Angular momentum commutes with Hamiltonian

[Feynmanfan]

How can I prove that the Hamiltonian commutes with the angular momentum operator?

In spherical coordinates it is straightforward but I'd like to understand the physical meaning of it.

Thanks.

 

[vanesch]

How can I prove that the Hamiltonian commutes with the angular momentum operator?
In spherical coordinates it is straightforward but I'd like to understand the physical meaning of it.
Thanks.

What hamiltonian ? The hamiltonian of the free particle ? Of the harmonic oscillator ? Of the hydrogen atom ?
All these do indeed commute with the angular momentum operator (Lx,Ly,Lz). But if you'd have a non-central potential, this would not be the case.
The physical meaning is this:
the hamiltonian is the generator of time translations (huh ? :-) Yes, that's the content of Schroedinger's equation: the time derivative of the state (wavefunction) is the hamiltonian applied to the wavefunction.
The angular momentum operator is the generator of space rotations.
If both commute, that means that it doesn't matter if you first rotate (a small bit) and then "translate in time" (advance a bit in time), or whether you do it in the opposite order (first advance a bit in time, and then rotate).
So that means that "energy" (the hamiltonian) is "conserved under rotations" (essentially that your problem is spherically symmetrical) ;
or:
that "angular momentum" (the angular momentum operator) is "conserved under time translations", meaning: angular momentum is constant during the motion: it is a constant of motion.
cheers,
Patrick.