How do I learn Quantum Mech and General Rel?

Factual Questions
1

I am a sort of dilettante science buff. I read lots of non-fiction and have a decent math background. My education at university was Computer Science although that was a while ago (I think it was the second Coolidge administration;))

Watching The Mechanical Universe from The Annenberg/CPB Project explain Newtonian mechanics tickled me no end. Yeah, I am a demi-geek. But it also got me thinking. Quantum Mechanics and General Relativity are the crowning achievements of twentieth century science and I want to learn them in their fullness. I’ve read Hawking’s and Gribbin’s and Walker’s versions that explain things in laymen’s terms but I want to go whole hog.

So to the question: What sorts of math skills do I need to acquire and who do you folks recommend to get me started. I figure this’ll take a few years but it’s sort of a hobby. At least I think so…

Thanks people.

2

I only know the basics, but:

Quantum: Linear Algebra. It’s all about the matrices.

General Relativity: Differential Geometry. All about the tensors. You could probably spend two years just learning the background math for this alone.

Someone with a more intimate knowledge can correct or refine these statements.

3

Huh. Interesting. I would agree that Linear Algebra is tremendously important for QM, but equally important, in my experience, is a good working background with partial differential equations, which is what you ultimately need if you want to solve problems in QM exactly. Most of the time, we just give up on doing that, and resort to linear algebra techniques, and there are certainly times when you can get the exact results in this way.

For GR, I guess I’m a little surprised that you find differential geometry so useful, Achernar. I’m not a relativist, but in practical terms, when I have worked problems in GR, I haven’t used my differential geometry much at all. Maybe that means I’m just doing things the hard way?

4

For a basic guide to special and general rel, this one isn’t bad.
Also, check the other stuff at physics.about.com, there is quite a lot of useful info there.
This is a fairly decent downloadable text. Contains stuff ranging right through from the basics to relativity, quantum physics and beyond.
Entertaining read, too.

5

For QM, learn Calculus, then see if you can get your hand on “Mathematical Methods in the Physical Sciences” by Boas. You’ll learn the practical side of probably 2 or 3 yrs worth of advanced Math from that book. We used it in College for a 1 yr course and it was great.

6

I took a quick peek there Starman and it looks like I have been started on my quest. I figured lineaar algebra was in my future, but then I pretty much enjoyed it the first time I went through it so maaybe a purchase of a Schirmers’ guide is coming up.

I took a class in DE in my math background but I can’t say that I enjoyed them that much. Maybe the second time will be the charm. Thanks for the help so far…

7

Yikes, I’m not a relativist either; perhaps I spoke wrongly. I learned about affine connections and things like that in a course called Differential Geometry, but perhaps this is not the best term to apply. What math would you say you use?

8

Oh, it’s quite true that you’re really using differential geometry when you’re doing GR, of course, but I don’t know that one actually needs to know the machinery. Maybe I’m being a little naive about the entire thing, but to me the actual applications in GR seemed to revolve more around differential equations than anything else. Perhaps someone like Chronos who actually does this sort of thing for a living could be more helpful, but I always thought of it as ultimately GR is just solving G[sub][symbol]mn[/symbol][/sub] =8[symbol]p[/symbol]T[sub][symbol]mn[/symbol][/sub], which is admittedly a fiendishly horribly awful equation to try to solve (note to the non-physicists in the audience: no, that really is an ugly equation, honestly. It just doesn’t look like it).

Of course, I also have this strong suspicion that all of physics, in the final analysis, is just applied DE’s, so maybe I’m biased here.

9

I used the same book my junior year with less stellar results…I’m guessing you mean learn differential equations, and then read Boas. I say this from experience because I never really got the differential equations well, which killed me trying to use Boas.

And to the OP. My BS in Physics leaves me with many more questions than answers to both QM and Relativity. If you want to learn it, plan on graduate school and you can’t learn it without the math.

10

For a quick into on the weirdness of Quantun, get George Gamow’s “Mr Tomkins” books. But if you really want to learn QM, you’ll need to get a college-level text. There are plenty of these, most titled some variation of “Quantum Mechanics” – Schiff, Cohen-Tannoudji, etc. There’s a classic text titled something like "Introduction to Modern Physics by Kennard, Richtmeyer, and Lauritsen, and I learned by another text of that name by McGreevey. Browse through a technical bookstore or any large bookstore, or online.

For General Rel., one highly-recommended text, which also makes a pretty good coffeetable by itself, is Misner, Thorne, and Wheeler’s Gravitation. I know of no one who has read this massive tome all the way through. It’s written with a quirky sense of humor, but is frequently hard going. There are other, shorter and more easily comprehended texts on the subject, but I can’t name any right now.

11

One book that seems to come up a lot for GR courses is Bernard Schutz’s General Relativity: A First Course; it’s certainly less daunting than MTW and the other commonly mentioned texts (Wald, Weinberg). I didn’t find it to be all that great, but then I took my first GR course before I was probably really ready for it.

Since you’ll have to work a lot of problems to really learn this stuff, you should also know about a great book called Problem Book in Relativity and Gravitation by Alan Lightman. Lots of worked problems at varying levels of difficulty.

As far as quantum books go, I’ve used and referred to most of the most popular ones (Schiff, Messiah, Sakurai, Cohen-Tannoudji, Baym), and the one I keep going back to is my undergrad text, Principles of Quantum Mechanics by Shankar. Very well written, with a lengthy mathematical introduction that’s probably sufficient on its own, without sending you to other lin al references.

12

To go “whole hog” the easiest way is to get a graduate degree in physics. If you want to do it the hard way, ie. on your own…

Math: Calculus, differential equations, and linear algebra are the essentials. Most of the rest you can pick up as you go from the physics texts you’ll be using.

