# What is the math knowledge necessary for starting Quantum Mechanics?

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?

• 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. Dec 15, 2011 at 1:39
• Related Math.SE question: math.stackexchange.com/q/758502/11127 Apr 20, 2014 at 7:16
• There are many different mathematical levels at which one can learn quantum mechanics. You can learn quantum mechanics with nothing more than junior high school algebra; you just won't be learning it at the same level of mathematical depth and sophistication.
– user4552
Sep 24, 2014 at 23:11
• Linear Algebra and Probability Theory are the most important. Apr 6, 2023 at 5:50

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.

• +1 for the book recommendation. This was the one I was taught with and it provided an excellent starting point. Dec 15, 2011 at 16:56

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.

• 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. Dec 6, 2011 at 22:37
• 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..... Dec 7, 2011 at 0:38
• 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. Dec 7, 2011 at 5:20
• 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. Dec 7, 2011 at 5:29
• @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 Dec 15, 2011 at 17:52

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.

Try these two lectures from Leonard:

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..

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.

• nice and cheap book
– user46925
May 31, 2015 at 10:26

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.