Rigorous transition from discrete to continuous basis

[Alex Cros]

Hi all,

I'm trying to find a mathematical way of showing that given a complete set $$\left |a_i\right \rangle_{i=1}^{i=dim(H)}∈H$$ together with the usual property of $$\left |\psi\right \rangle = ∑_i \left \langle a_i\right|\left |\psi\right \rangle\left |a_i\right \rangle ∀ \left |\psi\right \rangle∈H$$. Now, by letting the set $$\left | a_i \right \rangle_{i=1}^{i=dim(H)} → \left |a_i\right \rangle_{i=1}^{i=∞}$$ and $$\left |a_{i+1}\right \rangle = \left |a_i\right \rangle+\left |δ\right \rangle$$ as $$ δ→0$$ (in the sense of $$\left |a_{i+1}\right \rangle∈Neighborhood(\left |a_i\right \rangle)$$) we should obtain the familiar expression $$\left |\psi\right \rangle = ∫ da \left \langle a\right|\left |\psi\right \rangle\left |a\right \rangle ∀ \left |\psi\right \rangle∈H$$.
How could this be linked in a rigorous way without the usual "for the continuous case replace sum by integral".
Thanks in advance!

PD: Sorry for the latex form, writing in physics forums can be daunting without any packages...

 

[bhobba]

What you need to study is called Rigged Hilbert Spaces - which is graduate level math. It has been discussed here a few times eg:
https://www.physicsforums.com/threads/rigged-hilbert-spaces-in-quantum-mechanics.917768/

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:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

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:
https://www.amazon.com/dp/0821846302/?tag=pfamazon01-20

That will explain it all in terms that is mathematically exact.

Then, do some functional analysis. I suggest the following:
http://matrixeditions.com/FunctionalAnalysisVol1.html

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

 

[Alex Cros]

What you need to study is called Rigged Hilbert Spaces - which is graduate level math. It has been discussed here a few times eg:
https://www.physicsforums.com/threads/rigged-hilbert-spaces-in-quantum-mechanics.917768/

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:
https://www.amazon.com/dp/0521558905/?tag=pfamazon01-20

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:
https://www.amazon.com/dp/0821846302/?tag=pfamazon01-20

That will explain it all in terms that is mathematically exact.

Then, do some functional analysis. I suggest the following:
http://matrixeditions.com/FunctionalAnalysisVol1.html

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

Thank you so much man, that really helps and now my summer is going to be way more interesting!