Rigorous transition from discrete to continuous basis

In summary, the conversation discusses the topic of Rigged Hilbert Spaces in quantum mechanics and how it relates to distribution theory and functional analysis. The solution to this problem has been a challenge for many mathematicians, including Von-Neumann and Hilbert, and requires a deep understanding of these advanced mathematical concepts. Recommendations for further study and resources are provided to aid in understanding this complex topic.
  • #1
Alex Cros
28
1
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...
 
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  • #2
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
 
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  • #3
bhobba said:
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!
 
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Related to Rigorous transition from discrete to continuous basis

1. What is meant by a "rigorous transition" from discrete to continuous basis?

A rigorous transition from discrete to continuous basis refers to the mathematical process of converting discrete values or data points into a continuous function or curve. This involves finding a continuous function that closely approximates the discrete data and can be used to make predictions or calculations.

2. Why is it important to transition from discrete to continuous basis in scientific research?

Transitioning from discrete to continuous basis is important in scientific research because it allows for more accurate and precise analysis of data. Continuous functions can provide a more complete representation of the underlying patterns and relationships in the data, making it easier to draw conclusions and make predictions.

3. What are some common methods used for the rigorous transition from discrete to continuous basis?

There are several methods that can be used for the rigorous transition from discrete to continuous basis, including polynomial interpolation, spline interpolation, and Fourier series. Each method has its own advantages and limitations, and the choice of method will depend on the specific data and research goals.

4. What are some potential challenges or limitations of transitioning from discrete to continuous basis?

One of the main challenges in transitioning from discrete to continuous basis is the potential for overfitting, where the continuous function fits too closely to the discrete data and may not accurately represent the underlying trends. Other challenges may include choosing the appropriate method and determining the level of precision needed for the analysis.

5. How can the results of transitioning from discrete to continuous basis be validated?

The results of transitioning from discrete to continuous basis can be validated through various methods, such as cross-validation, where the data is split into training and testing sets. Additionally, comparing the results to other established models or conducting sensitivity analyses can help ensure the accuracy and reliability of the continuous function.

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