the possible pairs of eigenvalues quantum_mechanics_91da744f493baa4a3e8bf4c918afbab166da3e89.png, there corresponds only one basis vector.

If we still don't have a C.S.C.O., we can introduce another Hermitian operator quantum_mechanics_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png which commutes with quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png, and then try to construct unique triples, and so on. This can be performed an arbitrary number of times in an attempt to obtain a C.S.C.O.

A set of observables quantum_mechanics_3a13f03fecedaf69e8ac1f25f8061ab4699207a0.png is called a complete set of commuting observables if

  1. All the observables quantum_mechanics_3a13f03fecedaf69e8ac1f25f8061ab4699207a0.png commute by pairs
  2. Specifying the eigenvalues of all the operators quantum_mechanics_3a13f03fecedaf69e8ac1f25f8061ab4699207a0.png determines aa unique (to within a multiplicative factor) common eigenvector

If quantum_mechanics_b0a1135ca76f1edb00e87ccc300c90c3f189b2e3.png is a C.S.C.O., the specification of the eigenvalues quantum_mechanics_a57269d8e3f46e48b02127600a42a7d3ec5713d5.png determines a ket of the corresponding basis (to within a constant factor), which we sometimes denote by quantum_mechanics_1cc0ea548713d3b610eb0943e7bbc2ad5cc34102.png

Principles of Quantum Mechanics - Dirac

Notation

  • quantum_mechanics_9bbf5cf19b67af35dfc6cf52860fd3a99678a3f0.png denotes an eigenket belonging to the eigenvalue quantum_mechanics_4f8e6968304c8ecb49ad4814c6cc9b7dd90085c6.png of the dynamical variable or a real linear operator quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png
  • If quantum_mechanics_4f8e6968304c8ecb49ad4814c6cc9b7dd90085c6.png is an eigenvalue with multiple corresponding eigenkets, then quantum_mechanics_51951d9f7901c492eb75b825c8624b1a34f4a3d6.png denotes the ith corresponding eigenket

Definitions

Words

conjugate complex
refers to the complex conjugate of a number
conjugate imaginary
the bra quantum_mechanics_ac1a7b4181f9c0ff6925cf6daba4184f8f962503.png corresponding to the ket quantum_mechanics_943c3700edbf5c9e06ceb1641231eb84e5c7e5c2.png
commutative operators
quantum_mechanics_bd8f4504b409580f6e31d1f9b80fb5f3021e96a1.png

Linear Operators

Notation

  • quantum_mechanics_d292605accd02bab9bbeca076c99781cfbff4cf0.png defines applying an operator to a bra

Theorems

Orthogonality theorem

Two eigenvectors of a real dynamical variable belonging to different eigenvalues are orthogonal .

quantum_mechanics_ad69e00b1c650a3dea75b0690790d2dd4b7e1e5f.png

Existence of eigenvectors / eigenvalues
  • Simple case

    Assume that the real linear operator quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png satisfies the algebraic equation:

    quantum_mechanics_d361ed4f9cb98551d11481263fa536ed883e018c.png

    which means that the linear operator quantum_mechanics_34e892de3f8592bbf983050005bbb84b0c0726bc.png produces the result zero when applied to any ket vector or to any bra vector.

    Further, let the equation above be the simplest algebraic equation that quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png satisfies. Then

    1. The number of eigenvalues of quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png is quantum_mechanics_1dbee3eb7b7bd0453a691130ee5d0d2bc8d46f23.png
    2. There are so many eigenkets of quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png that any ket can be expressed as a sum of such eigenkets.
    • Proof of 2)

      Let quantum_mechanics_332b34b209746ba21b056af6ea46187724fa36c6.png be s.t.

      quantum_mechanics_64745b205b1e9362649e78362fbace7bc194d8ef.png

      then

      quantum_mechanics_3ed271a660c7b07ac20ec655e52b0a5a1b18c70f.png

      Consider the case where we substitute quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png with quantum_mechanics_42bc7a478fa4de5f1c8dd6c4ba5fe9ce03171ab2.png in the above expression. In this case, each term of the sum above will be of the form:

      quantum_mechanics_e2ce33b8597f6d3c33f80f45a0171d3b4db450d5.png

      where

      I'm not seeing why every term except quantum_mechanics_da01703b65ca3c301fb90f8291cc7b4ac6bfc83d.png should vanish. I get the fact that quantum_mechanics_da01703b65ca3c301fb90f8291cc7b4ac6bfc83d.png does NOT vanish, but why do all the other terms vanish?

Observables

Assumptions

If the dynamical system is in an eigenstate of a rea dynamical variable quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png, belonging to the eigenvalue quantum_mechanics_0706aa491aa964de9355e01bca7a92bcd0fbcc0a.png, then a measurement of quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png will certainly give as result the number quantum_mechanics_4f8e6968304c8ecb49ad4814c6cc9b7dd90085c6.png. Conversely, if the system is in a state such that a measurement of a real dynamical variable quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png is certain to give on particular result (instead of giving one or other of several possible results according to a probability law, as in the general case) then the state is an eigenstate of quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png and the result of the measurement isthe eigenvalue of quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png to which this eigenstate belongs.

I.e. we assume eigenstates to be the the case when a measurement of a real dynamical variable quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png is certain to give one particular result.

Equations

Q & A

p.34 Eqn. 41

See here. Why do the terms vanish?

Quantum Theory for Mathematicians

Notation

2. A First Approach to Classical Mechanics

2.5 Poisson Brackets and Hamiltonian Mechanics

Let quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png and quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png be two smooth functions on quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png, where an element of quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png is thought of as a pair quantum_mechanics_d2035ad918145ce752485962447d040403b0d4d6.png, with

  • quantum_mechanics_ece32690554d9921c4b444c836723b6c66a535fb.png representing position of a particle quantum_mechanics_1d3904b476eaf35b0be83fa6f3c286f2e6d13d4e.png representing the momentum of a particle

Then the Poisson bracket of quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png and quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png, denoted quantum_mechanics_b02fb00f6ed6cd2e97c0614984a8dba6e52c516c.png is the function on quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png given by

quantum_mechanics_a62ca084a0960500c4b7fd6196fedc3916f5f2de.png

For all smooth functions quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png, quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png and quantum_mechanics_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png on quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png we have the following:

  1. quantum_mechanics_c6fb2e8ded104a138ff21d2b2f0401c628904687.png for all quantum_mechanics_8e1920707fd0ecf563db5e5cf42dac8c316a64e9.png
  2. quantum_mechanics_d7394bc30a89962842d0fa47a046fc81a8dab97d.png
  3. quantum_mechanics_d29beed244d8f1224d2accc5ea569db24d66b06c.png
  4. Jacobi identity:

    quantum_mechanics_b092c77fb061c931c2cfcc965b2e38ee32a6915d.png

The position and momentum functions satisfy the following Poisson bracket relations:

quantum_mechanics_f24843e37ceff854750cb3ab06329f1da0d34b3b.png

If a particle in quantum_mechanics_004097ff73cb85a0f596c8a3b60218ece0e16be1.png has the usual sort of energy function (kinetic energy plus potential energy), we have

quantum_mechanics_3516119a9a885a2d79f790a438734d9d69ed9725.png

With the Hamiltonian, and as usual, having quantum_mechanics_3073e89d9ff81d21211ce0184b9502f6b28a10f4.png, we can write Netwon's laws as:

quantum_mechanics_6645940a53e18020f9ba4cfa924f52797be4a6f3.png

These equations we refer to has Hamilton's equations.

