A Completely Classical Polarized Electron Reflection Theory

Above, (section 2.6,) I suggested that it might be enlightening to develop a completely classical theory of polarized electron reflection; this will be done in this section. The classical condition for reflection from a bulk sample is rather simple. In terms of the dimensionless numbers defined in equations 2.4, 2.5, and 2.6, an electron is reflected if

$$x_1 \geq{} 1\textrm{,}$$ (2.14)

and is not reflected if

$$x_1 < 1\textrm{.}$$ (2.15)

This thesis is (section 2.2) concerned with circumstances where $\vert x_1\vert \ll{} 1$. In this case, classical theory predicts a zero intensity reflection coefficient. Therefore, the measured reflections presented in chapter 5, and those reviewed in section 2.10, are purely quantum effects.

This does not imply that the classical theory is of no interest; an important test of the plausibility of the quantum theory presented earlier in this chapter is its obedience to Ehrenfest's theorem, which states [45] that, when the wave-function varies in space much more rapidly than the potential, the predictions of classical mechanics and quantum mechanics are the same.

In this case, the potential includes a step. Therefore, the wave-function varies in space much more rapidly than the potential only as the energy of the incident electrons tends to infinity. In this case, both classical theory, and the quantum theory presented earlier in this chapter, predict a zero reflection coefficient, and Ehrenfest's theorem is obeyed.