Essential math for quantum mechanics

All the things you need to know.

Quantum mechanics — and by extension quantum field theory — is a subfield of physics concerned with studying the dynamics of single and many-particle systems at the quantum scale, often designated as “quantum systems” which can encompass fundamental particles (e.g. quarks, leptons, bosons), atoms and even molecules (which is the focus of quantum chemistry). At this scale, known as the “quantum scale”, physics behaves a little differently than in the macroscale, and because of how small and sensitive things are to their surroundings, it is very difficult to measure multiple characteristics of a quantum system at the same time. For this reason, we think of quantum mechanics as being a probabilistic science, as opposes to the deterministic science that is classical mechanics when we deal with things in every day life, whether it be a Tennis ball or a car.

In this article, I will give a brief overview of the kind of mathematics used (and needed) to understand models in quantum mechanics. I shall preface this by mentioning I do expect the reader to be familiar with linear algebra (at the very least matrix operations), some calculus and also complex numbers, for which I have written some articles on already and you can check on my profile.

Vector spaces and bra-ket notation

Vectors are essential to physics, and in quantum mechanics they are used for describing the discrete states a quantum system can be in. Let us recall an ordinary vector in 3 dimensions as having the following components:

where the bold-faced x, y and z are unit vectors pointing in each of those directions. The magnitude (or length) of that vector is given by the dot-product with itself:

More generally, the dot-product between two distinct vectors is a scalar value consisting of their individual magnitudes and the angle between the two:

which is equal to zero when the two vectors are perpendicular (because there is no way to project one vector unto another in that case). In quantum mechanics, it is customary to represent vectors in column form such as:

This way, it is easier to think of operations by using matrix algebra. For instance, the dot product (also called…