Originally posted by amolv06I barely passed linear algebra and differential equations...sorry!
I'm having a hard time proving that A is hermitian where A acting on f gives the integral from a to x of f(x'dx'. Could anyone help me out? I can't seem to force hermiticity after doing <g|Af> by parts, but I'm not sure if I've gone as far as I can. It's kind of hard to post the math I have tried here, but if anyone could give me some advice, I'd appreciate it.
Originally posted by KazetNagorra"How do you define your inner product?"
How do you define your inner product? If you use the L2 definition of an inner product, then A is not hermitian at all if I'm not mistaken... (take e.g. g(x) = x, f(x) = x^2, a = 0)
I could be wrong though, I'm a bit rusty on this subject.
Originally posted by amolv06If you want to show A is not hermitian, then showing a single example where it does not apply suffices.
"How do you define your inner product?"
Correct, using the L2 definition. Both f and g lie in Hilbert space, so they are complex. I must show that A is either hermitian or not hermitian more generally. I do not think the professor would allow us to select specific vectors. I think I've got it, though. A is non-hermitian, as <f|Ag>=-<Af|g>.
Originally posted by amolv06Oh I get what you mean, yeah I think you can see it that way.
I am sorry, this is all very new to me, but I don't quite understand. Perhaps this is just semantics, but wouldn't normalizability be a consequence of the convergence of the inner product of the wavefunction with itself, rather than the other way around?
Furthermore, I understand that when referring to a localized particle, the wave function vanishes o ...[text shortened]... is very possible, even likely, that I'm confusing some of the terminology and/or concepts.