# Teaching myself quantum mechanics in preparation for second year

1. Nov 29, 2012

### phosgene

So I've just finished my first year in physics and survived :) I have 3 or so months all to myself, so I figure I can spare a little bit of time to study some maths and physics in preparation for second year (the first semester course is purely quantum mechanics). I have Fundamentals of Quantum Mechanics by Shankar. I'm finding Shankar's explanation of bra-ket notation to be pretty confusing, especially since there aren't many problems or worked examples. All of the other resources I've found are way too simplistic to be of much help. So what I'm looking for is either something to supplement the mathematical section of Shankar, or a textbook that might be easier to follow. I'd very much appreciate any suggestions.

2. Nov 29, 2012

### Jorriss

Zettili Quantum Mechanics. Each chapter has a section on solved problems and there is a solid chapter on the math in QM including bra ket formalism.

3. Nov 29, 2012

### YAHA

The problem is not that you need a supplement to the mathematics section. It is that you need to understand the mathematics behind it on its own, even without physics. Study Linear Algebra as soon as possible since this is pretty much the only way to understand what Shankar is talking about.

I personally live and die by Zettili's book; however, his chapter on mathematics is not easy at all! In fact, I found that Shankar had a slightly better and more lucid exposition of bra-ket. Likewise, his Rotations chapter is also filled with similarly dense stuff.

There is really no easy way out (or should I say "in"?) here. You need the mathematics to understand anything in quantum mechanics.

4. Nov 29, 2012

### micromass

Staff Emeritus
Studying linear algebra (and in particular: inner-product spaces) will help you understanding bra-ket notation. So I suggest you do that.

5. Nov 29, 2012

### ahsanxr

Listen to micromass! I took my first Quantum class (from Griffiths) before having studied inner product spaces and was utterly confused about the formalism. Sure I could do all those integrals involved in calculating expectation values but I had no idea about the formal aspects of the theory and what it was based on. But once I took my PDE class which involved inner product spaces and how they are related to special functions etc, everything started making so much more sense.

I think learning bra-ket notation without having studied inner products and a bit about dual spaces from a Linear Algebra book (I recommend Friedberg) is bound to be confusing. Even if you do manage to get through it, you won't fully appreciate it (although it's not that big of a deal). So I suggest you put down Shankar for a while and learn those topics (the most important concepts will be inner product spaces, orthogonal bases and how it relates to fourier series and special functions, hermitian and unitary operators, diagonalization, the spectral theorem etc). Once you've done that, Shankar's first chapter will essentially be just you translating the math notation to QM notation. I also suggest Zettili. While Shankar's explanations are usually crystal clear, he does lack a bit on problems. Zettili should make up for that. Basically anything, as long as you aren't using Griffiths :).

6. Nov 30, 2012

### phosgene

Thanks everyone, I guess it's time for me to study some linear algebra :)

7. Dec 2, 2012

### ZombieFeynman

An exceptional book for getting your hands dirty with the mathematics behind physics is Hassani's Mathematical physics.

A good linear algebra text is Linear Algebra Done Right by Sheldon Axler.

Shankar is a very good book, though it can be rather advanced for a first go at quantum mechanics, especially if linear algebra is new to you. I may be in the minority, but I actually think Griffiths QM does a fair job at a first go. I think Zettili is very good book as well.

I think having a few books to attack QM is the right way of going about it, but not the only way. When I took is as a grad student I used to trifecta of Shankar, Merzbacher, and Messiah, often supplementing with Bohm's Quantum Theory and Dirac's Principles of QM. As an undergrad I solely used Griffiths and I actually learned a lot too.

8. Dec 2, 2012

### micromass

Staff Emeritus
Not saying the book is bad, but I always thought it is much better to read a math book rather than a mathematical methods or mathematical physics book. Then again, I'm a mathematician :tongue2:

9. Dec 2, 2012

### ZombieFeynman

Hehe, I think the distinction is what you want to learn the math for. If one wants to learn math for the sake of learning math, then learning from anything but a math book would be pretty silly. I think Hassani (and Byron and Fuller or Morse and Feshbach) is a good book for learning math directly to put it to use on some physics, while still keeping some rigor and thoroughness. I would contrast these types of math methods books with a book like Arfken, which I think does a particularly disastrous job.

Many physicists have the attitude towards math of "Get in, get out, do some physics." I'm not saying I personally feel this way, but I think it's not good to suffer from too much mathematical rigor-mortis.

If we physicists started proving the math behind the world too, instead of just using it, what would mathematicians do for a living? =P

10. Dec 2, 2012

### ahsanxr

Depends on the book. If its a book which takes the approach: "here's some math that occurs in physics, for category 1 type problems, perform these steps, for this type do this and so on", then yes I agree with you. However one simply does NOT lump Hassani's book into this category. It's certainly not as rigorous as a math book however the general approach that is taken is broad and unified, where definitions and theorems are clearly stated, and also proved if not too distracting. I think having these type of books gives you a good view of what's important and relevant and if you go through those chapters, it gives you a working knowledge of the topic. You can then zoom in and study in more detail with a pure math book if you want a deeper understanding.

Also, if every physicist took the approach you suggest, while it would probably be best in the long run, short-term they won't be able to move on with the physics. For example, I'm doing an independent study with the goal of trying to learn enough manifold theory to understand the symplectic formulation of Hamiltonian mechanics. While I've read parts of Lee's Smooth Manifolds book, and I think it's a great book, it moves a bit slow for my purposes. So I resorted to reading Hassani's and Schutz's books so that I make sure I can actually get to H-mechanics as opposed to me still learning how to prove theorems about manifolds at the end of the semester.

Last edited: Dec 2, 2012
11. Dec 2, 2012

### micromass

Staff Emeritus
I was joking. I'm no physicist so I shouldn't tell other physicists how to do their job :tongue2:

However, I do feel that if you want to get a deep and rigorous knowledge of some math, then you will need to go to math books. Once I've seen a physicist who found equations in QM that contradicted other equations. If they studied functional analysis, then they would have seen where they went wrong. I do realize that a functional analysis course would be mostly useless to most physicists though.

I'm sure Hassani's book is wonderful (although I personally didn't like it), but it doesn't really compare to fundamental math books such as Lee or Do Carmo.

Again, I'm not telling physicists what to do. If you want a working knowledge of the math fast, then a mathematical physics book is probably the best way to go. I'm not disagreeing there. I'm just saying that personally I would be completely unsatisfied with that approach and I would not feel confident enough to actually do research. But again, I'm a mathematician.

12. Dec 2, 2012

### ZombieFeynman

I know you were joking. I was just busting your chops.

I was a math major that defected (converted?) and I still have a soft spot for the pure stuff.

13. Dec 2, 2012

### micromass

Staff Emeritus
Kill the traitor!!!!!!

14. Dec 2, 2012

### Jorriss

Arfken & Weber is a pretty good book imo, but it serves a different role than Hassani. Arfken & Weber really is a cookie cutter methods book, that's not what Hassani aims to be.

15. Dec 3, 2012

### ahsanxr

Are you talking about something like this?

http://physics.stackexchange.com/questions/14116/whats-wrong-with-this-derivation-that-i-hbar-0/

Yeah I can see how functional analysis would have been able to help since they probably go a bit deeper than saying that $$<x|x'> = \delta(x-x')$$