I want to study quantum mechanics. But I don't know what and which topics of mathematics are required. I know a bit of differential calculus. But what else is needed to study a bit advanced quantum mechanics (not qft)? Please name a few books also if possible.
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$\begingroup$ I would suggest Lie/group theory. Nothing has helped my understandig of quantum mechanics more than Lie theory. $\endgroup$– TimCommented Jan 20, 2015 at 10:17
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$\begingroup$ @user70848 , the most intuitive book I know is "Lectures on Quantum Mechanics" by Gordon Baym, if you can get it. It speaks in a simple language. $\endgroup$– SofiaCommented Jan 20, 2015 at 12:13
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$\begingroup$ Possible duplicates: physics.stackexchange.com/q/16814/2451 , physics.stackexchange.com/q/5014/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Jan 20, 2015 at 18:13
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$\begingroup$ Chapter 1 of Shankar's "Principles of Quantum Mechanics" is a very complete and readable introduction to the mathematics needed in quantum mechanics that's suitable for advanced beginners and enough to read graduate level quantum mechanics texts. $\endgroup$– SchroederCommented Jul 25, 2017 at 21:38
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If you don't know the mathematical topics to study that might mean that you are a starter. And if you are planning to learn it by yourself, you might want to study demystified qm. It has a lot of errata in it. But it goes through each and every example in detail. It doesnt use differential integral though (at least till the chapter i have covered) . It basically switches between ket (Dirac) and matrice (Heisenberg) formulations.
Other than that i know that Griffiths which was already mentioned by Demosthene was in demand in undergrad schools in USA. That might be a good place to start as well.
Final reference would be MIT OCW website to watch videos. Well, you can find the videos on youtube too. This way you will listen to a real professor in a real class session and you can get the assignments and study materials on OCW.
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There is probably a closed question covering all this, if someone has the courage to go and dig it up.
Anyway, to study QM you need knowledge of differential calculus, matrices (and in particular Hermitian matrices), a good understanding of classical mechanics (Hamiltonian formulation, and the concept of "turning points").
To go further, consider Lie groups (and Lie algebras to go even further), operator theory, relativistic mechanics (i.e. special relativity), path integrals and a solid knowledge of complex analysis to go through all the mathematical methods used in QM.
Also note that there are two ways to approach QM, the physical one and the mathematical one. Of course, the mathematical one is much more rigorous, and is probably harder to understand if you start with it. The physical approach will slowly introduce the concepts, and explain why we need a theory of quanta (the downside to this is that mathematical aspects will often appear to come out of the blue - but this actually helps focusing on the physics).
My advice: pick up a book on QM aimed at undergrads (Griffiths, Gasiorowicz) and try to work your way through it. When you want to get really serious, pick a mathematical/theoretical physics book (Weyl, Kuhn, Teschl)