Expected Value of Hamiltonian in a Forced Quantum Harmonic Oscillator.

[Helvecius]

Homework Statement

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.

Homework Equations

The Hamiltonian will be $H(t) = \hbar$$ω$$(a^{\dagger}a+\frac{1}{2})$ $-f(t)(a+a^{\dagger})$, where I've transferred 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.

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?
Thanks in advance.

 

[NateD]

A:The solution is to use the adiabatic theorem. The difference between the energies at $t=+\infty$ and $t=-\infty$ is the same as the difference between the adiabatic eigenenergies of the two Hamiltonians. This can be proven using the adiabatic theorem.The adiabatic eigenenergies (and eigenstates) are given by the solutions to$$H(-\infty)\Psi_n^{(-\infty)} = E_n^{(-\infty)}\Psi_n^{(-\infty)},$$and similarly for the Hamiltonian at $t=+\infty$. Since the Hamiltonians are different, the eigenenergies will be different and the difference between them is$$E_n^{(+\infty)} - E_n^{(-\infty)} = \int_{-\infty}^{+\infty}\frac{d\langle\Psi_n|H'|\Psi_n\rangle}{dt}dt = \int_{-\infty}^{+\infty}\langle\Psi_n|H'|\Psi_n\rangle dt,$$where I have introduced the notation $H'=\frac{\partial H}{\partial t}$. Since $f(\pm\infty)=0$ it follows that $\langle\Psi_n|H'|\Psi_n\rangle=0$ for $t=\pm\infty$, so the integral is always positive.