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Harmonic osccillator: solution for A in Y'AY

  1. Nov 7, 2011 #1
    1. The problem statement, all variables and given/known data

    What is A in [itex]\bar{\varphi}[/itex]A[itex]\varphi[/itex], if

    [itex]\bar{\varphi}[/itex]A[itex]\varphi[/itex] = [itex]\frac{-ip(\tau)\dot{q}(\tau)}{\hbar}[/itex]+[itex]\frac{p^{2}(\tau)}{2m}[/itex]+[itex]\frac{m\omega^{2}}{2}[/itex]q[itex]^{2}[/itex]([itex]\tau[/itex])

    2. Relevant equations

    provided that [itex]\bar{\varphi}[/itex] and [itex]\varphi[/itex] are ladder operators of the form:

    [itex]\varphi[/itex] = [itex]\left(\frac{m\omega}{2\hbar}\right)^{1/2}[/itex][itex]\left(q\left(\tau\right)+\frac{ip\left(\tau\right)}{m\omega}\right)[/itex]

    [itex]\bar{\varphi}[/itex] = [itex]\left(\frac{m\omega}{2\hbar}\right)^{1/2}[/itex][itex]\left(q\left(\tau\right)-\frac{ip\left(\tau\right)}{m\omega}\right)[/itex]

    p is the momentum and q is the position in real space,

    3. The attempt at a solution
    A possible solution might be to extract all those variable to get A, but [itex]\bar{\varphi}[/itex] and [itex]\varphi[/itex] are operators, so i am in complete darkness here, i hope you have a possible solution to this, thank you.
  2. jcsd
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