Learning Quantum Mechanics: Classical vs Operator Based

[James Jackson]

I'm intregued as to what route people learned QM to whatever level you're at. I was taught the 'classic' way - classical mechanics, fields, mathematics, statistical physics etc first, before moving onto the derivation of the 1D Schrodinger equation for a free particle (without operators etc, the intuitive modifying of the classical wave equation to satisfy the de Broglie conditions). This, of course, leads to solutions of the wavefunction for infinite square wells, potential barriers etc. From there the 3D equation was discussed, followed by a first look at spherical harmonics in the solution for the hydrogen atom.

The next step was the move to an operator based approach, which is where the true beauty really begins to be seen - previously quoted results just 'dropping out' of the mathematics, something which, of course, becomes even more amazing when looking at QFT.

I feel that I would have preferred to be taught linear algebra first and start from an operator based approach, as this is the language of QM. Whilst starting with a purely calculus approach does indicate where things are going, I feel it hides the subtle beauty...

 

[ZapperZ]

I think my path kinda parallel yours. I did my undergraduate work at U. of Wis-Madison, and after the 2 semesters of Intro Physics + calc, there was a course in "Modern Physics" that used Tipler's textbook. While I've read about SR and QM from popular journals, this was the first formal introduction to both subjects. While it was more of a survey of modern physics very much similar Intro Physics, it was a valuable introduction on what was to come, even at the superficial level. So this was a class I took while taking the undergrad. classical mechanics and ODE/PDE in the same semester.

It was only 2 semesters after that (after E&M) that I took the "hard core" QM course. I waited till I had vector algebra, but unfortunately I did not take linear algebra (it would have been helpful). At that time, UW also did not have a course in mathematical physics, so I remember constantly struggling with all these new mathematics that I had to learn on the spot, concurrently with the physics. This is why I recommend all those math-physics texts in my journal.

Zz.

 

[vanesch]

I'm intregued as to what route people learned QM to whatever level you're at.

Well, I had my first contact with QM in second year of engineering, and the professor followed a horrible book, something of the kind "Quantum mechanics: theory and applications" by Amnon Yariv, which threw in a mixture of algebra, intuitive wave equation stuff and a lot of hand waving. When I complained about that, she told me that if I really wanted to learn it, I should read Cohen Tannoudji, which I did. However, I was still not satisfied, and then she advized me to read Messiah. Now, that, to me, is just a great book. I read it almost completely (except the last few chapters), which then gave me the right mindset to read Cohen Tannoudji. I read most stuff on my own, but I could ask my professor whenever I had some problems (in the mean time, her course for engineers looked quite childish :-)
Much later, I read Modern Quantum Mechanics by Sakurai. I think that's a book that gives a lot of insight in much fewer pages than Cohen Tannoudji.

cheers,
Patrick.

 

[James Jackson]

Ah, Tippler's book - the only book that can actually flow between floorboards.

If I remember correctly, we were introduced to Schrodinger's Wave Equation in a very simplified way at the end of a first year course on oscillations and waves. We then, in our 2nd year QM course, looked at it in more detail (separation of variables, eigenfunctions and eigenvalues, harmonic oscillator, angular momentum, Legendre functions, hydrogen atom and a first look at spin). It's in the 3rd year that it got more interesting, operators were introduced, we looked at spin orbit coupling in detail, basic spin statistics theorem, measurement problems, EPR etc. The optional 3rd year course covered the variation principle, pertubation theory etc.

Perhaps the most interesting (Annd useful from a mathematical point of view) was the option quantum computation and information theory course. As it was taught by mathematicians, physicists and computer scientists, it gave a detailed background of the 'why' of the maths, not just the 'what', and introduced such helpful things as the density operator.

4th year modules cover relativistic QM, beginning of gauge tranformations and all that jazz.

I agree that the intro using purely calculus does give a good precis of what's to come, but I wonder what ground is to be gained by not starting straight with linear algebra and deriving results, rather than stating them then coming back to the derivation later on.

Then again, I've always had this issue with the education system - my revision technique usually consists of teaching myself the course 'above' the one I'm doing to get a solid grounding of where things are coming from.

I just wonder how many other people would prefer teaching to be this way.

 

[dextercioby]

I've had a course on Atom's & Molecule's physics in the first semester of the second year.In the I-st & the II-nd semester of the third year,i've had the course on QM.Very well done,even though some chapters were missed due to lack of time.The course's syllabus on the internet (even though only in romanian and .doc file) is impressive.

But i'd say,QM made à la Craiova is okay.First semester's intro was done following Prugoveçki [1] and Akhiezer & Glazman [2].

Daniel.
----------------------------------------------
[1]J.Prugoveçki,"Quantum Mechanics in Hilbert Space",1971.
[2] Akhizer & Glazman,"Theory of Linear Operators in Hilbert Spaces".

