Math, Quantum Mechanics and Statistical Mechanics

[SiyumLeisho]

For the people who are in either of these fields, which math did you encounter most frequently in the field? Abstract Algebra, Analysis, Probability, Statistics, Calculus, or other? I have taken introductory courses for both topics at university and both seem to involve a wide range of math disciplines. I am pretty sure that I can't learn all of the branches of mathematics involved in quantum and stats mech. I would like to know which one I should focus on (and the reason why, if possible). Thank you in advance.

 

[VantagePoint72]

From your title, it's not entirely clear what you mean by 'the field'. I presume you mean quantum mechanics.

The most important branch of mathematics for the foundations of quantum mechanics (apart from the basic introduction to calculus that is the foundation of any education in physics) is linear algebra. For the applications of quantum mechanics, an understanding of (ordinary and partial) differential equations is invaluable. Knowledge of statistics is useful for more detailed applications of QM, but is not essential for understanding the basic ideas.

At the very least, it is absolutely necessary to know calculus and linear algebra to understand quantum mechanics, because they are the language in which the theory is formulated. The rest of the mathematical toolbox can be filled in as you go along, according to which areas of QM you find yourself most interested in exploring.

 

[Jorriss]

The math you need for quantum mechanics varies a lot depending on what you are planning on doing. I suppose there are three different areas of math needed.

1) For formalism centric quantum, one needs functional analysis and linear algebra.

2) For solving textbook style problems one needs ODEs, PDEs and linear algebra.

3) For solving actual quantum problems, one needs numerical linear algebra, PDE's, etc.

Thats just very broadly speaking. In reality, you'll need a mixture of all three (plus other areas such as algebra, for example).

For statistical mechanics, you run into a similar scheme. You'll need probability theory, numerical analysis courses, etc.

You can't just focus on one area of math, if you spend all of your time focusing on PDEs and none on linear algebra, you'll be stuck often. Luckily, you don't need expertise in any of these fields.

 

[SiyumLeisho]

I guess linear algebra it is then. Thank you for your replies.