Eigenfunctions, eigenstates and eigenvalues

[Harper]

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.

 

[Simon Bridge]

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: $\hat A = e^{b\frac{d}{dx}}$ ??

 

[Harper]

Yes you read it correctly and no I didn't write that out.

 

[Simon Bridge]

OK - so how does the eignenvalue equation work?
Notes on writing equations in PF: https://www.physicsforums.com/help/latexhelp/ ... since you are doing QM, LaTeX is an essential tool.

 

[Harper]

Hey thanks for the tool and I think I have it figured out. Thank you for the help.

 

[Simon Bridge]

Well done - what did you figure out (just for other people stuck on the same thing...)?