Borek said:
I was trying to help (elsewhere) a student with some QM related problem and I realized something. When discussing QM we underline the fact that electron doesn't behave as a particle, it behaves as a wave. Yet when we explain wave function we say something like "square of the wave function is a density probability of finding the electron" - which seems to suggests to students the electron is a particle that can be found in a given place with a given probability. I feel like it enforces thinking in terms of electron being a pointlike object and as such doesn't help students in understanding the subject.
Won't it be better to explain these things in terms of the charge density? Unless I am missing something very fundamental, charge density and probability density of finding electron are directly proportional to each other, so we are free to choose whether we call the square of the wave function "probability density" or "charge density", aren't we?
Sadly, fact that every QM textbook I know uses the same "probability density" approach probably renders the idea DOA.
(I am actually not sure where is the best place to discuss it, feel free to move the thread).
One must not explain in a wrong way first and then tell the students that it was wrong but put it in the right way from the very beginning. On the quantum level you must give up thinking in terms of classical particles as well as classical fields from the very first moment. To start the QM1 lecture, of course, you have to motivate this, and thus in the first lecture one should give a brief historical overview beginning with Planck's idea about the exchange of energy (and momentum) between the electromagnetic field and charged particles in terms of "quanta" ##\hbar \omega## and ##\hbar \vec{k}## respectively. In fact that's the right picture in some sense when it comes to relativistic QFT, which is the only still valid way to define "photons", but that cannot be the aim of the QM1 lecture.
Then unfortunately you have to go on with Einstein 1905, who interpreted the em. field in terms of a kind of particles. You can discuss the photoelectric effect and the Compton effect based on this heuristic picture, but you should also tell them that this is not how modern theory describes it. Nevertheless on the heuristic level it was an important step to realize that qualitatively the em. field has both "field (wave) aspects" and "particle aspects", which brought de Broglie to the idea that also for elementary particles (in his days electrons and protons) this might be the correct quantum description. Of course, you should again emphasize right away that this socalled "wave-particle dualism" is overcome with the modern quantum theory, which will be treated in your lecture now.
The next step in this approach is Schrödinger's work on the wave equation, and you should right away start with the non-relativistic treatment of free particles. Indeed, what you suggest was Schrödinger's original idea: You interpret the wave function ##\psi(t,\vec{x})## as a kind of classical field describing the particle, where ##|\psi|^2## is the density of particles. This is of course an analogy from the em. field, where ##1/2(\vec{E}^2+\vec{B}^2)## is the energy density of the field.
However, and this must be very carefully explained, this contradicts observations, because whenever you put a photo plate in the way of the moving electron it will give you a single spot but not a smooth density distribution, i.e., despite the fact that you use a field to describe the electron, again you find a particle aspect of it. The conclusion finally is the up-to-day valid interpretation of the wave function given by Born in a footnote in his famous paper on scattering processes (1926): Appropriately normalized, ##|\psi(t,\vec{x})|^2## is the probability distribution for finding the electron at position ##\vec{x}## at time ##t##.
Together with the Schrödinger equation that's the starting point for the entire modern formulation of QT. You'll find the usual operators (first in position representation) with an algebra which looks pretty similar to classical mechanics, the necessity of the wave functions forming a Hilbert space, the unitarity of time evolution (leading to the conservation of total probability, as it should be for a proper stochastic process) and so on. Finally everything ends up with Dirac's representation free formulation, operator algebras (first just the Heisenberg algebra generated from position and momentum, leading to the full non-relativistic description of a spinless particles), the commutator relations (which have their analogon in classical mechanics in the Poisson bracket formulation of Hamilton mechanics), and finally the importance of symmetry principles to get more general descriptions for particles with spin.
Then it's also natural to extend the entire business to many-body problems, including the important argument given in #2: The wave-function of an ##N##-body system is a function of time and ##3N##-dimensional configuration space and not some single-particle-density/continuum description. This rather follows from coarse graining in the sense of quantum-statistical mechanics, where you can derive several layers of semi-classical descriptions leading from the Kadanoff-Baym equations to transport equations and finally to ideal and viscous hydrodynamics as approximations to describe many-body systems in an efficient way, but the fundamental description is in terms of quantum theory, which is probabilistic in a fundamental way, and so far despite many attempts to overcome it over the last 9 decades, there's no working alternative formulation avoiding probabilities and statistics as fundamental ways to describe how matter behaves in Nature!
