# 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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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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