Eigenfunctions, eigenstates and eigenvalues

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Homework Help Overview

The problem involves the operator A_hat defined as exp(b*(d/dx)) and its relationship to the eigenstate ψ(x) with eigenvalue λ. The inquiry focuses on the behavior of ψ(x) as x increases by multiples of b.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the eigenvalue equation and its implications for the function ψ(x). There is an attempt to express the operator's action on shifted arguments of ψ(x) and to understand the dependence of ψ(x) on x.

Discussion Status

Some participants have provided guidance on writing the eigenvalue equation and the use of LaTeX for clarity. There is an acknowledgment of progress made by one participant, though details of their findings remain unspecified.

Contextual Notes

One participant notes the importance of correctly interpreting the operator and its action on the eigenstate, while another emphasizes the need for clarity in mathematical expressions within the context of quantum mechanics.

Harper
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Homework Statement


The problem states consider A_hat=exp(b*(d/dx)). Then says ψ(x) is an eigenstate of A_hat with eigenvalue λ, then what kind of x dependence does the function ψ(x) have as x increases by b,2b,...?

Homework Equations

The Attempt at a Solution


Started out by doing (A_hat)ψ(x+b), turned that into (A_hat)ψ(x)+(A_hat)ψ(b). Not sure where to go from there and/or how to incorporate λ.
 
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Did you write out the eigenvalue equation?
Note: you are asked what happens with ##\psi(x)\to\psi(x+nb): n=1,2,3,\cdots##
ie - how does ##\psi## depend on ##x##?

Do I read this correctly: ##\hat A = e^{b\frac{d}{dx}}## ??
 
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Yes you read it correctly and no I didn't write that out.
 
Hey thanks for the tool and I think I have it figured out. Thank you for the help.
 
Well done - what did you figure out (just for other people stuck on the same thing...)?
 

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