Can someone please explain overlap integrals to me?

[randybryan]

I'm studying for my Quantum Mechanics course and I've understood most of it so far, until we reached the quantum harmonic oscillator. Finding the solution was no easy feat and I have to say I only had a slender grasp of the derivation, but now I'm being given questions about certain energy states that need overlap integrals. I have nothing in my nots about them. I know they're the complex conjugate of one function multiplied by another function and is equivalent to the dot product of two vectors, but where does it come from and how is it applied to the Q harmonic oscillator?

 

[SpectraCat]

I'm studying for my Quantum Mechanics course and I've understood most of it so far, until we reached the quantum harmonic oscillator. Finding the solution was no easy feat and I have to say I only had a slender grasp of the derivation, but now I'm being given questions about certain energy states that need overlap integrals. I have nothing in my nots about them. I know they're the complex conjugate of one function multiplied by another function and is equivalent to the dot product of two vectors, but where does it come from and how is it applied to the Q harmonic oscillator?

Hmm ... normally one wouldn't talk about overlap integrals in the case of a single harmonic oscillator problem, right? Since all the eigenstates are orthogonal, all of the overlap integrals (which you characterized properly above, by the way), should either be 1 or 0, right?

So, are you by any chance working on a problem to calculate Franck-Condon factors? Or some other problem where you have multiple harmonic oscillators with different minima to deal with?