# Math needed for quantum mechanics

1. Jun 1, 2015

### micromass

Staff Emeritus
What math do I need to really understand quantum mechanics? Please advise!

It might be too much, but since this is my new hobby: are there any cool books that combine quantum mechanics and biology?

2. Jun 1, 2015

3. Jun 1, 2015

### aleazk

It really depends on how much you are interested in the (rigorous) mathematical foundations of QM. If you are only interested in learning the basics of QM and how to use it, then a solid knowledge of finite dimensional linear algebra supplemented with some (very basic) material from functional analysis is enough (of course, assuming you know calculus, differential equations, and all the usual stuff in the physicist's toolkit)

If you are really interested in the mathematical foundations, then you need some good amount of stuff: basics of general topology, basics on measure theory (including its application to probabilty theory; the quantum logic formulation of QM is a generalization of this), Banach spaces and algebras (including spaces of operators), Hilbert spaces (the basic stuff like Riesz's theorem and bases, but also the general theory behind bounded operators and densely-defined unbounded operators); spectral theory (the spectral theorem for unbounded self-adjoint operators and the theory behind it); Lie groups and harmonic analysis (including the imprimitivity theorem; most of the foundational issues related to the study of basic quantum systems, like localizable and covariant systems, can be reduced to the study and application of this theorem, i.e., the classification of representations of different Lie groups).

Most mathematicians are already familiar with all this material, so I don't think you will find anything new (I say this because, if I remember well, you are a mathematician)

4. Jun 1, 2015

### esuna

Schrodinger's "What is Life?" is essential if you are interested in the connections between physics and biology. Obviously we've come a long way since then however. I can dig up some other stuff I've come across later.

Is this a trick question? Because isn't the correct answer than no one "really" understands quantum mechanics?

If not, then linear algebra, probability theory, partial differential equations, operator theory, spectral theory, combinatorics, group theory, and probably more.

5. Jun 2, 2015

6. Jun 2, 2015

### Almeisan

Just complex numbers, matrices and their eigenvalues/vectors. Helps to know a bit about calculus, statistics and maybe waves.

If you know the popular physics and you see the math of discrete problems like polarization of spin up/down, you kind of can fudge how it 'feels' for continuous stuff requiring complex functions and partial diff eqs.

I mean, how much mathematical knowledge of QM do you really need as a lay person?
Do you really want to solve Schrodinger's for real-life problems?

7. Jun 2, 2015

8. Jun 3, 2015

### Shobah

As esuna noted it depends on what you mean by 'really' understanding quantum mechanics. A full treatment would probably require some incursions into quantum field theory, gauge theory, particle physics... as well, for which some understand of Lie algebras is desirable, and all of the things listed above. Quantum mechanics in itself is not so hard; a lot of QM courses at uni start off by establishing the linear analysis tools you need in the context of QM and go from there - introducing the Dirac formalism. Although the computations can get quite messy, the basic ingredients are simple - quantum simple harmonic oscillators, quantized angular momentum and perturbation theory (including time-dependant) will get you a long way into understanding QM.

I would have said that solid analysis skills (integration, Fourier transforms, solving DEs and PDEs, etc) will make you able to start learning QM effectively enough.

Example: I don't have any good textbooks in mind, but Prof. Tong's notes are available online and cover everything from introduction of the Dirac formalism through perturbation theory, all the way to quantum information, and recommend some books on the way.

http://www.damtp.cam.ac.uk/user/rrh/notes/pqm14_281014.pdf

9. Jun 5, 2015

### atyy

Physically, why is finite dimensional linear algebra everything one needs to understand the essence of QM? The answer is that it is believed that the Bell test with two spin 1/2 particles captures the essence of quantum phenomena: non-commuting operators and entanglement.

However, mathematically, apart from the well-known complaints against Dirac's version of QM, there seem to be some interesting problems like Tsirelson's problem, which is to about how much the Bell inequalities can be violated.
http://arxiv.org/abs/1008.1142
http://arxiv.org/abs/0812.4305