Best way to learn QM comprehensively

[twofish-quant]

Also, it's useful to get different textbooks, because different people will explain the same things in different ways, and having different explanations will help you see what is going on. If one textbook doesn't make sense, then find the explanation in textbook B, and then once you understand that, then reread textbook A.

 

[Fredrik]

Can one explain to me why in the US such a delicate subject as QM is taught in 2 ways: to undergraduates and to graduates ? What's the difference and why is it a need to teach in 2 different ways ? A serious course should not be taught unless some specific knowledge is already present in the student's head.

To answer the OP, in my humble opinion, to learn QM comprehensively means to adopt the following path for physics & mathematics:

- Classical mechanics with a course based on Goldstein (+ Landau).
- Classical electromagnetism and special relativity with a course based on J.D. Jackson (+ Landau).
- Mathematics courses on linear algebra, abstract algebra, multivariate calculus, ODE's + PDE's, complex analysis and Fourier calculus, functional analysis and group (representation) theory (including Lie algebras).

With this whole package, one's ready for a serious QM book, like Galindo & Pascual (2 volumes).
Having learned QM comprehensively, the student can go for General Relativity based on Wald's book and formal QFT inspired from Weinberg's 3 volumes.

I disagree with some of this. I dislike Goldstein's book more than anything else in physics. I hate the notation, his explanations, and even the smell of the second edition. There is however a third, so maybe it doesn't have to smell bad. Landau & Lifgarbagez looks good to me, but I haven't actually studied it. It should be mentioned that it's a short book that explains how to solve a wide range of problems in classical mechanics. This makes me think that it's a great choice for people who already understand the foundations of the theory and want to get better at doing problems, but not such a great choice who people who just want to understand the foundations of the theory and to be able to solve simple problems. I don't really have any recommendations of my own. I guess V.I. Arnold for the math nerds, but for typical physics students...I just don't know. Maybe L & L is the way to go.

I'm also not a fan of Jackson. It's an absurdly difficult book, and the payoff isn't good enough to justify the effort. At least not for future theorists and teachers. But maybe it is for future engineers and experimentalists. The reason I say this is that it's a book about how to solve every conceivable type of problem, but it doesn't explain the foundations very clearly. At least that's how I remember it, but it's been a long time since I was forced to study it. I don't know what to recommend for classical electrodynamics, but I've seen recommendations in other threads, so if I had been thinking about buying a book on this topic, I'd search for recommendations in the book forum.

I don't know enough about Galindo & Pascual to comment on that recommendation. I'll just say that for QM, I think Griffiths looks adequate as an introduction. I would however supplement it with "Lectures on quantum theory: mathematical and structural foundations" by Chris Isham. This book is an excellent supplement to whatever a student uses as his/her first book, because it's a short (and cheap) book about what the theory actually says, and not so much about how to calculate stuff.

I think the importance of differential equations is always overstated in these threads, while the importance of linear algebra isn't emphasized enough. It's not a bad idea to take courses on differential equations, but they're not nearly as essential as linear algebra. Students who don't know the basics of linear algebra well will have a lot of trouble with concepts like spin. For linear algebra, I recommend Axler. Some people don't like it, so I will also mention Friedberg, Insel & Spence as an alternative.

Complex analysis is also not essential. I think it's definitely worthwhile to take a course on that topic, but students who haven't can still learn QM and QFT.

I also like Wald and Weinberg, but someone who's going to study GR from Wald, should also read Lee's books on differential geometry and a good text on special relativity. I like the SR sections in Schutz's GR book.

 

[dustytretch]

Wow, I could not have asked for more, thanks for all the help on this!
Now I have an idea about how I am going to tackle the quantum world, the mathematics I need for the task and what texts to look into and not look into.
And also learning classical mechanics as a prerequisite.

My plan of action is to order the M boas book and learn what
maths is needed to use the book.
Then after that I will order one or more of the recommended introductory
texts on QM.

