Need a Linear Algebra Book for Quantum Mechanics?

In summary, the conversation discusses the differences between the spectrum of an operator in proof-based linear algebra and quantum mechanics. The individual is looking for recommendations on books that deal with linear algebra, specifically with vector spaces of infinite dimension, and functional analysis for quantum mechanics. Several books are suggested, including one that covers the necessary material on metric spaces and linear algebra, and another that presents the math in a rigorous way. The conversation also touches on the use of bra-ket notation and the availability of these books at the individual's university library.
  • #1
fluidistic
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I've had a proof based linear algebra course as a freshman, where I learned that the spectrum of an operator was the set of the eigenvalues of that operator. Now in quantum mechanics I learned that this isn't true and that the spectrum of an operator can contain infinitely more numbers. :confused::eek:
Also in my course I've never learned anything about vector spaces of infinite dimension.
I'm getting lost with the linear algebra part of QM. Could you please recommend me some book(s) that deals with linear algebra (better if it's aiming at physicists) with vector spaces of infinite dimension?
Thank you very much.
 
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  • #2
You're asking for books on functional analysis. Without making any assumptions about your knowledge of topology and real analysis, I can recommend this book: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20. It is light on topology and measure theory but it has the benefit of covering all the necessary material needed on metric spaces and LA. If you perhaps want a more concise summary of the functional analysis used in QM, take a look at chapter 1 of Ballentine.

EDIT: btw I remember you told me that your professor doesn't use bra-ket notation. Chapter 1 of Ballentine has the added benefit of showing you why it works (i.e. Riesz representation theorem) and then he proves a good number of theorems whilst using the notation so it should give you good exposure to the notation if that is still of importance to you.
 
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  • #3
WannabeNewton said:
You're asking for books on functional analysis. Without making any assumptions about your knowledge of topology and real analysis, I can recommend this book: https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20. It is light on topology and measure theory but it has the benefit of covering all the necessary material needed on metric spaces and LA. If you perhaps want a more concise summary of the functional analysis used in QM, take a look at chapter 1 of Ballentine.

EDIT: btw I remember you told me that your professor doesn't use bra-ket notation. Chapter 1 of Ballentine has the added benefit of showing you why it works (i.e. Riesz representation theorem) and then he proves a good number of theorems whilst using the notation so it should give you good exposure to the notation if that is still of importance to you.

Ok thanks a lot, I didn't even know about functional analysis. I've never taken a topology course, nor real analysis.
Unfortunately the library of my university lacks both books. If I buy from amazon (assuming I've no problem with the prices, which isn't a given), at best I'll have the book in about 1.5 month, basically too late for my course.
At least now I know where to look at: functional analysis.
About the bra ket notation you're right, my prof. doesn't use it in public. I've borrowed Sakurai and Messiah's books yesterday though in order to learn it.
 
  • #5
micromass said:
Another good book is Prugovecki: https://www.amazon.com/dp/0486453278/?tag=pfamazon01-20

I do think it's more a math books than a physics book, though. But it presents all the math is a nice and rigorous way.

Ok thank you very much. Apparently they have it at the library... going for it on Tuesday.
 

FAQ: Need a Linear Algebra Book for Quantum Mechanics?

What is linear algebra and how is it related to quantum mechanics?

Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. In quantum mechanics, linear algebra is used to represent and manipulate the state of a quantum system, as well as to solve equations and make predictions about the behavior of particles.

What are the key concepts and topics covered in a linear algebra book for QM?

A linear algebra book for QM would cover topics such as vector spaces, matrices, eigenvalues and eigenvectors, inner product spaces, and linear transformations. It would also include applications of linear algebra in quantum mechanics, such as the representation of quantum states and operators, and the solution of Schrödinger's equation.

Is prior knowledge of linear algebra necessary to understand quantum mechanics?

While some knowledge of linear algebra can be helpful in understanding the mathematical concepts in quantum mechanics, it is not necessarily a prerequisite. Many introductory textbooks on quantum mechanics also include a review of relevant linear algebra topics.

Are there any recommended linear algebra books specifically for quantum mechanics?

Yes, there are several popular textbooks that focus on the use of linear algebra in quantum mechanics, such as "Quantum Mechanics for Scientists and Engineers" by David A. B. Miller and "Quantum Mechanics: Concepts and Applications" by Nouredine Zettili.

How can I best use a linear algebra book for QM to improve my understanding of quantum mechanics?

To get the most out of a linear algebra book for QM, it is important to have a basic understanding of both linear algebra and quantum mechanics. It can also be helpful to work through practice problems and examples, and to apply the concepts learned to real-world situations. Additionally, seeking guidance from a teacher or tutor can also aid in understanding the material.

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