Momentum perturbation to harmonic oscillator

[m1rohit]

Homework Statement

the problem and a possible solution(obtained from a book) is attached as a pdf to the post.However Iam unable to understand it.Please download the attachment.

Homework Equations

equation no (2) in the pdf.Is there any use of space translation operator in here.

The Attempt at a Solution

I have solved this problem using perturbation theory.however Iam intrigued by the method used here.

 

[DrClaude]

It can be understood using the properties of the Fourier transform. Since $x$ and $p$ are canonically conjugate, transformation from $\psi(x)$ to $\psi(p)$ is done by a Fourier transform. And from their properties, it is known that shifting the zero in one variable will be equivalent to a complex phase shift in the other.

 

[m1rohit]

Thank you very much for your reply.can you refer me any link or book where i could get problems on translation operators.

 

[DrClaude]

I don't if it's the right level for you, but you can take a look at W. Greiner & B. Müller, Quantum Mechanics: Symmetries (Springer).