If quantum_mechanics_968a9d01800b28238a13cd3c80be094fa6b0d6d6.png is a solution of the Hamilton's equation, then for any function quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png on quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png, we have

quantum_mechanics_c149f41439cdc0421538a6a2e124f31a42ca2bba.png

Call a smooth function quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png on quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png a conserved quantity if quantum_mechanics_d2446ba272aa525fc36979a5bf9f87c8d5738a5f.png is independent of quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png for each solution quantum_mechanics_968a9d01800b28238a13cd3c80be094fa6b0d6d6.png of Hamilton's equations.

Then quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is a conserved quantity if and only if

quantum_mechanics_8917a7d320cd94640fa8de6a5e051ca3abade324.png

In particular, the Hamiltonian quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png is a conserved quantity.

Solving Hamilton's equatons on quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png gives rise to a flow on quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png, that is, a family quantum_mechanics_7f3c3dc0278d16aaa933a146524f7679ec6b2f86.png of diffeomorphisms of quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png, where quantum_mechanics_c07df4782e4b6e558eb0ffb1667fad9fc39448f9.png is equal to the solution at time quantum_mechanics_a270d2f0aac3268943aea13ffd8c0dd862578f2e.png of Hamilton's equations with initial conditions quantum_mechanics_d2035ad918145ce752485962447d040403b0d4d6.png.

Since it is possible (depending on the choice of potential function quantum_mechanics_d198c0d06b0da5525f9a966222601066509ac01d.png ) that a particle can escape to infinity in finite time, the maps quantum_mechanics_7f3c3dc0278d16aaa933a146524f7679ec6b2f86.png are not necessarily defined on all of quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png, but only on some subset therof.

If quantum_mechanics_7f3c3dc0278d16aaa933a146524f7679ec6b2f86.png is defined on all of quantum_mechanics_759d91cbd6055f733f91560e09ed06f5e5296dea.png we say it's complete.

The flow associated with Hamilton's equations, for an arbitrary Hamitonian function quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png, preserves the (2n)-dimensional volume measure

quantum_mechanics_69457bf54b15984899f54c9df6b7e17fc21531a7.png

What this means, more precisely, is that if a measurable set quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png is contained in the domain of quantum_mechanics_7f3c3dc0278d16aaa933a146524f7679ec6b2f86.png for some quantum_mechanics_570462f792cbe96a48effcfae87b005cefc599cb.png, then the volume of quantum_mechanics_d0af9035ef08ea71f8a34c8b42e94427d1d6a08f.png is equal to the volume of quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

3. A First Approach to Quantum Mechanics

3.2 A Few Words About Operators and Their Adjoints

  • Linear operator quantum_mechanics_2e030f1ad8ed697cdd8a012e20c9ef316bdaef36.png is bounded if

    quantum_mechanics_28509e0aca3342d7046c1bcad01c00b7c66014f4.png

  • For any bounded operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, there is a unique bounded operator quantum_mechanics_5a4a9389543149f2412dcad1b2b34f36d543803f.png, called the adjoint of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, such that

    quantum_mechanics_0fab1a7a45d84897fda06cb763b2953518ecd6d8.png

  • Existence of quantum_mechanics_5a4a9389543149f2412dcad1b2b34f36d543803f.png follows from the Riesz Theorem

For any bounded linear operator quantum_mechanics_2e030f1ad8ed697cdd8a012e20c9ef316bdaef36.png there is a unique bounded operator quantum_mechanics_5a4a9389543149f2412dcad1b2b34f36d543803f.png, called the adjoint of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, such that

quantum_mechanics_0fab1a7a45d84897fda06cb763b2953518ecd6d8.png

The existence of quantum_mechanics_5a4a9389543149f2412dcad1b2b34f36d543803f.png follows from Riesz Theorem.

We say quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is self-adjoint if quantum_mechanics_ce1000419cdb536263abaff08867483d5793df7e.png.

Further, if quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is a linear operator defined on all of quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png and having the property that

quantum_mechanics_7d4787cd5b4c417e15123ae088960820308e7b2f.png

then quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is automatically bounded.

This means that an unbounded operator cannot be defined on the entire quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png!

An unbounded operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is a linear map from a dense subspace quantum_mechanics_1bff19eaa88ccc9c63947a3aeddc17927dcddd47.png into quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png.

Then quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is "not necessarily bounded", since nothing in the definition prevents us from having quantum_mechanics_b2b93330e5cdf03d7d710603ff3a1de53b9267c2.png and having quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png be bounded.

For an unbounded operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png, the adjoint quantum_mechanics_5a4a9389543149f2412dcad1b2b34f36d543803f.png of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is defined as follows:

A vector quantum_mechanics_d8930f79d82c6c1e3efe7c0e23725f56f93d472c.png belongs to the domain quantum_mechanics_03a1c68878f419c70c8d6cda2f00f680a3827f58.png of quantum_mechanics_5a4a9389543149f2412dcad1b2b34f36d543803f.png if the linear functional

quantum_mechanics_3b748dfa61ee6affdc5fc865e345960b02956ad9.png

defined on quantum_mechanics_6a382c465e465094a1f65a0ba1a1f4cba0d5f92f.png, is bounded.

For quantum_mechanics_15337f7f4c4d8bd301e26f32cdec6897990a3b7f.png, let quantum_mechanics_e3bed1ab65e0acb5d4b923d293e2e8b15ac991f5.png be the unique vector quantum_mechanics_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png such that

quantum_mechanics_6c772da4d0a9559823beee2492773b6a5f9e64e7.png

Since quantum_mechanics_8dcb62109c3141b4b16f1d174fe1a61b7b519b47.png is bounded and quantum_mechanics_6a382c465e465094a1f65a0ba1a1f4cba0d5f92f.png is, by definition of a unbounded operator, dense, the BLT theorem tells us that quantum_mechanics_8dcb62109c3141b4b16f1d174fe1a61b7b519b47.png has a unique bounded extension to all of quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png.

Further, Riesz theorem then guarantees the existence and uniqueness of quantum_mechanics_4ef26ff1bd3f789ae16862cbfc47f13a48b0f7eb.png, the corresponding vector such that

quantum_mechanics_7dc9ef98e5c18b636c9fadebd69e677a129d19d5.png

Thus, the adjoint of a unbounded operator is a linear operator on quantum_mechanics_6a382c465e465094a1f65a0ba1a1f4cba0d5f92f.png.

