Is This the Correct Solution for Sakurai Ch 2 Problem 14.b?

[malawi_glenn]

Homework Statement

Consider a one-dim harm osc; start with the Schrödinger equation (SE) for the state vector, then derive the SE for the momentum-space wave function.

The Attempt at a Solution

My answer is this, all primed letters are numbers (as in sakurai notation). Its going to take a while for me to do all my steps, so before I do all that; I just want to see if someone here can confirm my answer as the right one (sakurai is a popular book to teach from). If it is now the correct answer, I will post all the steps I've done so you guys can put me in right direction. Thanx in advance physics fellows!

$$\left(\dfrac{(p')^2}{2m} - \dfrac{\hbar ^2 \omega ^2 m }{2} \dfrac{\partial^{2}}{(\partial p')^2} \right) \psi _{\alpha}(p') = i \hbar \dfrac{\partial}{\partial t}\psi _{\alpha}(p')$$

I have found many answers to sakurai on the web, but not to this one; and our teacher said that this was a good problem to do.

 

[cristo]

That looks correct to me.

 

[malawi_glenn]

That looks correct to me.

Thanx a lot for your answer!