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Angular momentum and Hamiltonian commutator

  1. Jan 11, 2010 #1
    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.
     
  2. jcsd
  3. Jan 11, 2010 #2

    Matterwave

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    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.
     
  4. Jan 11, 2010 #3
    This is just the equation of motion of an operator in the Heisenberg picture, isn't it?
     
  5. Jan 11, 2010 #4
    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?
     
  6. Jan 11, 2010 #5
    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
     
  7. Jan 11, 2010 #6
    Thanks for the link. I think my confusion arose because I'm used to thinking of the Hamiltonian in terms of the Schrodinger picture and have not gotten used to the Heisenberg formulation.
     
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