Please explain what this bra-ket expression means

[jbb]

I am reading a paper that uses a quantum mechanical notation that I do not understand. I have found a webpage that explains it, but I do not understand the explanation either:
http://chsfpc5.chem.ncsu.edu/~franzen/CH ...

(let Y represent the Psi)
Specifically, I understand what <Yn|A|Yn> stands for, but I don't understand the significance when the bra and the ket represent different wavefunctions: <Yn|A|Ym>.

The above website states "it gives the average of the physical quantity corresponding to operator A for the wave functions". This confuses me. The expectation value of a single wave function is a concept I understand, but what is the meaning of the expectation value of two wave functions simultaneously? Does it have to do with transition probabilities?

If you could explain this to me or give me a reference, that would be much appreciated.

 

[f95toli]

The expectation value of a single wave function is a concept I understand, but what is the meaning of the expectation value of two wave functions simultaneously? Does it have to do with transition probabilities?

Yes, it can of course have different meaning in different context (depending on what A is) but when e.g. calculating transition probabilites it is the matrix element that tells you the probability amplitude that a system in state Ym will make a transition in into state Yn when perturbed by the operator A.
In pertubation theory "A" is essentially the hamiltonian that couples the two system together and can e.g. represent coupling via the electric field.
Look up e..g "Fermi's Golden Rule" (I think there is a wiki page) if you want to know more.

 

[jbb]

Yes, it can of course have different meaning in different context (depending on what A is) but when e.g. calculating transition probabilites it is the matrix element that tells you the probability amplitude that a system in state Ym will make a transition in into state Yn when perturbed by the operator A.
In pertubation theory "A" is essentially the hamiltonian that couples the two system together and can e.g. represent coupling via the electric field.
Look up e..g "Fermi's Golden Rule" (I think there is a wiki page) if you want to know more.

Thank you very much! Fermi's Golden Rule explains all my questions.