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Question about the Heisenberg Picture

  1. Oct 15, 2011 #1
    1. The problem statement, all variables and given/known data

    I've seen this example for using the Heisenberg equation of motion to solve the Simple Hamonic Oscillator.


    However, if you were only interested in one variable, lets say position, on how the the position operator varies in time and its resulting expectation values: Would is it neccessary to find and solve the equation of motion for P as well?

    Also once one has the relevant equation of potion:

    [tex]\frac{dX(t)}{dt} = \frac{p(0)}{m} [/tex]

    Why is it so often solved by differentiating the equation a second time and solving, rather than just integrating?

    2. Relevant equations

    3. The attempt at a solution
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
  3. Oct 15, 2011 #2


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    The equations for the SHO are
    \dot{x}(t) &= \frac{p(t)}{m} \\
    \dot{p}(t) &= -m\omega^2x(t)
    \end{align*}Note that you have functions of time on both sides of the equations. You don't have p(0) on the right-hand side of the x equation. You differentiate one equation and substitute in the second one to get rid of one of the functions. The resulting second-order equation is straightforward to solve.
  4. Oct 15, 2011 #3
    Thanks for the reply, Vela.

    I was thinking about it earlier as I walked the dog, but hadn't got around to sitting down with it. I did wonder if assuming that the answer in the commutator was p(0) was unfounded.

    Glad to have someone confirm it and that'd I'm not a complete idiot, just a bit slow :D
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