Derive derivative matrix for radial direction
To understand any possible trajectory on the Weyl disc we need two derivative matrices. One in a radial direction and one in a \theta
direction. First, let's consider the radial direction.
In general with the change in \phi^r_x
multiple things happen with the wavefunction:
- Distribution changes its mean value
- The spin distribution changes.
- Distribution changes dispersions
\sigma_x^2
and\sigma_z^2
. - The main axis of the wavefunction change.
Since in general, the problem is nasty we will neglect spin fluctuations, so the quasiclassical minima do not change properties with \phi^r_x
. Then we can neglect effects (3) and (4), and consider (2) as nonfluctuating, fixed in the minima. Then the steps for derivation are:
- Construct
\psi
as a product of harmonic oscillator states and write an equation for spin components. - Take a derivative with a chain rule of the state.
- Take the derivative of the equation for the spin to figure out it's change due to
\phi_x^r
and put it back in in the full derivative in the previous step. - Project the derivative on the state
|\psi_+>
and|\psi_->
.