An unbounded operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is symmetric if

quantum_mechanics_e1b9480decbb215cefc93fa780ac6b2421d6afb1.png

The operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is self-adjoint if

quantum_mechanics_c2134d197b265f9825a896fe1fcb290b0abca99a.png

Finally, quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is essentially self-adjoint if the closure in quantum_mechanics_1a750127ed22b0a42fbdad89d932fee51ac2b4ff.png of the graph of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is the graph of a self-adjoint operator.

3.6 Axioms of Quantum Mechanics: Operators and Measurements

The state of the system is represent by a unit vector quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png in an appropriate Hilbert space quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png.

If quantum_mechanics_8da0fc3bf2a962165ca0a2c0af1935e76f2f1715.png and quantum_mechanics_71d1962b87cebf15fe77446a90bd14720ebe9def.png are two unit vectors in quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png with quantum_mechanics_24f5c6e358a3a922ad2d2b8bdd9c4c93ce72b6d2.png for some constant quantum_mechanics_52804b4df8339f1ece2c3ea87a14a556db8de929.png, then quantum_mechanics_8da0fc3bf2a962165ca0a2c0af1935e76f2f1715.png and quantum_mechanics_71d1962b87cebf15fe77446a90bd14720ebe9def.png represent the same physical state.

There is a more general notion of a "mixed state", which we will consider later.

To each real-valued function quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png on the classical phase space there is associated a self-adjoint operator quantum_mechanics_b186c269efdc7f5e61e6e9f4d2f3ab7a7cd0cda2.png on the quantum Hilbert space.

"Quantum Hilbert space" simply means "the Hilbert space associated with a given quantum system".

If a quantum system is in a state described by a unit vector quantum_mechanics_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png, the probability distribution for the measurement of some observable quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png satisfies

quantum_mechanics_5d926dbbe752fc4ae215795e9e4eaff1ede1306c.png

In particular, the expectation value for a measurement of quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is given by

quantum_mechanics_c9b108a893dd84026ca7febf6644a485a195ff21.png

Suppose a quantum system is initially in a state quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png and that a measurement of an observable quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is performed.

If the result of the measurement is the number quantum_mechanics_da9eb912c1f040970a272b334e2197c291627041.png, then immediately after the measurement, the system will be in a state quantum_mechanics_9c958f5e8ff0be415451976740075cc10facffab.png that satisfies

quantum_mechanics_f6cb8f580c35e373c5c2b89ad8f93147b0ecec5c.png

The passage from quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png to quantum_mechanics_9c958f5e8ff0be415451976740075cc10facffab.png is called the collapse of the wave function. Here quantum_mechanics_b186c269efdc7f5e61e6e9f4d2f3ab7a7cd0cda2.png is the self-adjoint operator associated with quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png by Axiom 2.

The time-evolution of the wave function quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png in a quantum system is given by the Schödinger equtaion,

quantum_mechanics_b50d1f8e349ab17fc705e5e4b76a014d66ab69f0.png

Here quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png is the operator corresponding to the classical Hamiltonian quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png by means of Axiom 2.

3.8 The Heisenberg Picture

In the Heisenberg picture, each self-adjoint operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png evolves in time according to the operator-valued differential equation

quantum_mechanics_b739da2f9ce3c5f79b720aba91d0c4e91308c325.png

where quantum_mechanics_c59f9d9aaf7268a3b1fbf8ccbe7f86998287d61c.png is the Hamiltonian operator of the system, where quantum_mechanics_ca1fd9734446b1235e0941b5bcaff30302543c4e.png is the commutation, given by

quantum_mechanics_c0fae0caa4fffc40a40b051acf7e0835ddf269e0.png

4. The Free Schrödinger Equation

Notation

  • quantum_mechanics_8b49f91be66d2b8ed9cefa172ce50561a0e2207f.png

4.2 Solution as a convolution

"Free" means that there is no force acting on the particle, so that we may take the potential quantum_mechanics_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png to be identically zero.

Thus, the free Scrhödinger equation is

quantum_mechanics_ea57476486db600fe059f1b359c7e3f9b1088d76.png

subject to an initial condition of the form

quantum_mechanics_72103605e63c70fc92fe4bc5e814a7c8bd74c0c6.png

Suppose that quantum_mechanics_2449140a739de934f4521413e504b4232c9c351c.png is a "nice" function, for example, a Schwartz function.

Let quantum_mechanics_66bc698c1fbcdd033efd3ce419c449e0628f999e.png denote the Fourier transform of quantum_mechanics_2449140a739de934f4521413e504b4232c9c351c.png and define quantum_mechanics_0b2dbda6c6b2e03029d35ac061998e011bd6b702.png by

quantum_mechanics_70a55612227d7f7ec1898da3c7eab147cb7d9566.png

where quantum_mechanics_522916bbc1ccbdf0d09e376482006ff5cf08adf0.png is defined by

quantum_mechanics_3b9a5aa68d389baef5cb2cae9c7ad629b3700754.png

Then quantum_mechanics_0b2dbda6c6b2e03029d35ac061998e011bd6b702.png solves the free Schödinger equation with initial condition quantum_mechanics_682af946e7c476ca4df177275a23f6401baaf148.png.

5. A Particle in a Square Well

6. Perspectives on the Spectral Theorem

Notation

  • quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png denotes the position operator given by

    quantum_mechanics_c3bef9f576129c4e66217d6014383cdeaefc1436.png

    acting on quantum_mechanics_10aa83fe36a1bf2282ee5cb8a1b374d26f955eec.png

  • quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is a self-adjoint operator
  • Borel set quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png of quantum_mechanics_492d525117d0dcc93d066c8759f46b98cf9980ca.png
  • quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png denotes the closed span of all eigenvectors of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png with eigenvalues in quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png
    • In cases where quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png does not have true orthonormal basis, quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png is called spectral subspace
  • quantum_mechanics_dde19b4acdaf917b72042dd7cc725403a0463a64.png is the orthogonal projection onto quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png
  • For any unit vector quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png, we have

    quantum_mechanics_211ab53c7f87c5dea43c51e04aa0d8a6a1aeadd1.png

  • Indicator function

    quantum_mechanics_288788467d621ba744a8751d353a4389da42bf02.png

Goals of Spectral Theory

  • Recall that if the eigenvalues are distinct and quantum_mechanics_541a8541573e130dc310c688d0524f6e8fc0a5e3.png decomposes as

    quantum_mechanics_c3dbc005ed5b869d05992f7ecfb8fdbfbc43c7f2.png

    the probability of observing the value quantum_mechanics_655a6d131d2c3d88a4b94088a733479ade484ce8.png will be quantum_mechanics_ebb0b624ca2cb116a8455adb23c4bb85cff4bab4.png., since quantum_mechanics_727c2ca589ec73bca35b2b7af691631743a96c39.png is just the projection onto quantum_mechanics_53a4bc93f41d7733fb9e4821fdadf52d69030950.png.

