Hilbert spaces and quantum operators being infinite dimensional matrices

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I just realized quantum operators X and P aren't actually just generalizations of matrices in infinite dimensions that you can naively play with as if they're usual matrices. Then I learned that the space of quantum states is not actually a Hilbert space but a "rigged" Hilbert space.

It all started because I though "well, if X and P are just infinite dimensional matrixes, can't I just multiply them as I would a normal matrix, by multiplying Xxx' by Px'x and then integrating over x'?". But then I realized I'd have to integrate the product of two delta functions which... Uh... Doesn't make a ton of sense. So I asked around the web and I learned that you can't really do that, because X and P are not naive generalizations of finite dimensional matrices, and also the space of states is not a real Hilbert space. But every quantum mechanics textbook I have seen doesn't seem to say much about this and they're kind of misleading. I kinda went down a rabbit hole these last few hours trying to wrap my head around what's going on but I feel like I lack a lot of knowledge.

Generally when I try to solve a problem I feel like I am shuffling symbols around in ways that seem visually right, but I don't really understand what's going on. For this reason I'd appreciate some recommendations of books that explore the math and mathematical methods behind QM in a more rigorous way (though I'd still prefer if it wasn't entirely theoretical and also taught you useful and efficient methods you can use to solve problems), because at the moment I'm kind of overwhelmed at all the info and I don't know where to start. There is a lot of stuff I have to study so I'd prefer if it wasn't some immense 800 page bible.
 

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  • #2
anuttarasammyak
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bra <x| and ket |x> notation invented by Dirac would help you to understand the cases of continuous eigenvalues. Mathematicians I met hate his idea but I think it quite natural and easy to understand. His textbook published in 1930 or J. J. Sakurai tells about it.
 
  • #3
Baluncore
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I think you are beyond one book and so will need to browse the library for chapters or articles that are new to you. I get several relevant results when I Google;
+"mathematics" +"quantum mechanics"
 
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bra <x| and ket |x> notation invented by Dirac would help you to understand the cases of continuous eigenvalues. Mathematicians I met hate his idea but I think it quite natural and easy to understand. His textbook published in 1930 or J. J. Sakurai tells about it.
Braket notation is what I learned, but I don't feel like I understand much about what is really going on beyond that.
 
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I think you are beyond one book and so will need to browse the library for chapters or articles that are new to you. I get several relevant results when I Google;
+"mathematics" +"quantum mechanics"
I was suggested the books by Michael Reed and Barry Simon on Functional Analysis. It's a multivolume series but apparently the first book is good for an intro, or so I was told. Does anyone have any opinions on that?
 
  • #6
George Jones
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Generally when I try to solve a problem I feel like I am shuffling symbols around in ways that seem visually right, but I don't really understand what's going on. For this reason I'd appreciate some recommendations of books that explore the math and mathematical methods behind QM in a more rigorous way
Braket notation is what I learned, but I don't feel like I understand much about what is really going on beyond that.
What is your background in pure mathematics? For example, have you studied real analysis?
 
  • #7
vanhees71
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The way to make Dirac's braket formalism rigorous is the "rigged Hilbert space" formalism. See, e.g.,

A. Galindo, P. Pascual, Quantum Mechanics, Springer Verlag, Heidelberg (1990), 2 Vols.
 
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  • #8
George Jones
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The way to make Dirac's braket formalism rigorous is the "rigged Hilbert space" formalism. See, e.g.,

A. Galindo, P. Pascual, Quantum Mechanics, Springer Verlag, Heidelberg (1990), 2 Vols.
Other physics references that have varying degrees of coverage of rigged Hilbert spaces include:

A. Capri, "Non-Relativistic Quantum Mechanics";
A. Bohm (not D. Bohm!), "Quantum Mechanics: Foundations and Application".

For a rigourous overview of rigged Hilbert spaces (Gelfand triples) and Dirac notation, I recommend highly sections 11.2, 11.3, and 12.2 from "Quantum Field Theory I: Basics in Mathematics and Physics (A Bridge Between Mathematicians and Physicists)" and subsection 7.6.4 from "Quantum Field Theory II: Quantum Electrodynamics (A Bridge Between Mathematicians and Physicists)" by Eberhard Zeidler.

The books by Zeidler are very long, but the sections I referenced have reasonable length. I don't have my copies home with me, and I don't remember the mathematical background assumed in these sections.
 
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  • #9
George Jones
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I was suggested the books by Michael Reed and Barry Simon on Functional Analysis. It's a multivolume series but apparently the first book is good for an intro, or so I was told. Does anyone have any opinions on that?

Even though rigged Hilbert spaces make rigourous the physicists' version of the maths used in quantum mechanics, it is not strictly necessary to move out of Hilbert . Reed and Simon advocate remaining in Hilbert space. From Reed and Simon: "There has arisen an extensive literature on a 'rigorous' Dirac notation which attempts to capture the flavour of bra and kets more fully. ... We must emphasize that we regard the spectral theorem as sufficient for any argument where a nonrigorous approach might rely on Dirac notation; thus, we only recommend the rigged space approach to readers with a strong emotional attachment to the Dirac formalism."

