Essential math for quantum mechanics

Oscar Nieves
7 min readJan 19, 2023

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

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Oscar Nieves

I write stories about applied math, physics and engineering.