Semiclassical Husimi functions for spin systems

We derive a semiclassical approximation to the Husimi functions of stationary states of spin systems. We rederive the Bohr-Sommerfeld quantization for spin by locating the poles of the corresponding local Green function. The residues correspond to the Husimi functions, which are seen to agree very well with exact calculations.

Figure 1

Exact and semiclassical Husimi distributions for the simple Hamiltonian $H=\omega \hslash J_z$.

Figure 2

Exact (top) and semiclassical (bottom) Husimi distributions for the 200th state of the LMG model, with $j=200,\omega =1$, and $\hslash \alpha =1000$.

Figure 3

Exact (top) and semiclassical (bottom) Husimi distributions for the 270th state of the LMG model. The parameters are the same as in the previous figure.