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, Danu Apr 2 '16 at 8:39

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  • 2
    $\begingroup$ 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 $\langle x(t) \rangle$. $\endgroup$ – Abhinav Mar 31 '16 at 21:31
  • $\begingroup$ @Abhinav First of all I´m sorry for the false tag of the question. $\endgroup$ – user326049 Apr 1 '16 at 5:25
  • $\begingroup$ 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? $\endgroup$ – 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.

  • $\begingroup$ 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?" $\endgroup$ – user326049 Apr 1 '16 at 5:35

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