Best way to learn QM comprehensively

[WannabeNewton]

Honestly I am still put off by the insane abstractness of the math in the level of QM dexter talked about. I am much more interested in the physical phenomena and the experiments for which you don't need such formidable weaponry. In GR even if the math is somewhat abstract it is still geometrical in nature and I can still appreciate the immediate relativistic experiments framed in that so and so abstractness. The Pascual level of QM on the other hand takes abstractness to a much crazier level for a physical theory imo and that's why I personally would rather just learn the immediate physics and the experiments for which the required math is, as Jorriss stated, not nearly as high level as dexter mentioned.

 

[atyy]

I think the important thing is to learn QM simply. Nielsen and Chuang give a very good simple minded exposition in their quantum computation text. There's also a very good course by Venkataraman Balakrishnan http://www.youtube.com/playlist?list=PL405247A063BECD66. The third volume of the Feynman lectures also gives the structural outline of QM.

At the more advanced level, there's a lot of nonsense, including a major conceptual error in Ballentine's famous review article.

I then picked up this: https://www.amazon.com/dp/0393091066/?tag=pfamazon01-20 and quickly realized I needed a better or wider knowledge of the language of maths beyond just simple calculus and need a better knowledge of vectors and applications of calculus and differential equations in physical systems.
I picked up this http://www.amazon.com/dp/0521679710/?tag=pfamazon01-20 and have glanced over it, but when the summer holidays start for me I will start getting more serious.

I agree with two-fish quant that Taylor and Fench is superb to start with. It's best to just get your feet wet, and learn from simple-minded standard undergrad sources. These give you the ideas that are very, very successful experimentally. I haven't read the book by Riley, but it looks like it covers the standard material. Again, although this is simple minded, why not? If one wants to learn absolutely correctly, one would learn nothing, since all the useful QFTs don't have rigourous constructions. Even approaching via effective field theory is not universally acknowledged to solve all problems. While the simple minded view may not be absolutely correct, it is certainly not misleading - no more than thinking of differentiation as speed, and integration as total distance traveled and the area under the curve - in the sense that physics with calculus is just calculationally more powerful, but not more conceptual than physics without calculus.

This is like saying one should never attempt to play an actual piece on piano until one has practiced enough to play Totentanz flawlessly. It's not realistic.

In my opinion, early exposure to quantum mechanics can provide much for a student:

1. Developing early and simple intuition about how the quantum world behaves

2. Preparing the student for difficult formal manipulations (in graduate courses) using easier formal manipulations in an exciting modern context

3. Giving a student an idea of what to pay attention to in advanced mechanics and electrodynamics courses (I know when I heard about hamiltonians in an upper division mechanics course after taking a griffiths level QM course that I should pay attention!)

IIRC, Barenboim was criticized for playing the Hammerklavier sonata as a young man because he didn't have enough maturity. I believe he said that it's wasn't going to get better by not playing it. Tovey, I think also said that no advice is mischievous as that which discourages young people from playing the great Beethoven sonatas because correctness is not enough. I think my favourite anecdote is Kovacevich saying Brendel taught him how to play some difficult passage - just leave out a few notes - the gain in musicality is much greater, and no one will hear the omission anyway (not sure I'm getting the protagonists right)!

 

[Lavabug]

My first exposure to QM was in my junior year, in a course based on Cohen-Tannoudji's book. The first week was dedicated to the experimental facts that led to its development, the postulates, and then straight into state vector formalism which was developed alongside wf notation, though we were made to see how the former was the more sensible option very early.

I already had two analytical mechanics courses and a MM course that covered transforms special functions and hilbert spaces at that stage, so it made sense to use the full machinery. I wouldn't have seen the point of covering things like deriving Ehrenfest's theorem or Hamilton's equations in QM if I hadn't seen them in mechanics beforehand. Knowing a bit of proper mechanics also motivates the QHO and how it differs from the classical mass on a spring. We also had to prove Newton's EOM in Dirac notation which I thought was a neat break.

Looking back it was a great idea on the part of the instructor because it was taught in the most generalized fashion possible. QM2 which I'm doing at a different university barely requires me to study anything new at all because of this. If I had studied from Griffiths I think I would've not retained much of it, most of my new peers who did use it didn't remember much.

But then again I adhere to the "shut up and calculate" philosophy, because I'm more interested in QM's applications. IMO an "intuitive" approach just means stating the same thing with more verbosity, which doesn't help me solve problems or understand at all. IMO there's no point in learning QM without the proper math because it's not a "touchy-feely" subject where you can "guess answers" based on a gut feeling. There's just no way you can explain what the spectrum of a QHO looks like with vector diagrams or thinking of micro/macroscopic forces or energy conservation like you could in much of classical physics.

 

[jesse73]

This is like saying one should never attempt to play an actual piece on piano until one has practiced enough to play Totentanz flawlessly. It's not realistic.

In my opinion, early exposure to quantum mechanics can provide much for a student:

1. Developing early and simple intuition about how the quantum world behaves

2. Preparing the student for difficult formal manipulations (in graduate courses) using easier formal manipulations in an exciting modern context

3. Giving a student an idea of what to pay attention to in advanced mechanics and electrodynamics courses (I know when I heard about hamiltonians in an upper division mechanics course after taking a griffiths level QM course that I should pay attention!)

To add to this. Learning is not a one off affair. People learn piece by piece by refining on previous refinements. Intuition is best developed as early as possible.

 

[Arsenic&Lace]

Is the amount of math described by dexter even required by the serious theorist? Maybe a mathematical physicist, but a typical theorist?

 

[Fredrik]

Is the amount of math described by dexter even required by the serious theorist? Maybe a mathematical physicist, but a typical theorist?

I think it depends a lot on what their specialty is. Some of the things he mentioned, in particular differential equations and electrodynamics based on Jackson, is probably of more use to some experimental physicists than to most theoretical physicists. And most physicists don't need more abstract algebra than you can learn from Wikipedia. So you might be able to do without some of the things on the list. On the other hand, he left out some stuff too. The big omission is differential geometry. This is essential for general relativity and for some areas of particle physics. And you may need a course on topology before you can study differential geometry.

Graduate students in theoretical physics are often required to study some advanced topics on their own, like complex manifolds or fiber bundles. This is going to be very hard if you don't have the mathematical maturity, so even if you're not sure that you're going to need each math subject on the list, it's still going to be good for you to study all of them.

Mathematical physicists definitely need a course on topology, a course on measure and integration theory, and probably several on functional analysis.