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Expected Value of Hamiltonian in a Forced Quantum Harmonic Oscillator.

  1. Apr 22, 2012 #1
    1. The problem statement, all variables and given/known data
    Given an initial [itex](t=-∞)[/itex] Fock state , [itex]\left|n\right\rangle[/itex], and a function [itex]f(t)[/itex], where [itex]f(±∞)=0[/itex], show that for a Harmonic Oscillator perturbed by [itex]f(t)\hat{x}[/itex] the difference [itex]\left\langle H(+∞) \right\rangle - \left\langle H(-∞) \right\rangle[/itex] is always positive.


    2. Relevant equations
    The Hamiltonian will be [itex]H(t) = \hbar[/itex][itex]ω[/itex][itex](a^{\dagger}a+\frac{1}{2})[/itex] [itex]-f(t)(a+a^{\dagger})[/itex], where I've transfered the constant of the creation and annihilation operators to the function.
    At [itex]t=-∞[/itex] the initial state is [itex]\left|\psi(-∞)\right\rangle = \left|n\right\rangle[/itex] and [itex]f(-∞)=0[/itex], so it's the unperturbed Hamiltonian and [itex]\left\langle H(-∞) \right\rangle = (n + \frac{1}{2})\hbarω[/itex].
    Also, in Heisenberg's picture, [[itex]a(t), a^{\dagger}(t)[/itex]] = 1.

    3. The attempt at a solution
    My professor has suggested me to solve the Langevin equation for expected value or use Heisenberg's picture.

    The first takes me to:
    [itex]\left\langle H(+∞) \right\rangle - \left\langle H(-∞) \right\rangle = ∫^{∞}_{-∞} \left\langle ∂_{t}H(t) \right\rangle dt[/itex], and I am unable to go any farther since we don't have any other information about [itex]f(t)[/itex].

    The last takes me to:
    [itex] i\hbar \frac{d a(t)}{dt} = [a(t),H(t)] = -f(t) + \hbarω a(t)[/itex], which can be solved as [itex] a(t) = a(t_{0})e^{-iω(t-t_{0})} + \frac{i}{\hbar}∫^{t}_{t_{0}} f(t')e^{-iω(t-t')}dt'[/itex]. I assumed that when I take [itex]t_{0} = -∞[/itex] the first term of the sum is [itex]a[/itex], the annihilation operator. The same can be done to the creation operator. However, how can I use these integrals to evaluate [itex]\left\langle H(+∞) \right\rangle[/itex]? How to deal with the the integration when I substitute t for +∞?
    I expected the time derivative of [itex]f(t)[/itex] to appear somehow, so that the boundary could be used, but I can't seem to get that. Will it ever appear?
    Another idea I had was that since the perturbation ceases to act upon the system I could use the adiabatic approximation, but that would mean the state is the same and the difference between the expected values is merely zero.

    Am I missing something? Is there a way to use the results I've already found?
    Thanks in advance.
     
    Last edited: Apr 22, 2012
  2. jcsd
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