  • In cases where quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png does not have a true orthonormal basis of eigenvectors, we would like the spectral theorem to provide a family of projection operators quantum_mechanics_dde19b4acdaf917b72042dd7cc725403a0463a64.png
    • One for each Borel subset quantum_mechanics_8bb977edb6466538b1ce951d2a306ca45fbdca10.png
    • Will allow us to define probabilities as in "standard" case above
  • Call these projection operators spectral projects and the associated subspaces quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png spectral subspaces.
  • Intuitively, quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png may be thought of as the closed span of all the generalized eigenvectors with eigenvalues in quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

Position operator

  • Has no true eigenvectors, i.e. no eigenvecors that are actually in quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png
  • If we think that "generalized eigenvectors" for quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png are the distributions given by

    quantum_mechanics_40c9ff36b6b3f45f23426d455fe14640c6a3bc00.png

    then one might guess that spectral subspace quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png should consist of those functions that are "supported" on quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, i.e. a superposition of the "funtions" quantum_mechanics_f16a2cf3e73f4e57d0ab467ef51f2b79c6b97713.png with quantum_mechanics_be68c0842e9dff3e237d78f2c99cd519bfeaddb9.png should define a function supported on quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

  • Spectral projection quantum_mechanics_dde19b4acdaf917b72042dd7cc725403a0463a64.png is then orthogonal projection onto quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png:

    quantum_mechanics_2f68d8b2ad6a2b3f97e097cfac1a27163960bc3c.png

    then

    quantum_mechanics_aff0fd497b8ebf2cb9f87e78a11fe2b37da52e15.png

  • The functional calculus of quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
    • If quantum_mechanics_891b1ef6039ca2569899c14f8c59b68d188919ca.png then we should have quantum_mechanics_f34232251b0b513c502eb8da813aaf537632b7ca.png

7. Spectral Theorem for Bounded Self-Adjoint Operators

Notation

  • quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is the separable complex Hilbert space
  • Operator norm of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is

    quantum_mechanics_d3a24c1fd226998385fec8f51b987756fa0ad313.png

    is finite.

  • Banach space of bounded operators on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png, wrt. operator norm is denoted quantum_mechanics_b8df6c238cc24d6136d9771635a522db8da23fdb.png.
  • quantum_mechanics_bacaf3d072bf37503881b213c59ab985a147c4f7.png denotes the resolvent set of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • quantum_mechanics_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png denotes the spectrum of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • quantum_mechanics_6f0ec29ab78652b52ddfd73d24562ff22c162642.png denotes the projection-valued measure associated with the operator self-adjoint quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png
  • For any projection-vauled measure quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png and quantum_mechanics_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png, we have an ordinary (positive) real-valued measure quantum_mechanics_9f3218af7dda488d97e5a5bad2ce36752723e71f.png given by

    quantum_mechanics_45f5aeddef60dacc01d05b6c29781cf4dff978d9.png

  • quantum_mechanics_a7106423e6a132cea5db947e9ecfdd0670890384.png is a map defined by

    quantum_mechanics_6abe9be4ce10bade2b415398dcf15084214fb0f0.png

  • Spectral subspace for each Borel set quantum_mechanics_8bb977edb6466538b1ce951d2a306ca45fbdca10.png

    quantum_mechanics_373d9a36ee8cf6cf00eb42701ffc550b99090aa0.png

    of quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png

  • quantum_mechanics_09f46c13958381645200ecbcff66fd4e1c27a547.png defines a simultanouesly orthonormal basis for a family quantum_mechanics_3a0c1b9067643a47e31b9d1663f95fcb4048db79.png of separable Hilbert spaces

Properties of Bounded Operators

  • Linear operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is said to be bounded if the operator norm of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png

    quantum_mechanics_d3a24c1fd226998385fec8f51b987756fa0ad313.png

    is finite.

  • Space of bounded operators on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png forms a Banach space under the operator norm, and we have the inequality

    quantum_mechanics_561a8ba71c40964399bf3f176c658613db0e73c9.png

    for all bounded operators on quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png and quantum_mechanics_9492d908c579945ca9e745601e13c2e7ae1ffd0e.png.

For quantum_mechanics_6f1378740525d29d0b8c054f3b69e304f3802989.png, the resolvent set of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, denoted quantum_mechanics_bacaf3d072bf37503881b213c59ab985a147c4f7.png is the set of all quantum_mechanics_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png such that the operator quantum_mechanics_8aa8508fd75eebefd63ac5fae6ce936ba1a4ba01.png has a bounded inverse.

The spectrum of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, denoted by quantum_mechanics_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png, is the complement in quantum_mechanics_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png of the resolvent set.

For quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png in the resolvent set of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, the operator quantum_mechanics_6a6699eb35f32545e8f4135014afa079b470f30e.png is called the resolvent of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png at quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png.

Alternatively, the resolvent set of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png can be described as the set of quantum_mechanics_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png for which quantum_mechanics_8aa8508fd75eebefd63ac5fae6ce936ba1a4ba01.png is one-to-one and onto.

For all quantum_mechanics_6f1378740525d29d0b8c054f3b69e304f3802989.png, the following results hold.

  1. The spectrum quantum_mechanics_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is closed, bounded and nonempty subset of quantum_mechanics_20a24d1ed449dcbb47090631b2f93ca1189c3c42.png.
  2. If quantum_mechanics_a52b5e19cdde324f29c071d482bdac1303cdfef8.png, then quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png is in the resolvent set of quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png

Point 2 in proposition:hall13-quant-7.5 establishes that quantum_mechanics_41a71a01903b8fd1d931aa867d5f0c3c4c4a5bf3.png is bounded if quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is bounded.

Suppose quantum_mechanics_6f1378740525d29d0b8c054f3b69e304f3802989.png satisfies quantum_mechanics_003302cbf4b60db1b6797b412be4667d012e73a6.png.

Then the operator quantum_mechanics_ceec04ccc2a5efb6b6e2591ac2dd0f80a6533df4.png is invertible, with the inverse given by the following convergent series in quantum_mechanics_b8df6c238cc24d6136d9771635a522db8da23fdb.png:

quantum_mechanics_c6ef2912da209c031545825c99f017458d8f7534.png

For all quantum_mechanics_6f1378740525d29d0b8c054f3b69e304f3802989.png, we have

quantum_mechanics_6de4c54c2068877096aa4c7abf31d295bdf3ba43.png

Spectral Theorem for Bounded Self-Adjoint Operators

Given a bounded self-adjoint operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png, we hope to associate with each Borel set quantum_mechanics_1be15bdaf9866f25469beab46b89e856207c81a6.png a closed subspace quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png of quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png, where we think intuitively that quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png is the closed span of the generalized eigenvectors for quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png with eigenvalues in quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png.

