Quantum Mechanics

Table of Contents

2. A First Approach to Classical Mechanics

2.5 Poisson Brackets and Hamiltonian Mechanics

Let quantum_theory_for_mathematicians_cdd1cc131da6040eca078917132a377727053c44.png and quantum_theory_for_mathematicians_bed561b338628a088ae69301d4bba9fd83a70bd2.png be two smooth functions on quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png, where an element of quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png is thought of as a pair quantum_theory_for_mathematicians_42f33ebffa4e702fd1795a5a88f1f588d9aee5b0.png, with

  • quantum_theory_for_mathematicians_7a9f00f52fc25b76ee5704ac6f6e9bc4da1d48a9.png representing position of a particle quantum_theory_for_mathematicians_66ec047e56f0c3896ce0dc73e002edb3ac6aeb8d.png representing the momentum of a particle

Then the Poisson bracket of quantum_theory_for_mathematicians_cdd1cc131da6040eca078917132a377727053c44.png and quantum_theory_for_mathematicians_bed561b338628a088ae69301d4bba9fd83a70bd2.png, denoted quantum_theory_for_mathematicians_7c683eac19f261cf7b95da263c85bc454f9dd25c.png is the function on quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png given by

quantum_theory_for_mathematicians_defe4db1a039e35a6e0ea32ae4e5b3ab9491b540.png

For all smooth functions quantum_theory_for_mathematicians_cdd1cc131da6040eca078917132a377727053c44.png, quantum_theory_for_mathematicians_bed561b338628a088ae69301d4bba9fd83a70bd2.png and quantum_theory_for_mathematicians_3807b8495f01004f1d88370e2f4fad2f5032db06.png on quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png we have the following:

  1. quantum_theory_for_mathematicians_53a7e73e2a6d41ba0213827ea061a0b99bb10114.png for all quantum_theory_for_mathematicians_a74fc6f54ff9249999f16a8b20feb0af8b5904c9.png
  2. quantum_theory_for_mathematicians_9b3fde7ac9d49d7d66442c5c5750187c2b696dd8.png
  3. quantum_theory_for_mathematicians_24d438ff565c623928198eeac6ceb752d86ba231.png
  4. Jacobi identity:

    quantum_theory_for_mathematicians_f517396cb350908b2ed880f499f5fb9989572b6f.png

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

quantum_theory_for_mathematicians_255a9388470bb1486e3c52546735080658399750.png

If a particle in quantum_theory_for_mathematicians_c56d2a2224fedeb46b49b50d1b41f11e963216ce.png has the usual sort of energy function (kinetic energy plus potential energy), we have

quantum_theory_for_mathematicians_61187d2699ac7f6b651a42690e270262b5586675.png

Wit the Hamiltonian, and as usual, having quantum_theory_for_mathematicians_d575f50ea5ac85241e593585b320abbd64f547f4.png, we can write Netwon's laws as:

quantum_theory_for_mathematicians_4e95eb5b8d81dcdeba50954a896bfcc75ed2487c.png

These equations we refer to has Hamilton's equations.

If quantum_theory_for_mathematicians_5e7abdc753e080fe11399d27da19afda6b63519f.png is a solution of the Hamilton's equation, then for any function quantum_theory_for_mathematicians_cdd1cc131da6040eca078917132a377727053c44.png on quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png, we have

quantum_theory_for_mathematicians_e1dfabb84cb5db436a41c17ec10d285fb0e6389a.png

Call a smooth function quantum_theory_for_mathematicians_cdd1cc131da6040eca078917132a377727053c44.png on quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png a conserved quantity if quantum_theory_for_mathematicians_17ff1f507c467ae97f8a86e36fb10c4310298392.png is independent of quantum_theory_for_mathematicians_eb5d809ed7c492fae7d4927a6fc9a5e22f9b3831.png for each solution quantum_theory_for_mathematicians_5e7abdc753e080fe11399d27da19afda6b63519f.png of Hamilton's equations.

Then quantum_theory_for_mathematicians_cdd1cc131da6040eca078917132a377727053c44.png is a conserved quantity if and only if

quantum_theory_for_mathematicians_2f5876724a6d410f92b6c894f78030c83de310e6.png

In particular, the Hamiltonian quantum_theory_for_mathematicians_f1330523c697c7f312e3dcdd0386cfbf0532a867.png is a conserved quantity.

Solving Hamilton's equatons on quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png gives rise to a flow on quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png, that is, a family quantum_theory_for_mathematicians_3c504a0fcc87372c36aeb59d5d4d8baff116b850.png of diffeomorphisms of quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png, where quantum_theory_for_mathematicians_be338838485761ff56b93fb61644a5d98e7ecd3e.png is equal to the solution at time quantum_theory_for_mathematicians_eb5d809ed7c492fae7d4927a6fc9a5e22f9b3831.png of Hamilton's equations with initial conditions quantum_theory_for_mathematicians_42f33ebffa4e702fd1795a5a88f1f588d9aee5b0.png.

Since it is possible (depending on the choice of potential function quantum_theory_for_mathematicians_ebac26f7eba81eb516f423f028596d8e6018aa7a.png ) that a particle can escape to infinity in finite time, the maps quantum_theory_for_mathematicians_3c504a0fcc87372c36aeb59d5d4d8baff116b850.png are not necessarily defined on all of quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png, but only on some subset therof.

If quantum_theory_for_mathematicians_3c504a0fcc87372c36aeb59d5d4d8baff116b850.png is defined on all of quantum_theory_for_mathematicians_12d78938abb996b94db18c7061a759e1f0360c09.png we say it's complete.

The flow associated with Hamilton's equations, for an arbitrary Hamitonian function quantum_theory_for_mathematicians_f1330523c697c7f312e3dcdd0386cfbf0532a867.png, preserves the (2n)-dimensional volume measure

quantum_theory_for_mathematicians_80dfc054080631a6ac5df16f5c40e2910c673389.png

What this means, more precisely, is that if a measurable set quantum_theory_for_mathematicians_fbe98aaf8359e7eed3ba031caeaf6a3c13ae8690.png is contained in the domain of quantum_theory_for_mathematicians_3c504a0fcc87372c36aeb59d5d4d8baff116b850.png for some quantum_theory_for_mathematicians_93a27e4563ca640c3a9531a5614f4d7b21c41949.png, then the volume of quantum_theory_for_mathematicians_a6de2e0eb9331e023e7cfa78fdf7552f43fed802.png is equal to the volume of quantum_theory_for_mathematicians_fbe98aaf8359e7eed3ba031caeaf6a3c13ae8690.png.

3. A First Approach to Quantum Mechanics

3.2 A Few Words About Operators and Their Adjoints