GR: The best intro book is by Hans C. Ohanian. Much easier to read and understand than MTW. (I’m not familiar with Schutz.)

QM: I really like the Liboff text. It’s used as an upper-level undergrad or first-year grad text. If you work your way through this book then you will know as much QM as most working physicists know. (NB: most working physicists are not real QM hotshots.) My favorite graduate-level text is by Leslie Ballentine. It’s one of the few QM texts that has extended discussion of the meaning and paradoxes of QM. He consistently uses a non-Copenhagen interpretation of QM, which I think makes more sense than the available alternatives. The 1st chapter is heavy on math, but it’s not really necessary to know what a “rigged Hilbert space” is to understand QM.

13

[Monty Burns]
Excellent!
[/Monty Burns}

My life is now set for the next few years. I live in North San Diego county but have cause in the next week to go down and visit my brother in SD proper. I’ll just have to print out this thread and browse around the SDSU used bookstores. Or maybe UCSD used bookstores would be better? Where the heck are those anyway? La Jolla, Mira Mesa?

At any rate you folks have come through as I know you would.

MM

14

I’ll second FriendRob on Ohanian (I assume you meant Ohanian and Ruffini…). I used it as an undergrad, found it to be pretty decent. Schutz was not a terrific book, in my experience.

As an undergrad, I used Eisberg and Resnick for my very first QM course; that might be a bit easier than starting with an advanced undergrad book. I’ve heard good things about both Shankar as well.

15

re: Boas

Billy:

It’s been awhile, but I do remember that all of us in that class had the same experience-- you don’t get the same theoretical understanding as you would from a "real’ math class, but you sure do learn a TON of math in a short time that you need to understand QM. Maybe we just had a good professor, but I don’t think he could’ve done it w/o that book.

16

The only math which you absolutely need to know for quantum is differential equations and matrix algebra (along with all prerequisites of those, of course: Multi-variable calculus, trig, algebra, etc.). For General Relativity, you’ll need to gain a very strong working knowledge of tensor analysis and differential geometry, but here I wouldn’t recommend taking those math classes. A good GR course or book will, of necessity, start by introducing you to all the math you need to know, and what exactly you need to know about it. A math course on the same subject, however, will spend a lot of time on concepts you’ll never have use for in physics, without covering the practical shortcuts which physicists use. If you have the soul of a mathematician, then you will walk away from a Differential Geometry course with a very deep and satisfying grasp of the material, but if you’re a physicist at heart, it’ll mostly just confuse or bore you. Quick litmus test: When you look at an expression like (dy/dx), do you see “A little piece of y, divided by a little piece of x”? If so, don’t bother with the diiferential geometry courses. A mathemetician would shudder at the thought of treating (dy/dx) that way. Like in quantum mechanics, matrix algebra and differential equations will be useful, but for rather different reasons. For physics prerequisites, a good E&M (Electricity and magnetism) course will also help prepare you for relativity.

There are also a lot of math subjects/classes which won’t be directly useful, but which will indirectly make the physics much easier. I highly recommend that you study abstract algebra/group theory at an undergrad level, for either quantum or GR, and some knowledge of probability/statistics will also help in quantum.

As for books, Liboff and Griffiths both have good undergraduate quantum books, and I would recommend either of them. For GR, MTW (Misner, Thorne, and Wheeler, mentioned above by CalMeacham, is sort of The Bible, and contains everything a person might possibly ever want to know about relativity, but it’s a pretty poor intro. I found that Schutz’s green book was much more accessible (and more portable! You’d get a hernia lugging MTW around), and it uses all of the same notational conventions as MTW, so it’ll make it easier to move up to that tome, should you so decide.

A final recommendation, by the way: It’s much more difficult to learn from a book (or books) than it is from a class. The best way to learn this material would be to go for a graduate physics degree, but short of that, most colleges will also let you sign up for (or at least sit in on) classes without enrolling for a degree. This’ll give you access to a professor to handle specific questions, and will also provide some guidance in assigning (and checking) homework and tests. Yes, classes cost money, but if you try to learn exclusively from the books, you’ll already be shelling out hundreds of dollars: College textbooks typically cost around $80 each.

17

It’s really only an introduction to Quantum Mechanics, but I think the particle adventure is pretty good. You may enjoy it. Go to:

http://particleadventure.org/
Steven

18

Thanks, Chronos. I was hoping you’d chime in. The thought of auditing classes hadn’t occured to me. It’s something I’m considering now, of course.

Just starting on the math is going to be fun. (No, really, I’m a math geek.) Well, that and the book shopping…

19

I enjoy reading Eisberg & Resnick at bedtime. It’s a little heavy on the verbage vs. equations, but the style is very easy to read and is stimulating. (BTW I’m not a physicist, though I have had higher math and junior-level engineering courses.)

  • sort of hijack *

I’m not affiliated with the internet book search service called “Bookfinder”, but I highly recommend it. It’s a good way to find used books and previous editions for fairly cheap.

20

While it may not be terribly relevant to someone learning this stuff off on their own, it’s worth mentioning that basic GR is usually regarded as significantly more advanced than basic QM. For example, in my case, as an undergraduate at the University of Edinburgh the set text at the start of second year was Alastair Rae’s Quantum Mechanics. It’s not terribly advanced, but if you master that, you have some claim to understand the subject. And everybody who got a physics degree had to have done so. On the other hand, only a few did GR in fourth year.
I’ll third the suggestion of Schutz - particularly because it’s an explicit attempt to do an undergraduate (in UK terms - Schutz is at Cardiff) version of MTW. Or at least a rewrite of Track 1.