There follow comments on some assumptions in this analysis. Firstly, there is 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; this limit has, in any case, been taken in the binomial expansion. The same convention, of neglecting this deflection, was adopted by Weber et al. [51], when they analysed the spin polarization of transmitted electron waves.
Secondly, there is the issue of assuming that all electron spin directions are eigen-states of the Hamiltonian. In this, I depart from the tradition of analysis of PNR, where matrices are [28,29] used to represent the Zeeman energy, and the reflection coefficient, without any need for this approximation. I also depart from the work of Weber et al. [51] 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. I propose the production of a more ``first-principles'' model, which will use the matrix representation of the reflection coefficients, and therefore capture the Larmor precession, and other spin-flip scattering effects. However, this model is not proposed as a replacement for the one presented here, but as a complement to it. What I suggest is to subject both models, along with a third, completely classical, reflection model (section 2.7,) to experimental data, and use the well-established [52] 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 that can be placed in each model.
One reason for not simply abandoning all but the most ``first-principles'' of the models is given by Anderson [53], who points out that any system, more complicated than a molecule of approximately four atoms, is almost never in an eigen-state of its Hamiltonian, with the result that the Schrödinger equation does not 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 that implements a Schrödinger equation with the most realistic Hamiltonian will always be the most useful, in describing the real behaviour of the system.
In addition, I have devised a very different argument [10] for not always preferring the most first-principles model.
Thirdly, it is worthwhile to comment 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. [51]. At first glance, the classical-field analysis above appears to be entirely elastic. However, it is capable of assimilating the effect of these processes, which would appear as imaginary parts in the electric potential and magnetic flux density.