# Linear algebra references for a deeper understanding of quantum mechanics

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?

• One of my favorite texts is "Linear Algebra Done Right" by Sheldon Axler. You can read a synopsis here: linear.axler.net – TravisJ Apr 18 '15 at 3:30
• Shankar has a pretty nice linear algebra/functional analysis introduction. – Cameron Williams Apr 18 '15 at 4:07

## 2 Answers

Ok, let's be clear on something first: Linear algebra per se is not really important in a serious study of quantum mechanics. This is because most of the spaces one studies in quantum theory-Hilbert spaces, Heisenberg state spaces,etc-and their linear mappings,are actually infinite dimensional function spaces. Linear algebra proper is really the study of finite-dimensional vector spaces. As such, it really functions foundationally in quantum theory, as calculus is a foundation for a study of metric and topological spaces. So what you really are asking for are some good linear algebra sources to strengthen your background, so you can go on and study functional analysis and operator theory, which are the actual-forgive the pun-basis for studying quantum theory.

My favorite general book on linear algebra is Linear Algebra: An Introduction by Charles Curtis. This book is not only eminently readable, it's a book that balances theory and applications better then just about any book out there. It covers not only all the basics-linear spaces, linear independence, span,etc.-it also covers many sophisticated topics that aren't usually covered in basic courses and which are really important for further study, such as multilinear algebra and dual spaces. For a general text, this one can be matched,but not beaten.

However, there's a book I think you really need to look at after that given your area of interest and focus. That's Peter Lax's Linear Algebra And It's Applications . Not only is it beautifully written for a second course on linear spaces and it's written by one of the greatest living mathematicians there is-the book was specifically designed to act as a precursor to a graduate course on functional analysis. As a result, it focuses more on the analytical aspects of linear algebra then the algebraic or geometric aspects. To that end,Lax focuses much more on spectral theory and matrix analysis then other texts do.For example,chapter 10 contains a rapid but very complete introduction to the inequalities of matrix analysis.It's a wonderful read and will prepare you wonderfully for not only functional analysis, but the use of linear algebra and matrices in quantum theory.

Of course, after that, you should go directly to study Lax's beautiful text on functional analysis-which contains a number of applications to quantum theory, such as scattering theory.

That should help you. Take care-and remember, read math books by mathematicians, not physicists!

• Thank you, I'll be sure to look those up! It was my understanding though some subfields of QM (ie quantum information) used exclusively finite dimensional spaces. Is that wrong? – jip Apr 18 '15 at 23:48
• Quantum information theory is not really quantum physics per se, it's more of a specialized application to coding theory. The states of quantum information theory just happen to be finite dimensional and this is really more a consequence of the "discrete" nature of quantum information bits. So it doesn't really follow the usual mathematics of quantum physics, although it's indirectly governed by them,of course. – Mathemagician1234 Apr 20 '15 at 3:53

Going in line with what Methemagician1234 has told you I will put on the table other books on functional analysis that make connections to quantum mechanics. However, being a quantum chemist by formation I recognize that a lot of research does not deal directly with the infinite dimensional Hilbert space, but with its simulation on a finite dimensional space. So, a deep understanding of linear algebra is not only useful for understanding the foundation of functional analysis but also for the computer modeling of quantum phenomena. These are the resources:

As for the linear algebra books:

• Linear Algebra for Quantum Theory, by Per-Olov Löwdin is an attempt to explain linear algebra to those learning quantum theory starting from a very low level. It is written by a physicist.
• It is also worth mentioning that John Watrous's textbook is an excellent free resource for the mathematics of quantum information theory – Omnomnomnom Aug 17 '18 at 21:45
• @Omnomnomnom Thank you for for your suggestion. If you don't mind I will edit the answer to include your contribution. – Zythos Aug 18 '18 at 10:26
• Of course that's fine. Another resource that springs to mind, by the way, is Nielsen and Chuang's Theory of Quantum Information. – Omnomnomnom Aug 18 '18 at 19:01