# Linear Algebra Reference for Quantum Mechanics Topics

• theophyman

#### theophyman

hello,
is there a reference book in linar algebra that covers topics found in studying quantum mechanics like: vector spaces, operators , matrix , eigenvectors and eigenvalues.
i mean not a physics book, i want a mathematics book that talks about these notions and others in an abstarct way (not dedicated to quantum mechanics)

thank you

Any book on linear algebra will do. My favorite is Axler.

Note that linear algebra is the mathematics of linear operators between finite-dimensional vector spaces, but we often have to deal with infinite-dimensional vector spaces in QM. The mathematics of linear operators between infinite-dimensional vector spaces is called functional analysis, but you can understand QM very well without ever opening up a book on functional analysis, because linear algebra will give you the right intuition about things. (A lot of results about finite-dimensional vector spaces hold for infinite-dimensional ones too, but the proofs are often much, much harder). If you want to check out a book on functional analysis, Kreyszig is probably the best choice. I haven't read it myself, but it's getting very good reviews.

thank you,
but isn't there a book that talk about vector spaces in an abstract manner like the mathematicians do?

Any book on linear algebra or functional analysis meets that requirement. (In particular the ones I recommended).

Edit: I realize now that you were answering Qubix, not me.

Take a look at
Bernard Friedman, "Principles and Techniques of Applied
Mathematics" (John Wiley and Sons, New York, 1956).
Don't let its age keep you away from it. The math has not changed in 50 years. It is still a great book today.

Any book on linear algebra or functional analysis meets that requirement. (In particular the ones I recommended).

Edit: I realize now that you were answering Qubix, not me.

yes i mean Qubix.
thank you for your explanation about algebra of finit and infinit dimensional spaces.

and the book of axler:Linear algebra done right is a great book. thank you.

Take a look at
Bernard Friedman, "Principles and Techniques of Applied
Mathematics" (John Wiley and Sons, New York, 1956).
Don't let its age keep you away from it. The math has not changed in 50 years. It is still a great book today.

thank you, but that book is not dedicated to linear algebra, perhaps the first two chapters are about linar algebra.

Any book on linear algebra will do. My favorite is Axler.

i have that great book in hands, and i am reading it, it covers most of the linear algebra found in quantum mechanics,and it treats linar algebra in abstract way, it has exercises with solutions (separate solutions manual). good book.

You could also look at Paul Halmos: "Finite Dimensional Vector Spaces". He had in mind when writing it proofs that would generalise readily to the infinite dimensional case whenever possible (and has also written a dedicated book on Hilbert Space).

thank you,
but isn't there a book that talk about vector spaces in an abstract manner like the mathematicians do?

Sheldon Axler - Linear Algebra done Right

or, if you really want to learn algebra the mathematician's way

Serge Lang - Algebra (but it's pretty difficult and deep, most of it is useless for quantum mechanics)

Serge Lang's 'Algebra' made me cry, wet the bed, and forget the FOIL method before my head just burst like a tick. I wouldn't wish that tome a physicist if they smacked me! :rofl:

If you seriously want the underlying maths you will need Affine spaces as well as linear ones.

Tensor Geometry: The geometric viewpoint and its uses

by C T J Dodson and T Poston

is the ideal book.

You could also look at Paul Halmos: "Finite Dimensional Vector Spaces". He had in mind when writing it proofs that would generalise readily to the infinite dimensional case whenever possible (and has also written a dedicated book on Hilbert Space).

yes , it is a great book too, i had found it in references section of a "quantum computation and information" book.

and i thank all of you who had contributed to this thread. your informations and suggestions are great.

yes , it is a great book too, i had found it in references section of a "quantum computation and information" book.

and i thank all of you who had contributed to this thread. your informations and suggestions are great.

Yup, nothing like a really hot reading list to get the blood pumping right?!

Sheldon Axler - Linear Algebra done Right

or, if you really want to learn algebra the mathematician's way

Serge Lang - Algebra (but it's pretty difficult and deep, most of it is useless for quantum mechanics)

yes, it is an advanced book,
Serge Lang has other books about linear algebra such as:"Linear algebra" and "introduction to linear algebra"