Testing idea of quasiclassics
If the idea that we can expand the Hamiltonian around quasiclassical minima is correct, then besides spin aligning along the x-axis as we increase \phi^r_x
we should also see how the mean of wavefunctions approaches each other. To imagine that visually try to execute in Mathematica:
Manipulate[
ContourPlot[(x - a)^2 + z^2/4 - 2 Sqrt[x^2 + z^2], {x,
0, +5}, {z, -10, +10}, Contours -> 100], {a, 0, 5}]
wherein both minima we expect that levels would be sitting there.
Numerically to see if such approximation is valid, we can evaluate $<\pm|\phi_x|\pm>$ and $<\pm|\phi_z|\pm>$ and plot those with respect to \phi^r_x
.