What is the math knowledge necessary for starting Quantum Mechanics?

There is a nice book with an extremely long title: Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles. It does the basics pretty well. Griffith's would be the next logical step. After that there is Shankar.

You don't need any probability: the probability used in QM is so basic that you pick it up just from common sense.

You need linear algebra, but sometimes it is reviewed in the book itself or an appendix.

QM seems to use functional analysis, i.e., infinite dimensional linear algebra, but the truth is that you will do just fine if you understand the basic finite dimensional linear algebra in the usual linear algebra course and then pretend it is all true for Hilbert Spaces, too.

It would be nice if you had taken a course in ODE but the truth is, most ODE courses these days don't do the only topic you need in QM, which is the Frobenius theory for eq.s with a regular singular point, so most QM teachers re-do the special case of that theory needed for the hydrogen atom anyway, sadly but wisely assuming that their students never learned it. An ordinary Calculus II course covers ODE basics like separation of variables and stuff. Review it.

I suggest using Dirac's book on QM! It uses very little maths, and a lot of physical insight. The earlier edition of David Park is more standard and easy enough and can be understood with one linear algebra course and Calc I, CalcII, and CalcIII.

Try Schaum's Outlines: Quantum Mechanics, ISBN 0-07-054018-7. You'll see the math there, but you'll need to do the deep background studies on all the math from Chapter 2.

Try these two lectures from Leonard:

https://www.youtube.com/watch?v=5UqDb2BcxZk

https://www.youtube.com/watch?v=2STsUIHCaLU

Also more at https://glenmartin.wordpress.com/home/leonard-susskinds-online-lectures/

PS:I dont have any physics and math background except a few basics. so I cant comment on if these were too basic for you..

Good question. I come from a non-physics background and I have to learn quantum mechanics for my future studies and unfortunately I couldn't find any thorough answer on stackex unlike Steve Denton's in here

This answer is Steve Denton's suggestion on Quora website.

You can tackle basic, nonrelativistic QM at an introductory level with just the following prerequisites:

Linear algebra (mainly vector algebra and matrix algebra, and especially including eigenvectors and eigenvalues, which are absolutely central to QM) Complex numbers (especially the representation and manipulation of complex numbers in terms of complex exponential functions, and the representation of waves using the same) Differential and integral calculus of a single variable, including ordinary differential equations Basic probability and statistics A lot of the specialized concepts and mathematical functions that crop up in elementary QM (e.g. operator algebra, Hilbert space, Hermitian conjugates, inner products, Hermite polynomials, delta functions, Dirac bra-ket notation, projection operators, etc.) will be introduced to you during your QM studies, so they are not prerequisites as such.

For intermediate/advanced level nonrelativistic QM, you will need a few additional things, as a minimum:

Partial differential equations Spherical polar coordinates (used a lot in atomic & nuclear physics) Special functions (e.g. Legendre polynomials and related functions) Complex analysis (particularly the calculus of residues - i.e complex integration) Green's functions Fourier analysis Group theory A good familiarity with classical analytical mechanics, both the Hamiltonian and Lagrangian formulations, and the Principle of Least Action would also be very worthwhile acquiring at this point, as they are absolutely central ideas and techniques in any advanced physics, and particularly quantum field theory.

For relativistic QM and quantum field theory, the main things you will need, as a minimum, are:

Calculus of variations, or variational calculus (and its applications, via the Principle of Least Action, in classical analytical mechanics) Functional integration Tensor calculus (in 4D Minkowski spacetime; full-blown general tensor calculus will not be needed for the most part, but some knowledge of both it and general relativity might occasionally come in handy, and will certainly be needed if you want to go into areas like quantum cosmology, string theory, quantum gravity, etc.)

In addition to that, having a solid knowledge of classical mechanics could help you since classical mechanics is mandatory in the Lagrange, Hamilton and Hamilton-Jacobi formalisms.