A particle of mass m in the one-dimensional harmonic oscillator is in a state for which a measurement of the energy yields the values hω/2 or 3hω/2, each with a probability of one-half. The average values of the momentum <p> at time t = 0 is √mωh/2. This information specifies the state of the particle completely. What is this state and what is <p> at time t?
H|E> = E|E>
|ψ(0)> = 1/√2(|0>) + 1/√2(|1>)
p = -i√mωh/2(a - aτ)
The Attempt at a Solution
The first step of the problem (I thought) was basically verifying that the expectation value of momentum on the initial state should satisfy √mωh/2. I began doing this by doing <ψ(0)|p|ψ(0)>. I changed the momentum operator into the energy eigenstate lowering/raising operators then began cranking out the algebra.
At the end I kept getting 0, which is obviously wrong since the problem states it should be √mωh/2. Can anybody tell me what I'm doing wrong? I have a feeling it depends on how I made my |ψ (0)> (listed in relevant equations). Is there a phase I need to be worried about? If there is, I'm confused why I need to add in a phase to this problem, so an explanation of that would be nice =). Thanks.
(all h's are supposed to be hbars).