nigelscott said:
Re: Reply #3 <x|y> and <x|p> are interpreted as inner products correct? Thus, I assume these can expanded in terms of integrals involving delta functions. <x|p> is easy to figure out but I don't understand how to get δ(x-y) for <x|y>.
You have hit upon something actually quite difficult here - the use of Delta Functions in QM. Basically they do not belong to a Hilbert Space but to something called a Rigged Hilbert Space and coming to grips with that is a whole other story - you need to be very adept at analysis and delve into tombs like Gelfand and Vilenkin - Generalized Functions - tough going even for guys like me into that sort of stuff - but it must be said not impossible if you persevere.
At the beginner level however its best to simply view Delta Functions as an ordinary function that for all practical purposes behave like a delta function - but really isnt. Under that view <x|y> = integral delta (x-x') delta (y-x') dx' = delta (x-y).
Basically the idea of a Rigged Hilbert Space is this. In a Hilbert space H bras and kets can be put into one to one correspondence ie its vectors and the linear functions defined on those vectors - called its dual -can be put in one to one correspondence. But if instead of a Hilbert space (which consists of all square integrable functions) you consider the space of all test functions (which is a subset of square integrable functions) then its dual T* (ie the space of all linear functions defined on that space) is much larger than H and includes stuff like the delta function. So what you have is T subset of H subset of T* - which is called a Gelfland Triple and the basis of Rigged Hilbert Spaces.
For a bit of an introduction to Rigged Hilbert Spaces check out (which also explains what a test function is if you do not know it):
http://en.wikipedia.org/wiki/Distribution_(mathematics )
http://www.abhidg.net/RHSclassreport.pdf
As to its physical meaning that is contained in a theorem you won't find mentioned that often in textbooks but IMHO is very fundamental - Gleason's Theorem which basically says Born's rule is the only way you can define probabilities on a Hilbert Space.
http://en.wikipedia.org/wiki/Gleason's_theorem
In fact from the assumption that QM is theory whose pure states are complex vectors all of QM basically follows. Check out:
http://arxiv.org/pdf/quant-ph/0111068v1.pdf
Its slightly different to the way I mentioned (from Gleasons Theorem) but it gives you the gist.
Thanks
Bill