Time Dependent expectation value in momentum space

[spaderdabomb]

Homework Statement

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?

Homework Equations

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).

 

[TSny]

The coefficients of |0> and |1> in |ψ(0)> need not be real. You can always choose an overall phase factor for the wavefunction to make one of the coefficients real, but you should allow for the other coefficient to be complex.

 

[spaderdabomb]

Ah thank you, I see it now =)