# Linear Algebra for Quantum Physics

A week ago I asked people on this forum what mathematical background was needed for understanding Quantum Physics, and most of you mentioned Linear Algebra, so I decided to conduct a self-study of Linear Algebra. Of course, I'm just 1 week in, but I have some questions.

1. How is this going to be applicable to quantum physics? I have learned about matrices (addition, subtraction, multiplication and inversion) and about how to solve multiple equations with 3 unknowns using matrices, and now I am starting to learn about vectors. I am just 1 week in, so this is probably not even the tip of the iceberg, but I want to know how this is going to help me.

2. Also, say I master Linear Algebra in general in half a year (I'm in high school but I'm extremely fast with maths), what other 'types' of math would I need to self-study before being able to understand rudimentary quantum physics mathematically?

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Quantum mechanics "lives" in a Hilbert space, and Hilbert space is "just" an infinite-dimensional vector space, so that the vectors are actually functions. Then the mathematics of quantum mechanics is pretty much "just" linear operators in the Hilbert space.

Quantum mechanics    Linear algebra
-----------------    --------------
wave function        vector
linear operator      matrix
eigenstates          eigenvectors
physical system      Hilbert space
physical observable  Hermitian matrix

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Hilbert spaces need not be infinite-dimensional. Every finite-dimensional, complex, inner product space is a Hilbert space. – joshphysics Aug 8 '14 at 2:17

Well, learn linear algebra. An advanced text (on linear algebra over "field" number systems) is these lecture notes [pdf] from UC Davis.

Once you get that done, you should study differential equations. Or if you want to skip ahead, perhaps Fourier analysis. A free reference would be my notes [pdf]. It's mildly physics-oriented, but connects the ideas back to linear algebra.

Quantum mechanics, when you boil it down, is Fourier analysis. (Instead of the "frequency domain" you have "momentum space", etc.)

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I suggest getting a used copy of Zetteli. Chapter 2 is a survey of the mathematical tools of QM, and the very beginning of Chapter 3 are the postulates of QM.

That will show you directly the math you need, and you can consult other books for more detailed explanations of the parts that give you trouble.

The matrices and vectors are important because they map very well to the mathematics of QM, and so they form the basic language in which QM is expressed. As you continue to study linear algebra, you will learn about eigenvectors and eigenvalues. Those are used to describe the measurement process, which is essential to QM.

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Well, if you want to gain any quantitative insights into QM, you'd have to pick up some calculus as well - mainly differential equations, and if you really insist, Fourier analysis too. I was taught decent basic calculus in high school, so you may already know some of the basics.

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The first major step would be calculus. Really just becoming familiar with integration and differentiation on all types of functions. From there a little knowledge on differential equations can go a long way. Knowing just this can get you solving some basic problems. "Early Transcendentals" by Thomas is a good calculus book. Then there are some nice math physics books that cover many different subjects from linear algebra to complex analysis. I don't particularly like this book but I use it, "Mathematical Methods in the Physical Sciences" by Mary Boas.

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Square symmetric matrices (or rather complex Hermitian ones) represent the observables of a quantum mechanical system. Their eigenvalues represent the possible observed values in ideal experiments. There is a basis of orthonormal eigenvalues, which allows you to write any state vector as a linear combination (superposition) of basis vectors. Squared absolute values of inner products define probabilities. Then one needs functions of matrices, in particular the matrix exponential, which gives the dynamics of a system, and an explicit solution of the Schroedinger equation in the case of an n-level system.

Thus you need to learn enough to be able to have a good grasp on these concepts: matrix, transpose, conjugate transpose, linear combination, basis, eigenvalue, eigenvector, inner product, matrix power series, matrix exponential. Wikipedia has good summarizing articles on each of these topics, to help you give an overview. You can skip other stuff, and get back to it in case you need it.

In analysis you need systems of linear differential equations with constant coefficients (these are related to the matrix exponential), and the Fourier transform. The latter involves integration in 3 dimensions, but again, you can skip a lot and turn back to things skipped once you need them.

Then you can look into various quantum mechanical texts or lecture notes, for example my online book http://lanl.arxiv.org/abs/0810.1019 - the first chapter of it should be understandable even with little prior knowledge, if you can tentatively accept concepts without a full understanding. My FAQ http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html might also be of help. By reading these and noting where you lose track you can find out what other concepts you need to make sense of your reading. This will tell you what else you need to learn. Ultimately, almost all of linear algebra and analysis is useful in quantum mechanics, but what and when depends on what you are interested in.

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