Thanks Timothy Treciokas

 

[Fredrik]

And also learning classical mechanics as a prerequisite.

It's not absolutely essential to study classical mechanics first, but I would still recommend that you do. You should at least make sure that you understand what classical mechanics is. You don't have to be able to solve difficult problems in classical mechanics.

My plan of action is to order the M boas book and learn what
maths is needed to use the book.

I haven't read Boas, so I don't know if that recommendation was good or bad, but my first thought is that it might be better to get a book on calculus (e.g. Lang), and one on linear algebra (e.g. Friedberg, Insel & Spence, or Axler).

 

[dustytretch]

it might be better to get a book on calculus (e.g. Lang), and one on linear algebra (e.g. Friedberg, Insel & Spence, or Axler).

Will these books as well as M boas require a prerequisite knowledge of calculus and linear
algebra to understand?

EDIT: I just had a look at 'A First Course in Calculus' - Serge Lang on amazon, it looks like it needs no prerequisite.

Thanks for the further suggestions!

 

[Jorriss]

It's not absolutely essential to study classical mechanics first, but I would still recommend that you do. You should at least make sure that you understand what classical mechanics is. You don't have to be able to solve difficult problems in classical mechanics.

This has been my experience. For basic quantum mechanics (like UD quantum I-II) classical mechanics was not super necessary. However, for more specialized quantum classes such as molecular spectroscopy I found out I was totally ignorant of how much classical mechanics I needed. Stuff I didn't even imagine needing like rotational motion of rigid bodies, systems of coupled oscillators, etc, etc.

In other words, don't slack on your classical mechanics but don't kill yourself trying to get graduate level mechanics before even seeing quantum.

 

[saim_]

Will these books as well as M boas require a prerequisite knowledge of calculus and linear
algebra to understand?

EDIT: I just had a look at 'A First Course in Calculus' - Serge Lang on amazon, it looks like it needs no prerequisite.

Thanks for the further suggestions!

I suggest the time honored and established way. Learn the math you are supposed to right now. Do your A-level math as quickly as possible; maybe a few months or even lesser. That will have some calculus in it. Then just brush up on some concepts A-level maths sometimes misses like Taylor series, vector spaces, just a knowledge of what Fourier series is, and what partial differentiation and multiple integrals "mean", so you don't get scared at their first sight. Don't go learning how to evaluate them and all the techniques; that takes entire courses. You can complete this in a year. Next: open a nice gentle QM text.

 

[chill_factor]

The biggest challenge in QM other than the concepts is that it requires you to master basic math. Not just know basic math, but MASTER basic math. if you see an integral you can't solve, you must know how to use completing the square, trig substitutions, integration by parts, etc. to turn it into a form you can either solve, or plug into the computer.

You must also master (not just know, but master) the concepts of physics such as orthonormality of wavefunctions that will otherwise cause a great deal of pain if you happen to forget it, or about how wavefunctions are continuous, their derivatives are continuous, and how to set up boundary condition problems.

There are a lot of people who just haven't mastered basic math (they might know it, but they haven't mastered it), and thus simply can't do quantum.

 

[Fredrik]

if you see an integral you can't solve, you must know how to use completing the square, trig substitutions, integration by parts, etc. to turn it into a form you can either solve, or plug into the computer.

I don't agree with this part. I don't even remember how to integrate sin^2, but I understand QM as well as anyone here. (Edit: I do know how to integrate it: Just type "integrate sin^2" into Wolfram Alpha ). I do however agree that someone who wants to understand QM should master the concepts. For example, you should have a perfect understanding of what an integral is, but you don't need to know how to integrate all sorts of weird functions. You should have a perfect understanding of partial derivatives, and of what a differential equation is, but you don't have to be able to solve all sorts of weird differential equations. (There is however no excuse for not being able to take partial derivatives of all combinations of elementary functions, because this is easy). You should have a perfect understanding of the basic concepts from linear algebra, like the relationship between linear operators and matrices.

 

[jtbell]

you should have a perfect understanding of what an integral is, but you don't need to know how to integrate all sorts of weird functions.

