## Expected Value of Hamiltonian in a Forced Quantum Harmonic Oscillator.

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

2. Relevant equations
The Hamiltonian will be $H(t) = \hbar$$ω$$(a^{\dagger}a+\frac{1}{2})$ $-f(t)(a+a^{\dagger})$, where I've transfered the constant of the creation and annihilation operators to the function.
At $t=-∞$ the initial state is $\left|\psi(-∞)\right\rangle = \left|n\right\rangle$ and $f(-∞)=0$, so it's the unperturbed Hamiltonian and $\left\langle H(-∞) \right\rangle = (n + \frac{1}{2})\hbarω$.
Also, in Heisenberg's picture, [$a(t), a^{\dagger}(t)$] = 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:
$\left\langle H(+∞) \right\rangle - \left\langle H(-∞) \right\rangle = ∫^{∞}_{-∞} \left\langle ∂_{t}H(t) \right\rangle dt$, and I am unable to go any farther since we don't have any other information about $f(t)$.

The last takes me to:
$i\hbar \frac{d a(t)}{dt} = [a(t),H(t)] = -f(t) + \hbarω a(t)$, which can be solved as $a(t) = a(t_{0})e^{-iω(t-t_{0})} + \frac{i}{\hbar}∫^{t}_{t_{0}} f(t')e^{-iω(t-t')}dt'$. I assumed that when I take $t_{0} = -∞$ the first term of the sum is $a$, the annihilation operator. The same can be done to the creation operator. However, how can I use these integrals to evaluate $\left\langle H(+∞) \right\rangle$? How to deal with the the integration when I substitute t for +∞?
I expected the time derivative of $f(t)$ 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?
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 Tags harmonic oscillator, perturbation, quantum mechanics