If you want to study the Hilbert space approach (as opposed to the rigged Hilbert space approach), then I recommend the beautiful "Quantum Theory for Mathematicians" by Brian Hall over Reed and Simon, even though I like Reed and Simon. Hall is a really wonderful book, but it has "the basics of L^2 spaces and Hilbert spaces" as prerequisites. Most of the functional analysis presented in the book is in chapters 6 - 10, through which the author gives several paths "I have tried to design this section of the book in such a way that a reader can take in as much or as little of the mathematical details as desired."
 
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  • #10
anuttarasammyak
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Braket notation is what I learned, but I don't feel like I understand much about what is really going on beyond that.
As for your concern in OP i.e. matrix representation of operators with continuous eigenvalues,say A,
[tex]A=\int da \ |a>a<a| [/tex]
[tex]A=\int \int dx' dx'' \ \ |x''><x''|A|x'><x'| [/tex]
where <x"|A|x'> is element of matrix in x representation.
[tex]<x''|A|x'>=\int da \ \ <x''|a>a<a|x'>[/tex]
 
  • #11
strangerep
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There is a lot of stuff I have to study so I'd prefer if it wasn't some immense 800 page bible.
Try ch1 of Ballentine, which is quite brief, and see whether that's at the level you're looking for.
 
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What is your background in pure mathematics? For example, have you studied real analysis?
I've done Calculus 1, 2 and 3 (uni level) and Compelx Analysis, I am just now getting into Real Analysis.
 
  • #13
Math_QED
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I've done Calculus 1, 2 and 3 (uni level) and Compelx Analysis, I am just now getting into Real Analysis.
Complex analysis before real analysis?
 
  • #14
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Try ch1 of Ballentine, which is quite brief, and see whether that's at the level you're looking for.
I'm browsing it now. I already know most of the stuff in it but it does have some things that I haven't seen in other books.
 
  • #15
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Complex analysis before real analysis?
Yeah that was the curriculum. I believe Calculus 1, 2 and 3 in my uni contains some stuff from real analysis and then the subject Real Analysis covers more advanced topics.
 
  • #16
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Other physics references that have varying degrees of coverage of rigged Hilbert spaces include:

A. Capri, "Non-Relativistic Quantum Mechanics";
A. Bohm (not D. Bohm!), "Quantum Mechanics: Foundations and Application".

For a rigourous overview of rigged Hilbert spaces (Gelfand triples) and Dirac notation, I recommend highly sections 11.2, 11.3, and 12.2 from "Quantum Field Theory I: Basics in Mathematics and Physics (A Bridge Between Mathematicians and Physicists)" and subsection 7.6.4 from "Quantum Field Theory II: Quantum Electrodynamics (A Bridge Between Mathematicians and Physicists)" by Eberhard Zeidler.

The books by Zeidler are very long, but the sections I referenced have reasonable length. I don't have my copies home with me, and I don't remember the mathematical background assumed in these sections.
Thanks, that's very interesting especially the QFT book, since I am interested in learning that too.
 
  • #17
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Even though rigged Hilbert spaces make rigourous the physicists' version of the maths used in quantum mechanics, it is not strictly necessary to move out of Hilbert . Reed and Simon advocate remaining in Hilbert space. From Reed and Simon: "There has arisen an extensive literature on a 'rigorous' Dirac notation which attempts to capture the flavour of bra and kets more fully. ... We must emphasize that we regard the spectral theorem as sufficient for any argument where a nonrigorous approach might rely on Dirac notation; thus, we only recommend the rigged space approach to readers with a strong emotional attachment to the Dirac formalism."

If you want to study the Hilbert space approach (as opposed to the rigged Hilbert space approach), then I recommend the beautiful "Quantum Theory for Mathematicians" by Brian Hall over Reed and Simon, even though I like Reed and Simon. Hall is a really wonderful book, but it has "the basics of L^2 spaces and Hilbert spaces" as prerequisites. Most of the functional analysis presented in the book is in chapters 6 - 10, through which the author gives several paths "I have tried to design this section of the book in such a way that a reader can take in as much or as little of the mathematical details as desired."
Thanks, but there's an issue. The issue is that I'm not very smart and I feel like if I try to learn two different things (the Dirac approach with rigged spaces plus the Hilbert space approach) I will get really confused. A lot of what people have already suggested focuses on the rigged approach, and I sorta need to learn the Dirac formalism better anyways for my classes, so idk if I can handle learning both. I feel I may get lost.
 
  • #18
PeroK
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Yeah that was the curriculum.
Do you mean complex numbers, rather than complex analysis?
 
