Ladder operater for momentum space wavefunction (harmonic oscillator)

[Ned Stark]

Homework Statement

I need to find the momentum space wavefuntion Phi(p,t) for a particle in the first excited state of the harmonic oscillator using a raising operator.

Homework Equations

Phi_1(p,t)= "raising operator" * Phi_0 (p,t)

The Attempt at a Solution

In position space, psi_1 (x) = a+psi_0(x), where a+=1/sqrt(2pi*omega*h-bar)*(-ip+mwx)

but in position space, x is replaced with i*h-bar*d/dp, and the operator for p is just p.

I tried inserting the new position operator in a+, but what I get is, what looks like a good solution, multiplied by -i.

I'm not sure if wave functions can have a factor of i out front (i think the can, please correct me)

I really just need help formulating this ladder operator in p-space, I can take care of the rest

 

[mark726]

.
Thank you for your question! I understand your interest in finding the momentum space wavefunction for a particle in the first excited state of the harmonic oscillator using a raising operator. Here is a step-by-step guide to help you formulate the ladder operator in momentum space:

1. Start with the position space wavefunction for the first excited state: psi_1 (x) = a+psi_0(x), where a+ is the raising operator.

2. Use the momentum operator, p, in place of x in the position space wavefunction: psi_1 (p) = a+psi_0(p).

3. Substitute the momentum operator, p, with i*h-bar*d/dp: psi_1 (p) = a+psi_0(i*h-bar*d/dp).

4. Apply the raising operator, a+, to the position space wavefunction: psi_1 (p) = 1/sqrt(2pi*omega*h-bar)*(-ip+mwx)*psi_0(i*h-bar*d/dp).

5. Simplify the expression by distributing the operator and substituting for omega and m: psi_1 (p) = -i/sqrt(2*h-bar*m*w)*(d/dp + m*w*p)*psi_0(i*h-bar*d/dp).

6. Finally, use the momentum space wavefunction for the ground state, psi_0(p) = 1/sqrt(2pi*h-bar)*exp(-p^2/2*h-bar*m*w), to obtain the momentum space wavefunction for the first excited state: psi_1 (p) = -i/sqrt(2*h-bar*m*w)*(d/dp + m*w*p)*1/sqrt(2pi*h-bar)*exp(-p^2/2*h-bar*m*w).

I hope this helps you in formulating the ladder operator in momentum space. Please let me know if you have any further questions or need clarification on any step. Good luck with your research!