We would expect the following properties of these subspaces:

  1. quantum_mechanics_0034ea3c2d2e7790dc6fc5d01fa373087764c66b.png and quantum_mechanics_d8f20735391c69f2f5d3cbef5ab2f9763906e406.png
    • Captures idea that generalized eigenvectors should span quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png
  2. If quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png and quantum_mechanics_b2db59aeb94cbc1050079892ff07b21b493513b7.png are disjoint, then quantum_mechanics_4d260d163c1ca5703a26575caa389279f323b315.png
    • Generalized eigenvectors ought to have some sort of orthogonality for distinct eigenvalues (even if not actually in quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png)
  3. For any quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png and quantum_mechanics_b2db59aeb94cbc1050079892ff07b21b493513b7.png, quantum_mechanics_174913feba6e8898a472d5701f79a93f0501a81f.png
  4. If quantum_mechanics_7b717a22c79b7fe462a2f6e59262bcd47c279f96.png are disjoint and quantum_mechanics_df90acaed9f72be569e8c617ec4f853fdbf46174.png, then

    quantum_mechanics_ca0ebb49296581806b4125482c87a3ca7557d1a8.png

  5. For any quantum_mechanics_2f3e6613e385d320fae4ea9bc3d9f6d2b98dd564.png, quantum_mechanics_f2abeffb62ce136d1d611fce9e18ab5401268307.png is invariant under quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png.
  6. If quantum_mechanics_5794b4b598546503ac0c75c52678d52924bab06f.png and quantum_mechanics_8f76cc6c958beef63e111c2d03351b495f1c9be0.png, then

    quantum_mechanics_8bc214043b20102a6b8f407acd00f146d87c9dfb.png

Projection-Valued measures

For any closed subspace quantum_mechanics_dae3bc011085c76ed31e2c12477df8c12e2be5da.png, there exists a unique bounded operator quantum_mechanics_a9090c77ce9916955c745920bf8f134a0932d59d.png such that

quantum_mechanics_3c5772dc8f9a99be724c5465ef4814314a7d7984.png

where quantum_mechanics_b3f96c59fbac27960c0cb850c195e991d0d611a4.png is the orthogonal complement.

This operator is called the orthogonal projection onto quantum_mechanics_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png and it satisfies

quantum_mechanics_c8d534bfc954eb1cd8309e527e64110d7eae66fa.png

i.e. it's self-adjoint.

One also has the properties

quantum_mechanics_09e577a3705b6751ab8fc42dc5782c83be94a2ef.png

or equivalently,

quantum_mechanics_c140603b627f8409f9553db1f1a50a7ba1d7cc8e.png

Conversely, if quantum_mechanics_a9090c77ce9916955c745920bf8f134a0932d59d.png is any bounded operator on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png satisfying quantum_mechanics_79b68e5d6f686efdffcaa316bf75359980f36ae1.png and quantum_mechanics_108c1dada9d7bfb68c4c0cc80ac783f01f005f6c.png, then quantum_mechanics_a9090c77ce9916955c745920bf8f134a0932d59d.png is the orthogonal projection onto a closed subspace quantum_mechanics_d09e558e8b16ae9aff8e7fd088d8e072a09a73f1.png, where

quantum_mechanics_43c9697fe40871ccfc7d1a535fdf2e65c40ff473.png

  • Convenient ot describe closed subspaces of quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png in terms of associated orthogonal projection operators
  • Projection operator expresses the first four properties of the spectral subspaces; those properties are similar to those of a measures, so we use the term projection-valued measure

Let quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a set and quantum_mechanics_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png an quantum_mechanics_dcb897d0137330758a9675e99ee9ec7f93c4d742.png in quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

A map quantum_mechanics_a0a5da18bdf68959502bd27fa465cf4fd1638785.png is called a projection-valued measure if the following properties are satisfied:

  1. For each quantum_mechanics_f933ceed1d0fd23b8d8bcc17b875afcab7246893.png, quantum_mechanics_aa40c294f2551eb7ac81f321ae78295cc29e6375.png is an orthogonal projection
  2. quantum_mechanics_a1a5a70f855c26c9d1ddb6800e67fef382077d41.png and quantum_mechanics_bd2540a215495b47c88ffabdbca250a91b4c7e75.png
  3. If quantum_mechanics_0ec37729c8e1877c8f8d7b82616cda3c21bdeace.png are disjoint, then for all quantum_mechanics_8929ebd59d804f0b4681a02e4604c24f91a5b1c5.png, we have

    quantum_mechanics_8edd46701e07126e9b4b4188706d8a52d70722c0.png

    where the convergence of the sum is in the norm-topology on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png.

  4. For all quantum_mechanics_7b5b94b41d5ca2143d494b5c5d4f8dc54de78521.png, we have quantum_mechanics_d5437014fbddb52800ab3bb2efc605db4be27491.png

Properties 2 and 4 in of a projection-valued measure tells us that if quantum_mechanics_4070caa01fede33fe80e39178121e1df2177be4e.png and quantum_mechanics_fa578000a65909574243704c580c4055ed5a76f0.png are disjoint, then

quantum_mechanics_7a7f4921b3bfe2110f5626699f51bd10eb594746.png

from which it follows that the range of quantum_mechanics_8245ccebfb693a435aa6f108d9d9dc6070492efe.png and the range of quantum_mechanics_afeb28fd2597645cb9ae05ae855d3dc7333f47a3.png are perpendicular.

Let quantum_mechanics_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png be a quantum_mechanics_dcb897d0137330758a9675e99ee9ec7f93c4d742.png in a set quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and let quantum_mechanics_741ea1e115f85e1054f975a0d447b2197d65f167.png be a projection-valued measure.

Then there exists a unique linear map, denoted

quantum_mechanics_ab18cdcf74e5352ed8866baa952744ce7de4a04d.png

from the space of bounded, measurable, complex-valued functions on quantum_mechanics_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png into quantum_mechanics_b8df6c238cc24d6136d9771635a522db8da23fdb.png with the property that

quantum_mechanics_01bd399f2fcbb91a39c3cbf7bd49d3713b014549.png

for all quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png and all quantum_mechanics_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png.

This integral has the following properties:

  1. For all quantum_mechanics_f933ceed1d0fd23b8d8bcc17b875afcab7246893.png, we have

    quantum_mechanics_af379a420d8c662e6b438ff66e9e578d04f45e71.png

    In particular, the integral of the constant function quantum_mechanics_4468973182b954eeeb1a22bfe0c5b928511fa9f2.png is quantum_mechanics_a16bd19472c2cac063ec2b5dedd92f5be05b1e0a.png.

  2. For all quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png, we have

    quantum_mechanics_67fe6795a644d0af0a7004acc6f62ce15e32c217.png

  3. Integration is multiplicative: For all quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png and quantum_mechanics_5eda5bc8724030dff96b49f83b0e9ba134ffc700.png, we have

    quantum_mechanics_99c4b3a8dadf2ccccc5af1b9014701daf3dc10c6.png

  4. For all quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png, we have

    quantum_mechanics_92d8d3be3502dbd1359d9552890057b3860ae53a.png

    In particular, if quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is real-valued, then quantum_mechanics_af6bbcea1e62b5768eda04cc20ea7f2f64d22eb3.png is self-adjoint.

