The QHO and Ehrenfest’s Theorem

One of the things we learned in Chapter 5 was that quantum mechanics expectation values follow classical laws (this is the root of Ehrenfest’s Theorem). In this activity, we will work through some of the mechanics of calculations for a QHO, and use them to verify Ehrenfest’s Theorem.

A particle in a harmonic well is found to be in the initial state:

\[\ket{\psi(t=0)} = A \left[\ket{0} + 2e^{i\pi/2}\ket{1}\right]\]
  1. We need the time evolved state to calculate any expectation values. Determine $\ket{\psi(t)}$. Make sure that your state is normalized.
  2. Determine the expectation value of the position, $\braket{x}$. There are a number of ways to do this, choose one that makes most sense to your group. Which approach did you choose and why?
  3. Determine the expectation value of the momentum, $\braket{p}$. Again, there are a number of ways to do this, choose one that makes most sense to your group. Which approach did you choose and why?
  4. Show that your expressions satisfy Ehrenfest’s Theorem.