Fourier analysis and continuous spectra

[ThereIam]

So I've been self-studying from Griffiths Intro to QM to get back in shape for graduate school this fall, and I guess I'd just like some confirmation that I'm on the right track...

So while I am sure there are many other applications, the one I am dealing with is eigenfunctions of an operator having a continuous spectrum of eigenvalues, making the eigenfunctions non-normalizable.

So from the top, qualitatively, as I understand it, essentially how this works if one has solved the eigenvalue equation for a particular operator, and this yields an eigenfunction which is at first glance non-normalizable (not in Hilbert space). However, one notes that some linear combination of the eigenvectors CAN be normalized, and the "linear combination" manifests as an integral over some function (called the "expansion coefficient") * the eigenvectors, all of which we want to be equal to some ψ(x). I guess we're trying to figure out a way to express ψ(x) as a linear combination of it's eigenvectors (which in the case of QM would tell you a lot about the state of the system ψ(x) represents)

Let's say the eigenvectors are of the form e^(ikx) so
ψ(x) = ∫Θ(k)*e^(ikx)dk

Then in terms of Fourier stuff, we're seeking to exploit the fact that e^(-imx) is orthogonal to e^(ikx) unless m = k. So we say
Θ(k) = ∫ψ(x)*e^(-ikx)dx

And that Θ(k) can be plugged back into the first equation for ψ(x)...

More generally, one the point is just that we're exploiting the orthogonality of the e^\pmkx or the orthogonality of whatever function is present in the integral in order to find the expansion coefficient... right?

Am I hopeless?

 

[strangerep]

More generally, one the point is just that we're exploiting the orthogonality of the e^\pmkx or the orthogonality of whatever function is present in the integral in order to find the expansion coefficient... right?

That's the idea.

I think you would enjoy reading about the "Rigged Hilbert Space" formulation of modern quantum mechanics. Ballentine has a gentle introduction in ch1 of his textbook. A more extensive introduction can be found in this paper by Rafael de la Madrid. If you want even more, search for his PhD Thesis on Google Scholar.

Am I hopeless?

I see no reason whatsoever for that conclusion.