I promised to comment on some assumptions in this analysis. Firstly, there's the matter of modelling the electrons as a plane wave. This is equivalent to neglecting the deflection of the electrons by the Lorentz force, which means taking the limit of weak magnetic fields; something we've done in the binomial expansion anyway. The same convention of neglecting this deflection was adopted by Weber et al. [17], when they analysed the spin polarization of transmitted electron waves.
Secondly, there's the issue of pretending that all electron spin directions are eigen-states of the Hamiltonian. In this we depart from the tradition of analysis of PNR, where matrices are [2,3] used to represent the Zeeman energy, and the reflection coefficient, without any need for this approximation. We also depart from the work of Weber et al. [17] on electron transmission: they regard the Larmor precession, which is a manifestation of the fact that not all spin directions are eigen-states of the Hamiltonian in a magnetic field, as crucial in determining the transmitted polarization. We intend to produce a more ``first-principles'' model in the near future, which will use the matrix representation of the reflection coefficients, and therefore capture the Larmor precession, and other spin-flip scattering effects. However, we don't intend to devise this model as a replacement for the one presented here, but as a complement to it. What we'd like to do is subject both models, along with a third, completely classical, reflection model, to experimental data, and use the well-established [14] methods of Bayesian statistics, first to infer the parameters of magnetic flux density, electric potential, and layer thickness, for each model, then to judge the relative confidence which we have in each model.
One reason for not simply abandoning all but the most ``first-principles'' of the models is given by Anderson [1], who points out that any system, more complicated than a molecule of four atoms or so, is pretty well never in an eigen-state of its Hamiltonian, so the Schrödinger equation doesn't describe the state of the system. This is because the tunnelling-like processes, which would otherwise collapse the system into an eigen-state of its Hamiltonian, are very slow for complicated systems: often very slow compared with the age of the universe, and certainly very slow compared with the rate of occurrence of measurement-like interactions with the outside world, which collapse the system into eigen-states of operators other than the Hamiltonian. Therefore, it can't be guaranteed that the model which implements a Schrödinger equation with the most realistic Hamiltonian will always be the most useful in describing the real behaviour of the system.
I might be inclined to add to this a very different argument [8] for not always preferring the most first-principles model, but this isn't the time or the place for my speculations on mathematical philosophy. Anyone who has a burning desire to hear them can find them via the reference on the slide.
Thirdly, it's worth commenting on the effect on the polarization of transmitted waves, due to spin-dependent loss of electrons to inelastic processes, which was noted by Weber et al. [17]. At first glance, our classical-field analysis appears to be entirely elastic. However, it is capable of assimilating the effect of these processes, which will appear as imaginary parts in the electric potential and magnetic flux density.