 

[James Jackson]

P.S. My main texts have been Rae, Mandl and then Quantum Computation and Quantim Information by Nielsen and Chuang, which is actually very readable as an intro to QM in a linear algebra framework by itself.

 

[James Jackson]

So you have approached it from the abstract mathematic way, rather than the 'classic' UK way of teaching it?

 

[dextercioby]

The first lecture was called "PreHilbert spaces.Hilbert spaces".And the first chapter (almost a full semester) was called "The Mathematical Foundations of Quantum Mechanics".The second "The Postulates of Quantum Mechanics" which were presented in the Dirac/traditional formulation in the Schrödinger picture.

Daniel.

 

[Meir Achuz]

I have always thought that QM should be taught from Dirac's book, after getting a good enough math background. (Of course, I wasn't taught it that way.) I rate QM books by how well they present Dirac's treatment. The real problem is teaching Classical before QM. As I read all the "paradoxes" on this website, I realize that all of them come by trying to fit QM into CM language. It's like trying to understand Newton in Aristotle's terms. The same holds true for Relativity. One theory is right and the other theory is wrong (although there are limits in which it gives a good approximation). Since most students are taught the linear way, it is important to remember to forget many classical concepts in understanding QM and SR.
Anyway, that's my sermon for today.

 

[James Jackson]

I agree in part, but, and this is a big but, without a gounding in classical physics, how would students know about conservation laws, angular momentum, work and energy etc etc. These are all still fundamental concepts in QM...

 

[Morbius]

I think my path kinda parallel yours. I did my undergraduate work at U. of Wis-Madison, and after the 2 semesters of Intro Physics + calc, there was a course in "Modern Physics" that used Tipler's textbook.

Zapper,

I also had a class in "Modern Physics" that used Tipler's book - but taught
by Norman Tepley instead of Paul Tipler.

Prof. Paul Tipler taught my first course in "Quantum Mechanics"
[Fall term 1973 at Oakland University, Rochester, Michigan ]

Dr. Gregory Greenman
Physicist

 

[quantumdude]

After 3 terms with Halliday and Resnick I learned from...

Junior Year:
Intro Quantum Physics I,II: Brehm and Mullin
Applied Atomic and Nuclear Physics: Eisberg and Resnick, Fermi (Nuclear Physics)
Quantum Chemistry: Lowe (book by same name)

Brehm and Mullin [BM] was user-friendly, but wayyyy to wordy. But I liked that I was able to plow through the exercises after reading the book. Eisberg and Resnick [ER] is parallel to [BM], but has an edge: It covers perturbation theory. [ER] also covers some interesting topics such as a derivation of the radiation from a circulating charge. Lowe's book I used while in a graduate level chemistry course (but the QM was upper-level undergrad physics). It was interesting to see the difference in the problems and methods that chemists are interested in (basically approximating spectra for complex atoms and molecules).

Senior Year
Quantum Mechanics I: Cohen-Tannoudji, Vol. I

I hated this book. It makes Brehm and Mullin look terse.

First Year Grad School
Quantum Mechanics II: Sakurai, Modern Quantum Mechanics
Statistical Mechanics: Huang

By the time I started grad school the department had switched QM books. This is still my favorite nonrelativistic QM book. Taking QM II with it made me want to re-take QM I. But my school was very expensive, so instead I read Chapters 1 and 2 on my own.

I did not like Huang's book at all, and according to every recommendation I receive Pathria is at least an order of magnitude better.

Second Year Grad School
Quantum Mechanics III: Sakurai, Advanced Quantum Mechanics
Nuclear and Particle Physics (Reading Course): Satchler; Halzin and Martin
Quantum Field Theory (Reading Course): Bjorken and Drell, vols I and II

I enjoyed Sakurai's book, but not as much as Mod. QM. I would have preferred one that is 1.) more modern and 2.) makes a distinction between contravariant and covariant vectors. It really is a pain to have to switch notation to compare results with other books. We covered the entire book in 1 semester, and it was a fairly good introduction to relativistic QM and QFT.

QM III is the end of the line at my school, and I wanted to do subatomic theory. So I did the 2 reading courses, one with an experimental particle physicist and the other with a theoretical particle physicist. I enjoyed Halzen and Martin very, very much. I also had a real appreciation for Bjorken and Drell after having gone through Sakurai.

I plan to take myself through this stuff again in the near future (I'm beefing up my math right now). With the wisdom of experience, I will choose Sakurai once again to review nonrelativistic QM. You definitely can't go wrong with that. I'd like to also read Pathria for stat mech. And for QFT I will first go through my new copy of Zee's book, after which I will purchase the Weinberg set. Then I plan to move ahead to strings via Zweibach and then Polchinski, if I can handle it.

