Energy and position measurements of the QHO

You have a collection of identically prepared quantum harmonic oscillator systems. Measuring the energy of those system yields $\dfrac{\hbar\omega}{2}$ and $\dfrac{3\hbar\omega}{2}$ in equal amounts.

You measure the position of these systems and find the mean position changes in time: $\braket{x} = -\sqrt{\dfrac{\hbar}{2m\omega}}\sin \omega t$.

We will use this information to determine the expectation value of the momentum, $\braket{p}$.

  1. The energy measurements are time independent, so we can determine the coefficients for $\vert \psi(0)\rangle$. Based on our energy measurements, write down $\vert \psi(0)\rangle$.
    • Are you able to determine the relative phase of the two states based on energy measurements? Why or why not?
  2. Introduce a phase for each state (e.g., $e^{i\theta_0}$ for $\ket{0}$). Now using the Schroedinger equation, time evolve the state to find $\ket{\psi(t)}$.
  3. To find the relative phase, let’s use the expectation value of the position, which has no overall phase. Compute the expectation value of the position, $\braket{x}$, given your time evolved state.
    • There are a number of way to do this calculation. What is the best approach to doing this calculation? Why?
    • Given our , what must be the difference between the phases be (i.e., $\theta_0 - \theta_1$)?
  4. Now, find $\braket{p}$. Again, what is the best approach for doing this?