  • #19
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Do you mean complex numbers, rather than complex analysis?
The class is called complex analysis, in Greek at least. It covers complex numbers, analytic functions, complex integrals, Taylor and Laurent series, residue theorem, etc.
 
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  • #20
George Jones
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I feel like if I try to learn two different things (the Dirac approach with rigged spaces plus the Hilbert space approach) I will get really confused. A lot of what people have already suggested focuses on the rigged approach, and I sorta need to learn the Dirac formalism better anyways for my classes, so idk if I can handle learning both. I feel I may get lost.
Yes, I thought you were more interested in rigged Hilbert space approach. I made post #9 for two reasons.

First, in your original post you wrote

Then I learned that the space of quantum states is not actually a Hilbert space but a "rigged" Hilbert space.
I wanted to point out that isn't strictly true, and that it is possible to take the space of quantum states to be a Hilbert space (by using wave packets, etc.).

Secondly,

I was suggested the books by Michael Reed and Barry Simon on Functional Analysis. It's a multivolume series but apparently the first book is good for an intro, or so I was told. Does anyone have any opinions on that?
Not only is it possible to ignore rigged Hilbert spaces, this is what Reed and Simon does. Since I thought that you were interested in the rigged Hilbert space approach (which is quite cool), I didn't want you to invest blood, sweat, and tears slogging through the somewhat difficult Reed and Simon only to find that it didn't do what you wanted. On the off chance that you were interested in the Hilbert space approach, I gave what, in my opinion, is a better reference, Hall.
 
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  • #21
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Yes, I thought you were more interested in rigged Hilbert space approach. I made post #9 for two reasons.

First, in your original post you wrote



I wanted to point out that isn't strictly true, and that it is possible to take the space of quantum states to be a Hilbert space (by using wave packets, etc.).

Secondly,



Not only is it possible to ignore rigged Hilbert spaces, this is what Reed and Simon does. Since I thought that you were interested in the rigged Hilbert space approach (which is quite cool), I didn't want you to invest blood, sweat, and tears slogging through the somewhat difficult Reed and Simon only to find that it didn't do what you wanted. On the off chance that you were interested in the Hilbert space approach, I gave what, in my opinion, is a better reference, Hall.
Thank you, that makes sense!
 
  • #22
vanhees71
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Thanks, but there's an issue. The issue is that I'm not very smart and I feel like if I try to learn two different things (the Dirac approach with rigged spaces plus the Hilbert space approach) I will get really confused. A lot of what people have already suggested focuses on the rigged approach, and I sorta need to learn the Dirac formalism better anyways for my classes, so idk if I can handle learning both. I feel I may get lost.
I don't know, why one should learn QT not using the rigged-Hilbert space formulation (I'd learn it in the sloppy physicists' way first anyway to get a feeling for the math before I get lost in too much rigor). The pure Hilbert-space formalism is more complicated if you want to do it right.
 
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  • #23
George Jones
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@AndreasC, you might note want to look at these notes on Dirac notation.

http://web.mit.edu/8.05/handouts/jaffe1.pdf

These notes are formal, but not rigourous (by mathematicians' standards), and make no mention of rigged Hilbert spaces. For gentle (a relative term!) introductions to rigged Hilbert spaces, I would recommend Capri or Ballentine (see above). Other folks might have differing opinions.

Sometimes too much mathematical rigour leads to rigor mortis (paralysis) in physics. Having said that, I personally love some of the uses of pure abstract mathematics in physics.
 
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  • #24
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@AndreasC, you might note want to look at these notes on Dirac notation.

http://web.mit.edu/8.05/handouts/jaffe1.pdf

These notes are formal, but not rigourous (by mathematicians' standards), and make no mention of rigged Hilbert spaces. For gentle (a relative term!) introductions to rigged Hilbert spaces, I would recommend Capri or Ballentine (see above). Other folks might have differing opinions.

Sometimes too much mathematical rigour leads to rigor mortis (paralysis) in physics. Having said that, I personally love some of the uses of pure abstract mathematics in physics.
I browsed through the first chapter of Ballentine, it was interesting and I definitely learned some new things. I'm a bit busy right now because I have to learn a bunch of subjects in a few days because I have exams and I didn't bother much with some things but after that I'll come back to it and pay a bit more attention and also read some of the other suggestions. From what I saw Ballentine looks like a really cool book, although I've heard some people trashing its treatment of some subjects?
 
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  • #25
strangerep
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From what I saw Ballentine looks like a really cool book, although I've heard some people trashing its treatment of some subjects?
There was once a time, when I'd waste time, arguing with "some people" about this. But whenever I challenged them to formulate their disagreements into proper physics papers and submit to reputable journals, they never did so. YBTJ.

I learned a LOT from Ballentine (as you can probably guess from my signature quotes). :oldbiggrin:

I've also learned a LOT from various other textbooks. It's wise to study multiple accounts of any given subject.
 
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