By Property 1 and linearity, integration wrt. quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png has the expected behavior on simple functions. It then follows from Property 2 that the integral of an arbitrary bounded measurable function quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png can be comptued as follows:

  1. Take sequence quantum_mechanics_dfdd090ed57eac18401044d91801cceb018fdda4.png of simple functions converging uniformly to quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png
  2. The integral of quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is then the limit, in the norm-topology, of the integral of the quantum_mechanics_dfdd090ed57eac18401044d91801cceb018fdda4.png.

A quadratic form on a Hilbert space quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png is a map quantum_mechanics_61d679c639259b6b94fc80216239d7f6172c8628.png with the following properties:

  1. quantum_mechanics_967d8ef613988996b3544790631d6f61800cb25c.png for all quantum_mechanics_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png and quantum_mechanics_6a8bfebe144986d9a65acc5ce1b692fe4b0ba201.png
  2. the map quantum_mechanics_d5bbcf122da3a01863ca032167777ae5952befb0.png defined by

    quantum_mechanics_5818fea04112a5c12e71114cf1056826fc355db6.png

is a sesquilinear form.

A quadratic form quantum_mechanics_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is bounded if there eixsts a constant quantum_mechanics_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png such that

quantum_mechanics_be605b31d8337da565b44fedc1c3a59d694580bb.png

The smallest such constant quantum_mechanics_9ccec48caa132706c57c7e0e22c7558bf8f1a48b.png is the norm of quantum_mechanics_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png.

If quantum_mechanics_9fe46e6ccfd065fb50f2eeb9cbdc65835e9ec321.png is a bounded quadratic form on quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png, there is a unique quantum_mechanics_6f1378740525d29d0b8c054f3b69e304f3802989.png such that

quantum_mechanics_da54f04d317ef544c5a94c1050ebbfc6ff03bbb1.png

If quantum_mechanics_6065f2b22f6f529ab9631ce6799c1acc8b5f4650.png belongs to quantum_mechanics_492d525117d0dcc93d066c8759f46b98cf9980ca.png for all quantum_mechanics_196ddbe8da3548ec409c5b6a8dce31fc9543fba0.png, then the operator quantum_mechanics_e46729bc781c25bbc7120ee2892cc1c0215af7da.png is self-adjoint.

Spectral Theorem for Bounded Self-Adjoint Operators: direct integral approach

Notation
  • quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a quantum_mechanics_026b58b7d0ddfb732bc0a0630d01a61cc001cc27.png measure on a quantum_mechanics_dcb897d0137330758a9675e99ee9ec7f93c4d742.png quantum_mechanics_6ac37f3dbecbda6686c66a4e8881e71672d3a77a.png of sets in quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png
  • For each quantum_mechanics_11d540a6fa79f689fd95ec9cc15f1a16b185f327.png we have a separable Hilbert space quantum_mechanics_ad92e251486f17ae5d412863e949703d030ab96e.png with inner product quantum_mechanics_5cf50851931c4ff5b29313fd071a153f4cfd31ba.png
  • Elements of the direct integral are called sections quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png
Stuff

There are several benefits to this approach compared to the simpler "multiplication operator" approach.

  1. The set quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png and the function quantum_mechanics_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png become canonical:
    • quantum_mechanics_0754fda19bdabf81a4ee9c8074f7d0e2eb7518e3.png
    • quantum_mechanics_15ff1fdcb5b3d2a6babc4e830ddddb341c5ad11e.png
  2. The direct integral carries with it a notion of generalized eigenvectors / kets, since the space quantum_mechanics_ad92e251486f17ae5d412863e949703d030ab96e.png can be thought of as the space of generalized eigenvectors with eigenvalue quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png.
  3. A simple way to classify self-adjoint operators up to unitary equivalence: two self-adjoint operators are unitarily equivalent if and only if their direct integral representations are equivalent in a natural sense.

Elements of the direct integral are called sections quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png, which are functions on quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png with values in the union of the quantum_mechanics_ad92e251486f17ae5d412863e949703d030ab96e.png, with property

quantum_mechanics_3d84f93b5eb82c0dc254de2f53d71bb93ff1413d.png

We define the norm of a section quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png by the formula

quantum_mechanics_abb8f82664f4596d837d635bb8ca9b4cf1e3305e.png

provided that the integral on the RHS is finite.

The inner product between two sections quantum_mechanics_f3527ff67a51b84614a04556c727c038d0a7a4a9.png and quantum_mechanics_500fcd494df13a53a23eaadcb27199fe812b739b.png (with finite norm) should then be given by the formula

quantum_mechanics_7cf7a77520496bb7a3141669e6eee41c82383921.png

Seems very much like the differential geometry section we know of.

  • quantum_mechanics_ad92e251486f17ae5d412863e949703d030ab96e.png is the fibre at each point quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png in the mfd.
  • quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png is the mfd.

First we slightly alter the concept of an orthonormal basis. We say a family quantum_mechanics_3e6ce93c408fe9d4e69bfa680cd40284daf65fe2.png of vectors is an orthonormal basis for a Hilbert space quantum_mechanics_41a6ef8338fde1b69adbb704a2beeef54fb51bab.png if

quantum_mechanics_b251db11124517f8526c5eb3710365d2ddcec92a.png

and

quantum_mechanics_e99a98f43b79cf22e02cde11866e3b572a51d375.png

This just menas that we allow some of the vectors in our basis to be zero.

We define a simultanouesly orthonormal basis for a family quantum_mechanics_3a0c1b9067643a47e31b9d1663f95fcb4048db79.png of separable Hilbert spaces to be a collection quantum_mechanics_09f46c13958381645200ecbcff66fd4e1c27a547.png of sections with the property that

quantum_mechanics_3bfc12c9980fd658c433aaffc4e4f7b46fd5e449.png

Provided that the function quantum_mechanics_645a681c7a3e7c03cdb7683b97799acb64223b3d.png is a measurable function from quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png into quantum_mechanics_4d2f45153627eceed2f100b44db54377e2c0033b.png, it is possible to choose a /simultaneous orthonormal basis quantum_mechanics_fca298202d9cbc0fff267e972b581c767a8e9729.png such that

quantum_mechanics_5dcbdf7e585d622ba988829f0662654134875684.png

is measurable for all quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png and quantum_mechanics_ad63c81b09a45c2afd6c497ff38a542e26c1423a.png.

Choosing a simultaneous orthonormal basis with the property that the function

quantum_mechanics_ceffe25faec7fd033ecf07006d37be84758ae57e.png

is a measurable function from quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png into quantum_mechanics_4d2f45153627eceed2f100b44db54377e2c0033b.png, we can define a section to be measurable if the function

quantum_mechanics_1a9d4158ea58f7f1b68e7a5d3c096bd225348fc4.png

is a measurable complex-valued function for each quantum_mechanics_d8797d487554fc78e460f5843167fcb01b53e76b.png. This aslo means that the quantum_mechanics_53a4bc93f41d7733fb9e4821fdadf52d69030950.png are also measurable sections.

We refer to such a choice of simultaneous orthonormal basis as a measurability structure on the collection quantum_mechanics_3a0c1b9067643a47e31b9d1663f95fcb4048db79.png.

