lavinia said:
I am just starting out in quantum mechanics. The approach has been to talk about Hermitian operators as obervables acting on a complex vector space of states.Mathematically this is simple stuff, linear algebra. The Shroedinger equation makes it a little more difficult as does the use of symmetries to find degeneracies. But net net this is easy mathematics. Basic Quantum Mechanics is not difficult because of the mathematics.
Classical phase space is just a manifold of points. This is a straight forward generalization of our visual picture of the space around us. Quantum Mechanical phase space is a complex vector space, and points in the phase space can be linearly combined to make new states. I think for anybody this is difficult to intuit. Also the whole deal with eigen states as the results of observations. We end up with a mathematical formalism that is interpreted in terms of weird experiments. That is hard.
In Mathematics unintuitive ideas come to life through examples and applications to other areas of mathematics. That is how I want to learn Quantum Mechanics. In this context the historical approach has got to be enlightening. I don't agree that it should not be taught. I want to know it.
Another conceptual problem with Quantum Mechanics is that it is taught deductively from a set of Axioms (as is Special Relativity). These Axioms can seem like pure formalism and the exposition of the subject a syllogism. That can not be a good way to teach. That is not the way mathematics is taught nor is it the way mathematical ideas are discovered. This again urges the historical approach at least to gain insight into why these Axioms are necessary and how they were thought of.
I am a little surprised by learning Physics. It is a strange experience. It seems that Physical theories(unlike most mathematical theories which is what I am familiar with) are axiomatic systems from which physical phenomena are revealed through pure deduction. This again argues for understanding why these axioms are assumed.
I have the opposite impression. Modern mathematics textbooks are very often in an awful "Bourbaki style". It's all well ordered in terms of axioms, definitions, lemmas, and theorems with proofs, but no intuition whatsoever. I think in math the intuition comes from the applications, and physics is of course full of applications of interesting math issues (from analytical geometry via calculus to group theory and topology nearly everything interesting in math finds applications in physics).
Concerning the historic approach, I'm a bit unsure. From my own experience, I don't like it at all. In high school we learned "old quantum mechanics" first. This is physics of a transition era from classical ways of thought, which were proven wrong by observations at the time (e.g., the very familiar stability of matter is a complete enigma to classical physics as soon as you know that atoms consist of a positive charged particle (now called atomic nucleus) surrounded by negatively chargend electrons, as was revealed by the famous gold-foil experiment by Rutherford. The collisions of ##\alpha## particles (He nuclei) were perfectly described by classical scattering theory in the Coulomb field (that's why the corresponding cross section is named after Rutherford). So far so good, but then you had to understand the bound state making up an atom itself, and this caused a lot of trouble since on the one hand Bohr had just to solve the classical motion of a charged particle running around the nucleus under the influence of the mutual electrostatic interaction and assuming the quantization of the action, which was an ingeneous ad-hoc extrapolation of Planck's treatment of black-body radiation. Lateron the model was refined by Sommerfeld. However it was contradicting classical electromagnetics according to which there should be some bremsstrahlung and in a very short time the electrons should crash into the nucleus, i.e., the atom should be unstable. Instead of the bremsstrahlun, however, what's observed are clean pretty narrow spectral lines whose frequency was given by the distance between the energy levels known from the Bohr-Sommerfeld model, which is totally ununderstandable from classical physics. Last but not least the chemists knew that hydrogen atoms are spherical and not little flat disks.
Learning the old quantum mechanics first, cements very wrong pictures in the students's mind whic h have to be unlearnt and corrected when you want them to understand modern quantum theory. As you rightly realized the problem is indeed not the math. Taking the wave-mechanics approach for a long time of the QM 1 lecture you deal just with a scalar field. Compared to electromagnetics, where you deal with a lot of scalar and vector fields which are coupled via Maxwell's equations, it's a piece of cake to deal with a single pseudoparabolic pde, known as the Schrödinger equation. The problem is that you have to build the entire intuition behind this math via this math itself. There is no correct intuitive picture from our classical experience, which ironically becomes a pretty tough question to answer, why it is right (the answer is decoherence, and the course-graining concept of quantum statistical physics, and after you swallowed this many loose ends of classical statistics like the Ehrenfest paradoxon the question of the absolute measure of entropy, etc. are solved either).
On the other hand, you are right in saying that learning about the historical development of theories can help a lot to gain intuition, and particularly for QM 1 I'd not know how to teach it without a (however brief) historical introduction about how the idea to have the wave-mechanics approach or, in my opinion even better, the representation free Hilbert space formulation a la Dirac. I've never taught QM 1, but I guess, I'd give a brief introduction about the historical development, starting with Planck's black-body law, then talking about "wave-particle dualism" (sparing out however, the completely misleading picture of photons being "particles" in any classical sense, and the photoelectric effect does NOT prove the necessity of photons, i.e., the necessity of quantizing the em. field, which is anyway not discussed in QM 1) and finally getting via the wave-mechanics heuristics a la Schrödinger as quickly as possible over to Dirac's very neat mathematical heuristics to the Hilbert-space ##C^*##-algebra approach, where first the commutation relations can be reduced to the Heisenberg algebra of position and momentum operators, i.e., a canonical quantization approach.
I've once taught QM 2, where I emphasized the role of symmetries, which is in my opinion the right intuition for modern physics anyway. The ##C^*## algebra of non-relativistic QM is then very clearly deduced from the ray-representation theory of the Galileo group, which then helps very much to understand, by the notion of mass is so different in Galilean physics (central charge of the Galileo algebra) compared to Minkowskian physics (Casimir operator of the proper orthochronous Poincare group). Given the evaluations of this lecture by the students, this approach seems to be not too bad since they liked it, although they argued that one better keeps the discussion of the Galilean part shorter and rather go farther in the relativstic part, whic h they found much more interesting. Of course, I did not teach the wrong wave-mechanics interpretation of relativsitic QT a la Bjorken-Drell vol. I. This is really too much sacrifice to the historical approach, since once the Dirac formalism is motivated in QM 1, there is no reason, not to start relativistic QT right away as a local quantum field theory.
I think the best way in a good balance between the historical and the deductive approach to teach theoretical physics can be found in Weinberg's textbooks, where he usually has a historical introduction but then develops the theory itself in the deductive way emphasizing the mathematical structure and its meaning in context of the physics. I can only recommend to read his "Lectures on Quantum Mechanics" and of course "Quantum Theory of Fields".