That's what tables of integrals are for. An important skill here is to be able to recognize when you have an integral that's "close" to one in the table, and which substitution to make to get your integral to match the one in the table.

 

[genericusrnme]

Yeah, I agree with the above, knowing how to integrate functions isn't really as important as knowing WHAT an integral is.
Linear Algebra and group theory is what you really need.

Knowing that the wavefunction should be at least C^1 isn't even that important, you can understand QM without even thinking about that.

Modern QM is linear algebra, linear algebra and a little more linear algebra.

 

[dustytretch]

Okay, so a introductory book on linear algebra and on calculus is in the pipeline.
Try and 'master' linear algebra and basic math.
After that, try getting a good introductory text on QM and the M boas book on mathematics in physics.
This sound alright to you guys.

Oh and seriously, I can't be grateful enough for all the help, I owe you
guys.

Timothy Treciokas

 

[dustytretch]

I managed to snap up a copy of Serge Lang's 'A first course in Calculus' for £1.14 plus £2.80 postage for an ex-library copy on amazon!

 

[jtbell]

Yeah, I agree with the above, knowing how to integrate functions isn't really as important as knowing WHAT an integral is.

And being able to set up the proper integral for solving a problem, from a verbal or pictorial description of the situation.

 

[dustytretch]

Well, it is almost a year since I first posted this, I have come a long a bit.

I taught myself calculus from the Serge Lang book and now know it to the point of definite integrals and and indefinite integrals.
Knowing basic calculus really helps on a universal level, not just being able to
understand certain things in electrical engineering such as resonator circuits or
differential equations in physics but understanding what is behind certain things
and how things are worked out in such a large amount of subjects.
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.

 

[radium]

Even though it's probably not absolutely necessary to have taken classical mechanics before quantum mechanics, you will appreciate quantum mechanics much more if you have a background in classical mechanics. There are many fundamental ideas in quantum mechanics that have direct parallels in classical mechanics, like commutators, and Heisenberg's equation of motion and poisson brackets. Symmetry in quantum mechanics is incredibly profound and is much easier to see when you have experience with the classical analogue of a system. For example, you know from classical mechanics that angular momentum is the generator of rotations which is the same in quantum mechanics and gives rise to many amazing results.

 

[andrewkg]

Also instead of just buying books on the subject on reading you really should use what a poster above stated MIT OCW it is a fantastic thing, all of the lectures, notes, etc. doing something like that provides the opportunity for a regular schedule which is good for getting that math down for latter subjects in physics. In fact I'm using them now to learn those subjects. Also there is KhanA but theirs are not so great after trig, at lease from some of the reviews I've read.

 

[jesse73]

OCW is the older MIT courseware project.

edX is a newer improvement but it doesn't have as many courses.

 

[ZombieFeynman]

Can one explain to me why in the US such a delicate subject as QM is taught in 2 ways: to undergraduates and to graduates ? What's the difference and why is it a need to teach in 2 different ways ? A serious course should not be taught unless some specific knowledge is already present in the student's head.

To answer the OP, in my humble opinion, to learn QM comprehensively means to adopt the following path for physics & mathematics:

- Classical mechanics with a course based on Goldstein (+ Landau).
- Classical electromagnetism and special relativity with a course based on J.D. Jackson (+ Landau).
- Mathematics courses on linear algebra, abstract algebra, multivariate calculus, ODE's + PDE's, complex analysis and Fourier calculus, functional analysis and group (representation) theory (including Lie algebras).

With this whole package, one's ready for a serious QM book, like Galindo & Pascual (2 volumes).
Having learned QM comprehensively, the student can go for General Relativity based on Wald's book and formal QFT inspired from Weinberg's 3 volumes.

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!)

 

[Jorriss]

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.

It's more than that. There is absolutely no reason every student needs that much math for QM ever.

There are people who use QM every single day in their work but know nearly nothing of abstract algebra or functional analysis.

 

[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.