Given two measurable sections quantum_mechanics_f3527ff67a51b84614a04556c727c038d0a7a4a9.png and quantum_mechanics_500fcd494df13a53a23eaadcb27199fe812b739b.png, the function

quantum_mechanics_137dcf00cafc0c51510ef0d86141544e8bf82a55.png

is also measurable.

Suppose the following structures are given:

  1. a quantum_mechanics_026b58b7d0ddfb732bc0a0630d01a61cc001cc27.png measure space quantum_mechanics_35993036aec555a7097e8f33d1a67704e95153bf.png
  2. a collection quantum_mechanics_dd225d091a1ecdc6b815c3e87ac3def0c038b97c.png of separable Hilbert spaces for which the dimension function is measurable
  3. a measurability structure on quantum_mechanics_dd225d091a1ecdc6b815c3e87ac3def0c038b97c.png

Then the direct integral of quantum_mechanics_ad92e251486f17ae5d412863e949703d030ab96e.png wrt. quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png, denoted

quantum_mechanics_739aeef9a24d3699d121546ebb65ae5cf50381d0.png

is the space of equivalence classes of almost-everywhere-equal measurable sections quantum_mechanics_aa1698cb8ee1665238ec3e91824191643c62ee93.png for which

quantum_mechanics_71ff5e135eaa1ea78cb0e8fed6afb680be5d4b1f.png

The inner product quantum_mechanics_29f3145d892590ef40dd2f24b099e3ebb901a92e.png of two sections quantum_mechanics_f3527ff67a51b84614a04556c727c038d0a7a4a9.png and quantum_mechanics_500fcd494df13a53a23eaadcb27199fe812b739b.png is given by the formula

quantum_mechanics_5268dcad2c0282114fa787261678cfe6dae7bfbf.png

8. Spectral Theorem for Bounded Self-Adjoint Operators: Proofs

Notation

  • quantum_mechanics_6f1378740525d29d0b8c054f3b69e304f3802989.png with spectral radius

    quantum_mechanics_1ad0084a5a98f689b6ead58822b0ad0a2ed345a5.png

Stage 1: Continuous Functional Calculus

Stage 2: An Operator-Valued Riesz Representation Theorem

Let quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png be a compact metric space and let quantum_mechanics_b3ca7489571b2d31006c791334301d8f88436b6f.png denote the space of continuous, real-valued functions on quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png.

Suppose quantum_mechanics_ce7fb4eec15bb6f3e1b146dc1f73a301cd6686f5.png is a linear functioanl with the property that quantum_mechanics_f89b13726ed6d17890417243b11a3a86724def99.png is non-negative if quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is non-negative.

Then there exists a unique (real-valued, positive) measure quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png on the Borel sigma-algebra in quantum_mechanics_aa07b3a8458adb2855b54064282b1f340d44fbd4.png for which

quantum_mechanics_8ba8b057f37fa0ceaf7b8171442b33ef8cc95980.png

Observe that quantum_mechanics_acb0106122b7f90d9bd5639367a141a7e53d8327.png is a finite measure, with

quantum_mechanics_1a8f75cb4f0c986c071c686a803ffb46a7e6a63c.png

where quantum_mechanics_7d116d2259707306a9272c99ee5b455da5ea6e09.png is the constant function.

Practice problems

"Point-potential"

We have a potential of the form

quantum_mechanics_3cddfd370648d0f33bf4cd77309c980e31643d0c.png

which gives us the TISE:

quantum_mechanics_9e803f3937800d425a3f32545ef40f18516db773.png

Then we impose the following conditions on the solution quantum_mechanics_a284dbf7473556cebbba8621e88420c98f5232ad.png :

  • continuity at quantum_mechanics_5c302d31faee1dc741905749ed4f6a8e7454408b.png, i.e. require quantum_mechanics_521e17f773e1219a365d1e489f5a03bc56efba8a.png
  • quantum_mechanics_868b0040da432813efd87426d574deed35a49c30.png

The continuity restriction does NOT necessarily mean that the derivative itself is continuous!

If we then integrate both sides of the TISE above over the interval quantum_mechanics_37cfd5efbb6156137f735745e145d03795b9394c.png taking the limit quantum_mechanics_d1b02fae817763a9940d653a2ee018c296e7869b.png:

quantum_mechanics_4d1b0f636e038042b230e095c2c94be29100e3ae.png

which gives us

quantum_mechanics_788678918a1dfa07b0ece4a70b37ae8864701019.png

And since quantum_mechanics_8552d546fce1d58f5cbbd8e979104ec437020c46.png, we get:

quantum_mechanics_6723afcb2b888476bbae1b81423932fccd8fbec6.png

Then we compute the derivaties and take the limits, giving us something better to work with!

Harmonic Oscillator

Consider a potential energy of the form quantum_mechanics_2fcab0a66b894b138eb27e9569e4736b6116b495.png. The classical energy is then:

quantum_mechanics_eb7c8671db40323b5ae008de3c02c7b5720b2209.png

Using the Hamiltonian operator quantum_mechanics_14a40f189f2ce698341c03cd5c099a337431c49f.png and the momentum quantum_mechanics_cae61a421aad1a957b1b74ee29450b7b5fcb7418.png . We can then write the Schödinger equation, letting quantum_mechanics_373ae25c9741b3510810cd66e71bfbdcc28acd92.png and quantum_mechanics_d52922e0ac80aaa088f7858022e0ef7155a64854.png for convenience, as:

quantum_mechanics_1cabd8116440bcb99920202239fc1958d531b1da.png

or equivalently,

quantum_mechanics_4d1b13c223e20e0bccc5e68e4633d34bf93fbaf8.png

Goal is the to find solutions quantum_mechanics_a284dbf7473556cebbba8621e88420c98f5232ad.png that are physical and suitable for all quantum_mechanics_7a84c9a383f9772338016d101ccc096be06af784.png .

One approach is to perform the following:

  1. Obtain solution for limiting behavior
  2. Perform power series expansion

Our requirements for the solution:

  • single valued over the region (well-behaved)

First consider when quantum_mechanics_41a34521ddd4db2803a88b231a673d6b75eb4f1e.png is large and quantum_mechanics_11a526123021a7cffc729b5c46b2ad608f14d8b4.png dominates with quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png being negible by design . Then

quantum_mechanics_14cae9b329be7aef7018b62bbb9ea18ebf660799.png

which we can guess ourselves to a solution of

quantum_mechanics_d31eb749ac6ded20527cd7dd1659cc20a01677a5.png

to which we then note that quantum_mechanics_cd8ee9d875636248fd19006f1148c76ee376c803.png is not satisfactory given that it diverges (due to restricting out attention to square-integrable functions only). Hence,

quantum_mechanics_16520ef9a7337170ffb4f0da709b3eb4735232d4.png

Why does it make sense to consider the limit behavior?

Due to the continuity imposed on the solution / wave function quantum_mechanics_a284dbf7473556cebbba8621e88420c98f5232ad.png we now that the solution to the limiting behavior is related to the solution across the entire domain of quantum_mechanics_a284dbf7473556cebbba8621e88420c98f5232ad.png.

