Solve Eigenvalue Problem for Displacement Operator D

[quasar_4]

Homework Statement

The displacement operator D is defined by the equation D f(x) = f(x + a). Show that the eigenfunctions of D are of the form phi = exp(Bx)*g(x) where g(x+a) = g(x) and B is any complex number. What is the eigenvalue corresponding to phi?

Homework Equations

Postulates of quantum mechanics?
Physicist version of the eigenvalue problem?

The Attempt at a Solution

Unfortunately I'm used to tacking eigenvalue problems from a more mathematical standpoint, i.e., considering whether a linear operator is diagonalizable, finding the characteristic polynomial, checking dimensions of eigenspaces, etc. I'm completely new to QM and don't understand how the eigenvalue problem has changed (plus all the operators are hermitian, so don't we already know the thing is diagonalizable from spectral theory?). I think with this book's terminology, eigenfunction = eigenvector (the vectors are themselves functions, right?), but I'm still confused as to how we find the eigenvectors first and use them to obtain eigenvalues.

With the linear momentum operator, one can turn the eigenvalue problem into a simple ODE and solve for the eigenfunction, but I'm not sure what to do here. I'm not even sure how to start. Do you set this up as D phi(x) = phi(x +a) = f phi(x) and try to solve for phi? I'm so confused. Any help would be great!

 

[CompuChip]

Well, I'd start by just checking that the given functions actually are eigenfunctions, from which you will get the eigenvalue for free.

The question I can't answer right away, is why these are all. You could try to show that they form a complete basis for the set of functions D acts on?