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.