Driven harmonic oscillator [closed]

Given the Hamiltonian of a loaded particle

$$\hat H = \frac{\hat p^2}{2m}+eE(t) \hat x + \frac{1}{2}m\omega^2 \hat x^2$$ show that

1. The time dependent expected values $\langle \hat p\rangle$ and $\langle \hat x\rangle$ satisfy the classical laws of motion

2. Discuss the time dependence of the expected values for the case $E(t) = E_0 \sin(\Omega t)$.

I used Ehrenfest's theorem $(d/dt)\langle \hat A \rangle = (i/h) \langle [\hat H,\hat A]\rangle + \langle (\partial/\partial t)(\hat A)\rangle$ to show 1) but I don't know how to do 2). I know the definition of expected values $\langle \hat A \rangle = \langle \psi|A \psi \rangle$ where $\psi$ is the state of the particle.

closed as off-topic by John Rennie, Kyle Kanos, ACuriousMind♦, user36790, DanuApr 2 '16 at 8:39

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – John Rennie, Kyle Kanos, ACuriousMind, Community, Danu
If this question can be reworded to fit the rules in the help center, please edit the question.

• Welcome to Physics.SE! Check out the site's policy on homework questions. For (2), it seems to me that you can simply substitute for $E(t)$ in $\langle p(t) \rangle$ and $\langle x(t) \rangle$. – Abhinav Mar 31 '16 at 21:31
• @Abhinav First of all I´m sorry for the false tag of the question. – user326049 Apr 1 '16 at 5:25
• The problem with your hint for me is, and yes this sounds really dump, which state $\psi$ should I use in the expression for $<\hat p>$?It seems to me that I can´t simply use a gaussian wave? – user326049 Apr 1 '16 at 5:33

Ehrenfest's theorem proves that expectation values of $\hat{x}$ and $\hat{p}$ obey classical equations of motion in general. This is true regardless of whether the system you are considering is that of a driven harmonic oscillator or not.

The difference in the case of a driven harmonic oscillator, is that solutions to the Schrodinger Equation are proportional to coherent states. The expectation values of $\hat{x}$ and $\hat{p}$ in coherent states obey the equations of motion for a classical harmonic oscillator.

• 1) ok thats the same result that I received thank you 2) And this is the point in the problem that confuses me: "must I solve the Schrödinger equation for this point?" – user326049 Apr 1 '16 at 5:35