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Physics
Quantum Physics
Rigorous transition from discrete to continuous basis
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[QUOTE="bhobba, post: 6014997, member: 366323"] What you need to study is called Rigged Hilbert Spaces - which is graduate level math. It has been discussed here a few times eg: [URL]https://www.physicsforums.com/threads/rigged-hilbert-spaces-in-quantum-mechanics.917768/[/URL] The above gives a non-rigorous presentation of what you want as well as a link to a very rigorous PhD thesis on it - not to be touched until you are advanced in the area of math known as functional analysis. First though you need to understand distribution theory - for that I recommend: [URL]https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20[/URL] PLEASE PLEASE - I do not know how strongly I can recommend it - get a copy and study it. It will help you in many areas of physics and applied math in general. It makes Fourier transforms a snap, for instance, otherwise you become bogged down in issues of convergence - the Distribution Theory approach bypasses it entirely. Once you have done that go through the following: [URL]https://www.amazon.com/dp/0821846302/?tag=pfamazon01-20[/URL] That will explain it all in terms that is mathematically exact. Then, do some functional analysis. I suggest the following: [URL]http://matrixeditions.com/FunctionalAnalysisVol1.html[/URL] Unfortunately these days only available as an ebook - but still my favorite. Sorry this question has no easy answer. You are to be congratulated for attempting it. The solution defeated the great Von-Neumann and Hilbert - it took the combined efforts of other great 20th century mathematicaians to crack it - namely - Gelfland, Grothendieck, and Schwartz (probably others as well - it was a toughy) to crack it. Thanks Bill [/QUOTE]
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Quantum Physics
Rigorous transition from discrete to continuous basis
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