Momentum perturbation to harmonic oscillator

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Homework Help Overview

The discussion revolves around a problem related to momentum perturbation in a harmonic oscillator, with a focus on understanding the application of the space translation operator as presented in an attached PDF. The original poster expresses difficulty in comprehending the solution provided in the document.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the relationship between the Fourier transform and the properties of canonical conjugate variables, questioning the implications of shifting variables and the corresponding effects on wave functions.

Discussion Status

Some participants have offered insights into the mathematical properties involved, while others seek further resources to deepen their understanding of translation operators. There is an ongoing exploration of the concepts without a clear consensus on the best approach.

Contextual Notes

The original poster has referenced a specific equation from the PDF and is particularly interested in the role of the space translation operator, indicating a potential gap in foundational knowledge or context that may affect their understanding.

m1rohit
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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.
 

Attachments

Last edited:
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It can be understood using the properties of the Fourier transform. Since [itex]x[/itex] and [itex]p[/itex] are canonically conjugate, transformation from [itex]\psi(x)[/itex] to [itex]\psi(p)[/itex] 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.
 
Thank you very much for your reply.can you refer me any link or book where i could get problems on translation operators.
 
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).
 

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