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Could someone experienced in the field tell me what the minimal math knowledge one must obtain in order to grasp the introductory Quantum Mechanics book/course?

I do have math knowledge but I must say, currently, kind of a poor one. I did a basic introductory course in Calculus, Linear algebra and Probability Theory. Perhaps you could suggest some books I have to go through before I can start with QM? Thx.

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It's easier to learn something if you have a need for it, so you might use your interest in QM to inspire yourself to learn the math. –  Mike Dunlavey Dec 15 '11 at 1:39
    
Related Math.SE question: math.stackexchange.com/q/758502/11127 –  Qmechanic Apr 20 at 7:16
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I depends on the book you've chosen to read. But usually some basics in Calculus, Linear Algebra, Differential equations and Probability theory is enough. For example, if you start with Griffiths' Introduction to Quantum Mechanics, the author kindly provides you with the review of Linear Algebra in the Appendix as well as with some basic tips on probability theory in the beginning of the first Chapter. In order to solve Schrödinger equation (which is (partial) differential equation) you, of course, need to know the basics of Differential equations. Also, some special functions (like Legendre polynomials, Spherical Harmonics, etc) will pop up in due course. But, again, in introductory book, such as Griffiths' book, these things are explained in detail, so there should be no problems for you if you're careful reader. This book is one of the best to start with.

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+1 for the book recommendation. This was the one I was taught with and it provided an excellent starting point. –  qubyte Dec 15 '11 at 16:56
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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.

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Dirac's book is readable with no prior knowledge, +1, and it is still the best, but it has no path integral, and the treatment of the Dirac equation (ironically) is too old fasioned. I would recommend learning matrix mechanics, which is reviewed quickly on Wikipedia. The prerequisite is Fourier transforms. Sakurai and Gottfried are good, as is Mandelstam/Yourgrau for path integrals. –  Ron Maimon Dec 6 '11 at 22:37
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There is a story about Dirac. When it was proved that parity was violated, someone asked him what he thought about that. He replied "I never said anything about it in my book." The things you mention that are left out of his book are things it is a good idea to omit. Path integrals are ballyhooed but are just a math trick and give no physical insight, in fact, they are misleading. Same for matrix mechanics. Those are precisely why I still recommend Dirac for beginners... I would not even be surprised if his treatment of QED in the second edition proved more durable than Feynman's..... –  joseph f. johnson Dec 7 '11 at 0:38
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Matrix mechanics is good because it gives you intuition for matrix elements, for example, you immediately understand that an operator with constant frequency is a raising/lowering operator. You also understand the semiclassical interpretation of off-diagonal matrix elements, they are just stunted Fourier transforms of classical motions. You also understand why the dipole matrix element gives the transition rate without quantizing the photon field, just semiclassically. These are all important intuitions, which have been lost because Schrodinger beat Heisenberg in mass appeal. –  Ron Maimon Dec 7 '11 at 5:20
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Your comment about path integrals is silly. The path integral gives a unification of Heisenberg and Schrodinger in one formalism, that is automatically relativistic. It gives analytic continuation to imaginary time, which gives results like CPT, relativistic regulators, stochastic renormalization, second order transitions, Fadeev Popov ghosts, supersymmetry, and thousands of other things that would be practically impossible without it. The particle path path integral is the source of the S-matrix formulation and string theory, of unitarity methods, and everything modern. –  Ron Maimon Dec 7 '11 at 5:29
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@RonMaimon I have had to teach stochastic processes and integrals to normal, untalented folks. IMHO, stohastic processes count as probability theory, one of the trickiest parts, and path integrals are no help for beginners here either. It is still better for the beginning student to not take a course in probability and let what they learn about the physics of QM be their introduction to stochastic processes...I mean, besides what they already learned about stochastic processes from playing Snakes and Ladders. This is part of my theme: learn the physics first, and mathematical tricks later –  joseph f. johnson Dec 15 '11 at 17:52
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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.

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