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Homework Statement
A particle is in the ground state of a simple harmonic oscillator, potential → V(x)=[itex]\frac{1}{2}[/itex]mω[itex]^{2}[/itex]x[itex]^{2}[/itex]
Imagine that you are in the ground state |0⟩ of the 1DSHO, and you operate on it with the momentum operator p, in terms of the a and a† operators. What is the eigenstate? Then, how would you represent this operation in terms of a probability density in momentum space?
Homework Equations
p=i[itex]\sqrt{\frac{\hbar m ω}{2}}[/itex](a†-a)
Fourier Transform, x→p-space
The Attempt at a Solution
p|0⟩=i[itex]\sqrt{\frac{\hbar m ω}{2}}[/itex]|1⟩
So now the particle is in the first excited state of the SHO
But I don't understand the next part of the problem. How can I represent this terms of probability density in p-space?
I can perform the Fourier Transform and find that,
[itex]\Phi[/itex](p,t)=[itex]\sqrt{2}[/itex]([itex]\frac{\pi \hbar}{m ω}[/itex])[itex]^{\frac{1}{4}}[/itex] e[itex]^{\frac{-p^{2}}{2 \hbar m ω} - \frac{i t ω}{2}}[/itex]
And the probability density is,
|[itex]\Phi[/itex](p,t)|[itex]^{2}[/itex]=2[itex]\sqrt{\frac{\pi \hbar}{m ω}}[/itex] e[itex]^{\frac{-p^{2}}{\hbar m ω}}[/itex]
I don't believe I understand what this part of the question is asking. Any suggestions/ideas would be greatly appreciated.