 

[dextercioby]

Wow,Tom that's an impressive curricula you got there.I bow to you. Anyway,i still liked my education on theoretical physics and am willing to improve it with each day.

Daniel.

 

[pmb_phy]

I learned from Haliday and Resnick during Physics III. Then I went to a "Modern Physics" class for two semesters - Schroedinger equation "derivation" etc. Then in grad school we used "Quantum Mechanics," by Cohen-Tannoudji, et al. where its heavy on operators and bras and kets etc. Not to mention many other things I've read on the way.

Pete

 

[James Jackson]

Despite what texts people use, do you feel you learned the best way you could? Some have started from the abstract mathematics of Hilbert spaces, others (including myself) started from some basic results from Schrodinger's wave equation.

From what I can see, there are two routes, one followed more often than the other (you quess which...). Firstly, start with looking at solution of the wave equation in position representation, then introduce operators and see where it 'really' comes from. Secondly, introduce the abstract mathematics, linear algabra etc and derive the results the people on the other strand are told.

Considering the number of replies, I think this is an interesting point...

 

[dextercioby]

My course on QM was compulsory.The same things were taught to the future VI-VIII-th grade teachers,to the future high-school teachers,to the future experimentalists,theorists,shop-assistants,bank manager or who knows what other job you can get with a degree in phsyics.

In the seminars/exercise sessions,we'd do the same creepy exercises,starting with 1D problems solved with SE up until fine structure in the H atom,shapes of atomic and molecular orbitals etc.

There wasn't actually too much left for the student's research and documentation job,simply because in my country we lack the bibliography in physics...

Daniel.

 

[pmb_phy]

Despite what texts people use, do you feel you learned the best way you could? Some have started from the abstract mathematics of Hilbert spaces, others (including myself) started from some basic results from Schrodinger's wave equation.

From what I can see, there are two routes, one followed more often than the other (you quess which...). Firstly, start with looking at solution of the wave equation in position representation, then introduce operators and see where it 'really' comes from. Secondly, introduce the abstract mathematics, linear algabra etc and derive the results the people on the other strand are told.

Considering the number of replies, I think this is an interesting point...

How can one say how things would be if things were different? I myself prefer to learn several aspects and paths to end results.

Pete

 

[Dr Transport]

As a 2nd semester sophomore: An Intro to Quantum Physics, French and Taylor

Junior: Intro to Modern Physics, McGervey

Graduate student: Quantum Mechanics, Gordan Baym; Quantum Field Theory, Mandl
(my second time around) Quantum Mechanics, Liboff; Quantum Mechanics, Cohen and Tannouji
(I didn't learn anything new in my second graduate program taking QM)

 

[James Jackson]

It's called conjecture, Pete!

 

[masudr]

I learned straight from wikipedia and used definitions and theorems on hilbert spaces and bra-ket approach, as well as operators etc. you will obviously think that i do not have a sufficient knowledge to say i "know" qm, but i think I'm at least half the way there, and besides we get to do a proper course on it next year, in the 2nd year of my undergrad course.

 

[pmb_phy]

It's called conjecture, Pete!

Yes James. I understand that. What I meant was that we only end up with guesses. We will never know if those guesses are accurate. E.g. I find it impossible for me to make that guess in my case. If you were to point a gun at my head and force it out of me then I'd say I would have liked the Cohen-Tannoudji method (operators, Hilbert spaces etc.) Yet I also know that I never like to avoid the development of a subject from an historical standpoint. At this point the trigger is pulled.

Pete

 

[James Jackson]

I totally agree, I enjoy the historical aspect of it greatly too, seeing the path that lead to fully fledged modern QM, but that doesn't mean to say it needs to be taught in a mathematically historical way...

In some cases, it's downright confusing. If I recall correctly from A-Level, in Physics they teach the Bohr model as the quantum mechanical description of the atom, whilst in Chemistry they're graphing probability densities for the s, p and d orbitals of the hydrogen atom. Not quite joined up there...

Anyway, I would've preferred starting with the abstract, but I do enjoy my mathematics.

 

[ZapperZ]

In some cases, it's downright confusing. If I recall correctly from A-Level, in Physics they teach the Bohr model as the quantum mechanical description of the atom, whilst in Chemistry they're graphing probability densities for the s, p and d orbitals of the hydrogen atom. Not quite joined up there...

I actually do not mind going over the Bohr model. However, the way I've seen it done is that both the text and the instructor did not explicitly specify that this is an EARLY, primitive model to explain the atomic energy spectrum, and that while it was reasonably accurate for the H atom at that time, it failed with other atoms and certain has been replaced with a more accurate description. This I've not seen done in a very clear fashion.

