By azerIO


2011-11-10 22:02:32 8 Comments

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?

5 comments

@Dexters 2015-05-31 02:31:15

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..

@G. R. Wilbourn 2014-09-24 22:39:15

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.

@user46925 2015-05-31 10:26:02

nice and cheap book

@joseph f. johnson 2011-12-05 22:56:41

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.

@Ron Maimon 2011-12-06 22:37:37

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.

@joseph f. johnson 2011-12-07 00:38:24

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.....

@Ron Maimon 2011-12-07 05:20:34

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 2011-12-07 05:22:59

The anecdote about P violation is that many people said that P was conserved just as a matter of logical necessity, in several crappy quantum mechanics books. Dirac pointed out that he never said P was a fundamental symmetry in his book, and indeed, he didn't make that wrong argument. Dirac's treatment of symmetries is very good, the only place his book lacks is in the motivation for canonical commutation. Historically, he got this from Heisenberg's interpretation of the old quantum rule. The modern version is Schrodinger's. But I still think Heisenberg's way is most convincing.

@Ron Maimon 2011-12-07 05:29:10

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 2011-12-07 05:32:50

In particular, consider just one path-integral dominated result--- the operator product expansion. The Heisenberg operator algebra is obviously useful in field theory, but the commutation relation is not manifestly covariant. What are the covariant operator relations? These are all from the path integral, these are operator products. The OPE is central to 2d quantum field theory, which is essential both for strings and statistical mechanics. The path integral is the correct formalism for quantum mechanics, and to leave it out is like leaving out Newton's laws from classical mechanics.

@Lagerbaer 2011-12-15 05:29:48

@Ron Be that as it may, that's hardly something someone starting QM can or should be able to stomach.

@Ron Maimon 2011-12-15 10:47:09

@Lagerbaer: You are right. But a path integral, with emphasis on stochastic processes, is accessible even to a non-quantum student. The quantum version is straightforward once the stochastic version is internalized.

@joseph f. johnson 2011-12-15 17:52:11

@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

@Ron Maimon 2011-12-15 18:52:21

@joseph f. johnson: I am not sure what to say--- if you teach stochastic processes to "normal folks" (and I am not sure where you would find other-than-normal folks, because these people don't exist), you have to say at some point that the X(t) times dX/dt depends on the time order, and the commutator (the difference of the two orders) is 1. This is the Heisenberg commutation relation in path integral form, and it is also the Ito lemma. The relation between the two is completely intuitive. But you said a moment ago you think it's a mathematical trick, which it isn't.

@madtowneast 2011-11-11 07:04:10

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.

@Physicsworks 2011-11-10 22:25:35

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.

@qubyte 2011-12-15 16:56:45

+1 for the book recommendation. This was the one I was taught with and it provided an excellent starting point.

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