From this solution we can then derive more solutions using reduction of order, i.e. we multiply our previous solution with some arbitrary function quantum_mechanics_b4625aa4535b6123d93c130d4dc4772a395916cd.png, which gives use

quantum_mechanics_da7a51dcac7f5ae77c01d85d38dc773bb7fcca39.png

Taking the derivatives and substituting into the original differential equation we get:

quantum_mechanics_35910406573d2189867f2de26e53362542598f45.png

To make the connection with a "special" function in a few steps, let's make the substituion quantum_mechanics_e41fb976cc13edb2eaf6ad5377faac6f65754fd9.png and replace the function quantum_mechanics_b4625aa4535b6123d93c130d4dc4772a395916cd.png with quantum_mechanics_016c1d438ba804d2e533748e1e78f7c18040cc3c.png to get the following equation:

quantum_mechanics_b2356235c94206e47abb43ab2f80f8ece8c244fa.png

We can now represent quantum_mechanics_016c1d438ba804d2e533748e1e78f7c18040cc3c.png as a power series:

quantum_mechanics_fa8958301160391e54688d3bb22ebb77da483887.png

where we've made the assumption that this series converges for all quantum_mechanics_42b688f7ebfaa0e3791010e7725767cadf2ad420.png and that the function is well defined.

Physically, we know the above has to be case for the solutions to make sense and satisfy our overall constraint that the integral of the probability density must be finite: quantum_mechanics_6f2b3ab3bca9f330062b632106f4dc15507a22f6.png

Differentiating quantum_mechanics_016c1d438ba804d2e533748e1e78f7c18040cc3c.png and substituting back into the differential equation we (eventually) obtain the recursion formula:

quantum_mechanics_14691ee5fa0dddca3c52c138455746f7ae1f1cb7.png

And thus we have a solution! BUT we have a serious problem :

  • We assumed the series of quantum_mechanics_b4625aa4535b6123d93c130d4dc4772a395916cd.png to be convergent for all quantum_mechanics_7a84c9a383f9772338016d101ccc096be06af784.png which is not generally true in the recursion relation for all values quantum_mechanics_df08dda73af29a2bd1a17f98fe28db12eb64c4ee.png (the energy values) and quantum_mechanics_216baa1dc1da1eb3f1765b89622830a7cef9cccc.png (potential well curvature)
  • The function is square integrable if quantum_mechanics_b4625aa4535b6123d93c130d4dc4772a395916cd.png is truncated and does not contain an infininte number of terms in its series expansion

Thus, when we then require the

quantum_mechanics_9cd6e53399a93427b31068ccf585176fa6fe1239.png

it can be seen from the recursion relation that the series expansion is finite (and hence converges quantum_mechanics_8c20a46998940f387b38b9a36f77ced3cab9a42e.png ) if

quantum_mechanics_f8c0700405d6db1af0a22bd2306fc06e9176d010.png

or the energy values have the folloiwgn discrete spectra,

quantum_mechanics_187596ad787c002c670366668aef9b4dab7644ab.png

Whiiiich we can finally write as:

quantum_mechanics_7e33eeb681f19208fc71302b4c36639c6f407a5d.png

Short problems

Eigenvalue expansion of in angular momentum basis

The wavefunction of a particle at quantum_mechanics_9b50c3c5095e3def3ea0a38afe0602245b66e068.png is known to have the form

quantum_mechanics_cc914c407cc377ecce2d7340c71b010a3eb34e02.png

where quantum_mechanics_93369077affb352dbdb97e8b3182fd50784f2b14.png is an unknown function of quantum_mechanics_7bec91615d02a5342b34270ae4a669947491ecaf.png.

What can be predicted about the results of measuring:

  1. the z-component of angular momentum?
  2. the square of the angular momentum?

Hint: expand quantum_mechanics_2b2f2d027876c7cf5b5ea8891b87c8f419e31743.png in eigenfunctions of quantum_mechanics_c76feac5aa2f03f374763eddf78d6e3d69c645fd.png, which are of the form quantum_mechanics_03efe1f5b73e821cccf2da34bfa3dd83592b11e6.png, where quantum_mechanics_729d5e401b71d8ba960b263198a701fda1b79bb3.png.

Answer

Observe that

quantum_mechanics_dc391e7144c3186be8f879d725db77b590f6bcbc.png

Thus,

quantum_mechanics_b86667941008b51da75e906430609a22e9117ddf.png

Thus, when measuring quantum_mechanics_c76feac5aa2f03f374763eddf78d6e3d69c645fd.png, we can obtain either quantum_mechanics_6a323b2b527554a3fc02f00f2ae786c47e5105b1.png or quantum_mechanics_54971581e05407d8d05b20c2b045090875de3223.png with equal probability.

We can't really say anything about quantum_mechanics_0ef7976ae2bbe1274e36283db47447bb7fb7bef5.png, without knowing more about quantum_mechanics_839db719d09fe31a80daf18e813c32c2a928339a.png, other than observing that quantum_mechanics_91f1891383afa92640c4c6b0541fad9c22066077.png since quantum_mechanics_09812afc0e2df840113e6b6ba663dae2256a0849.png.

Constructing matrix for angular component of a system with quantum_mechanics_526aebf00d04b1bf139a69e10980dcdc1c6e0747.png

Construct the quantum_mechanics_5043061a7b1626659d6aaa3d4432ce6fc4bc83cf.png matrix which represent the Cartesian component in the z-direction of the angular momentum for a system with quantum_mechanics_526aebf00d04b1bf139a69e10980dcdc1c6e0747.png.

Answer

For quantum_mechanics_526aebf00d04b1bf139a69e10980dcdc1c6e0747.png, quantum_mechanics_37b9c81d1b9a43ca90ac4242d36681d0a41f7735.png, and the matrix-elements of quantum_mechanics_eab1858cf96b1ed3dcebeed8406e98845e73ef5c.png are given by

quantum_mechanics_1ce45d343b975ebbbfe9c05d1ba633366a290866.png

Thus, we have

quantum_mechanics_55e2c9e8ccd97746d842aae520da6ba9456496ee.png

where we use quantum_mechanics_579cfebc0c7c469b1f114b8ab49a6a2f14ca40f1.png instead of quantum_mechanics_88ff129d7bbaa864ac622e34ebba97801e661a0b.png to indicate that quantum_mechanics_eab1858cf96b1ed3dcebeed8406e98845e73ef5c.png only corresponds to this matrix in this specific representation / space (Cartesian coordinates in this case).

Q & A

Simplification of potential well problem due to direction of travel

Footnotes:

1

Is is true that the Hilbert space quantum_mechanics_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png and its dual space are isomorphic; however, we have taken for the wave function space quantum_mechanics_92757546978c1aad3468f8a69e8d334af2bfadac.png is a subspace of quantum_mechanics_5d229ea7e71b918baa91f8bbda3a25e08b64ceb5.png, which explains why quantum_mechanics_55d98540d635c58892332f4474b86442a69cd958.png is "larger" than quantum_mechanics_92757546978c1aad3468f8a69e8d334af2bfadac.png.