Angular momentum and Hamiltonian commutator

[blagershod.smee]

Hello,

Is it generally the case that [J, H] = dJ/dt?

I saw this appear in a problem involving a spin 1/2 system interacting with a magnetic field.

If so, why?This seems like a very basic relation but I'm having a bit of brain freeze and can't see the answer right now.

 

[Matterwave]

You can just work out the commutator yourself by decomposing J and H into component operators. It won't work if J is spin, but you can try it for L.

 

[jensa]

Hello,

Is it generally the case that [J, H] = dJ/dt?

I saw this appear in a problem involving a spin 1/2 system interacting with a magnetic field.

If so, why?This seems like a very basic relation but I'm having a bit of brain freeze and can't see the answer right now.

This is just the equation of motion of an operator in the Heisenberg picture, isn't it?

 

[blagershod.smee]

This is just the equation of motion of an operator in the Heisenberg picture, isn't it?

Okay, so in the Heisenberg scheme I can say that for any operator O, [O, H] = dO/dt?

Is this a postulate or derived from something else?

 

[jensa]

Okay, so in the Heisenberg scheme I can say that for any operator O, [O, H] = dO/dt?

Is this a postulate or derived from something else?

It's a dynamical equation which is equivalent to the Schrödinger equation. In the Schrödinger picture the states change with time, in the Heisenberg picture the operators change with time.

http://en.wikipedia.org/wiki/Heisenberg_picture

 

[blagershod.smee]

It's a dynamical equation which is equivalent to the Schrödinger equation. In the Schrödinger picture the states change with time, in the Heisenberg picture the operators change with time.

http://en.wikipedia.org/wiki/Heisenberg_picture

Thanks for the link. I think my confusion arose because I'm used to thinking of the Hamiltonian in terms of the Schrödinger picture and have not gotten used to the Heisenberg formulation.