How are Banach and Hilbert spaces applied in quantum mechanics?

[thephystudent]

As a european bachelor student in physics, i can follow a theoretical math course next year about banach and hilbert spaces. How useful are those subjects for physics?

 

[micromass]

The answer to your question depends on a lot of factors. First of all, it depends on what you mean with physics. For example, if your interests are in experimental physics, then the course will be completely useless for the rest of your carreer. Second of all, it depends on what you mean with useful.

The subject of Banach and Hilbert spaces are used a lot in quantum mechanics. If you want to study theoretical quantum mechanics, then these subject will most likely pop up. If you want to get a good foundation in these topics, and if you want to see how these mathematics objects work, then taking the course is a good option. If you just want to do physics and if you are ok with taking some things for granted, then I'm sure you will do fine without such a theoretical math class.

Be aware that you are taking a theoretical math class. The focus will be on mathematical rigor and proving (a LOT of) theorems. Do NOT underestimate such a course. Typically, they will not care about physical applications, so you might want to look these up yourself. Remember: just because it seems without applications doesn't mean that this is true. The opposite is true, actually, I claim that most results in the course actually have some kind of application in physics and have some physical significance.

Here are some examples of physical applications of the course (take this with a grain of salt, I am not a physicist).

• In the beginning of QM, there were two popular formulations which seemed to work. Nobody knew which one was correct. It was eventually discovered that both theories are actually the same thing. This is because $\ell^2$ and $L^2$ are isomorphic Hilbert spaces.
• A popular device in QM is of course the Bra-ket notation, where you have bra's <a| and kets |a>. But what are these things actually and why does this notation work? This is answered (essentially) by the Riesz representation theorem that classifies the continuous functionals on a Hilbert space.
• In QM, we interpret $|<a,b>|^2$ as some kind of probability. But why does this work? For example, why do those probabilities sum to 1? This is answered by the Parseval equality.

If you think about working in theoretical QM, then you should absolutely take this course as it will be helpful (but again, it is not easy).