I find it a good topic to go over because it illustrates to the students how an idea progresses and how knowledge is refined. A model that works sometime but not other times implies that there's something missing. But one usually start from some premise and improves on that. NO ONE comes out of nowhere and produces a ready-made and complete "grand unified theory". One would hope that such lessons will prevent budding quacks - but then again, most quacks do not enroll in physics classes such as these anyway.

Zz.

 

[vanesch]

I have always thought that QM should be taught from Dirac's book, after getting a good enough math background.

Well, I already said how I got into first contact with QM in my second year of engineering (that's 2 years after high school). The problem was indeed the math: although in parallel we got a quite good course on complex analysis and in the first year we got a good course on linear algebra, the true hilbert space stuff and functional analysis I met only MUCH MUCH later, after I graduated as an engineer, and I went to do my masters in physics, where I got a course on mathematical physics dealing rigorously with Hilbert spaces and so on. That was 4 years after my first contact with QM ! Although the mathematical physics course often gave examples in quantum mechanics, it wasn't a QM course per se, it was more a mathematics course.

The real problem is teaching Classical before QM.

The problem is more (the way I was taught the stuff) that I only had an analytical mechanics course in parallel with the QM course, so it is only towards the end of that course that the full machinery of lagrangians, hamiltonians, Poisson brackets and so on is put forward. The first part of the analytical mechanics course was more an application of Newton's laws to several classical problems (celestial mechanics, rigid body equations of motion...).
Another problem was that a rigorous course on electrodynamics and classical optics was only taught in the 4th year of engineering (2 years after the QM course). The basic EM course in the first year wasn't up to the level. In fact, often I used my intuition develloped in QM to get a feel for the EM or optical problems! Same with statistical physics, which was only taught for the first time in the 4th year (in the 2nd and 3rd years we got a sophisiticated course on classical thermodynamics - with applications to chemistry, but no link with statistical mechanics).

So I think I did about everything in the wrong order :-)

cheers,
Patrick.

 

[ZapperZ]

So I think I did about everything in the wrong order :-)

cheers,
Patrick.

Well, now that explains it then!

<runs and hides>

:)

Zz.

 

[vanesch]

Well, now that explains it then!

<runs and hides>

Where's the smiley for Zeus throwing his lightning ?

 

[gvk]

People around the world learn and learned QM using different textbooks. For example, I learned QM using Landau & Lif****z v.5 book. Those books were and remain so popular that students call them Landafshetz text. But despite their quality or popularity many students (if not all) were not satisfied with one-two sources and try to read as much as possible the original authors (Bohr, Heisenberg, Schrodinger, Dirac, Born, etc.). This way was, in my opinion, unavoidable because QM can not be tested by our own experience (we are 'classical' devices) and reading arguments of creators of QM is very important. I agree it's not shortest way to learn QM, but if you can not find a good textbook, this is only possible way to get a good understanding QM. Instead of this,
in my time, there were many professors who advocated something like this:
"go and 'to use to work with Schrodinger's equation' and get some solutions first, and then you will reach a feeling of understanding QM". It is interesting, how offen you listen it now?

 

[Kane O'Donnell]

Has anyone mentioned Liboff yet? Only a mid-level text, but very good. It's the recommended reference for our 2nd year QM course here at Newcastle, although after first year none of the courses have fixed texts.

 

[James Jackson]

I found Rae a good book for an introduction, but hardly uses dirac notation. Mandel was a lot better in that regards, and more 'formal' than Rae.

 

[gvk]

There is very good book on QM to the students who ready and want to solve ~200 QM problems, and on this way reach a feeling of understanding QM. It's "Practical Quantum Mechanics" by Siegfried Flugge, Springer-Verlag, Berlin 1971. All the problems have solutions with needed theoretical material.

 

[dextercioby]

A very good book,indeed.However,i think the interested (PhD) student would benefit more from solving (or at least trying to solve) the problems in Sakurai's book.

Daniel.

 

[Kane O'Donnell]

Dude, you *love* that book! This is like the hundredth thread you've mentioned it in! I'm starting to wish we had a copy at Newcastle.

Kane

 

[dextercioby]

You don't...? : You don't know what you're missing,though.

Daniel.

P.S.I'm a bigger fan of Galindo & Pascual's book.

 

[quantumdude]

Dude, you *love* that book! This is like the hundredth thread you've mentioned it in! I'm starting to wish we had a copy at Newcastle.

You're talking about Sakurai? If so: GET IT! GET IT! GET IT!

Seriously, I read that book for pleasure.

It is written with the purpose of graduating students to QFT, and it does a great job at it.

 

[Kane O'Donnell]

You don't...?

Nope, we don't. I should probably get the Physics Society here to buy a copy if it's that good

Kane