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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.

    http://en.wikipedia.org/wiki/Heisenberg_picture#Commutator_relations"

    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

    vela

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    The equations for the SHO are
    \begin{align*}
    \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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