Where can I find resources to learn the math behind quantum mechanics?

[PhysicsDad]

Hello all,

I am an amateur "physicist" just tinkering and experimenting with my son, and as I read the posts about different concepts, I can get the general idea of what is being explained. However, when it comes to the math (equations for the wave functions, calculating eigenstates, etc.) I am totally lost, and I was wondering if anyone had suggestions on a book or website that I could use to try and understand how to do these calculations. I know algebra and I have taken some calculus, though I haven't used it in years, and I am unfamiliar with a lot of the terms and symbols used in these equations. Any source to help me get a grip on the mathematics would be great.

 

[Jolb]

Since you mention wavefunctions and eigenstates, I would say that most introductory Quantum Mechanics books include some very good but quick-and-dirty treatments of the requisite math: Griffiths' "Introduction to Quantum Mechanics" gives most of the baby-steps in most of its derivations and has a good appendix on linear algebra; Shankar's "Principles of Quantum Mechanics" has a great chapter called "Mathematical Introduction" and some more useful appendices.

But for general physical math references, I would recommend Mary Boas' "Mathematical Methods in the Physical Sciences" for the stuff that follows right after calculus/differential equations, and for more advanced stuff I'd recommend Arfken's "Mathematial Methods for Physicists."

If you have any specific questions, it might help if you posted them to get an idea of the level you're at--those references are about at the level of a sophomore undergrad who's taken calculus and differential equations.

 

[Chopin]

To get started with the most basic concepts in quantum mechanics (e.g. plane waves, the Schrödinger equation in a potential well, and the energy levels of the hydrogen atom) you'll mainly need to know a lot of calculus. The basic integral and differential topics to start, and then eventually on to multivariate calculus (e.g. partial derivatives, volume integrals, curl and divergence). You'll also need a basic understanding of differential equations, since most of the fundamental equations in QM are expressed in terms of them (e.g. the Schrödinger and Dirac equation).

If you want to move further up the ladder, eventually you have to become fairly comfortable with linear algebra. All of QM is really an exercise in linear algebra, although it can be cleverly hidden behind simple calculus for a while, so you can get a ways in before you really have to start worrying about that.

As far as how to learn these topics, textbooks are probably the most straightforward way. I'm not sure what the best books available are for those topics, but all of the stuff I mentioned above is commonplace enough that it should be covered well in just about any textbook, so finding one that's well-reviewed online should be enough. You can also look around online for write-ups or videos (some of those topics probably have videos on the Khan Academy site, although I've never been there so I'm not sure). Finally, if you can get far enough into formulate some concrete questions, you are very welcome to post them here at PF, and someone will be glad to help you with whatever you're stuck on.

Good luck! The math can be pretty daunting at first, but if you work slowly and carefully enough it does all make sense eventually.

 

[WannabeNewton]

It depends on how in-depth a knowledge of the mathematics behind the subject you want. If you want to have just enough knowledge to ostensibly understand the physics then yes the mathematical methods books will work just fine. Otherwise just get a real linear algebra textbook e.g. Axler: https://www.amazon.com/dp/0387982582/?tag=pfamazon01-20

The title of the book speaks for itself.

 

[micromass]

It depends on how in-depth a knowledge of the mathematics behind the subject you want. If you want to have just enough knowledge to ostensibly understand the physics then yes the mathematical methods books will work just fine. Otherwise just get a real linear algebra textbook e.g. Axler: https://www.amazon.com/dp/0387982582/?tag=pfamazon01-20

The title of the book speaks for itself.

Well, if you only want a superficial knowledge of the mathematics, then math methods books are good. If you really want to understand why the math is the way it is, and what the pitfalls are, then you need an actual mathematics book.

 

[Jorriss]

Math method books are really useful and they serve a very necessary purpose in physics.

 

[micromass]

Math method books are really useful and they serve a very necessary purpose in physics.

Sure, a math methods book like Reed & Simon is really useful.

 

[Jorriss]

Sure, a math methods book like Reed & Simon is really useful.

I would say a book like Simon & Reed is, if anything, a mathematical physics book. It's goal is to teach the underlying math and formalism used in physics.

Arfken & Weber on the other hand is a cook book of how to solve physics problems. It isn't trying to show the structure of math, if it was, it would have to be 10x's as long.

 

[micromass]

I would say a book like Simon & Reed is, if anything, a mathematical physics book. It's goal is to teach the underlying math and formalism used in physics.

They call themselves a math method book

 

[Jorriss]

They call themselves a math method book

It says "Methods of Mathematical Physics."

Who knows who won this one.

 

[PhysicsDad]

Wow, that got a lot of responses real fast. Thanks everyone, I'll start with some of these suggestions and definitely come back for advice if/when I get stuck.

Thanks all

 

[Jorriss]

Wow, that got a lot of responses real fast. Thanks everyone, I'll start with some of these suggestions and definitely come back for advice if/when I get stuck.

Thanks all

Sorry, it was off topic. Ignore our mentioning simon & reed and arfken & weber.

Boas, math methods is more appropriate if you want a methods book.

 

[Chronos]

I agree with Chopin, you will be lost in QM without good knowledge of differential equations. Linear algebra is fundamental to QM, and usually a corequisite for introductory QM courses.