Bohr-Sommerfeld Quantization of Pseudospin Hamiltonians

It is shown here how to map the problem with pseudospin $J$ into an equivalent one in which $\frac{1}{J}$ plays the role of $\ensuremath{\hbar}$ and canonical variables exist at the classical level. Bohr-Sommerfeld quantization of the equivalent theory is found to produce a spectrum in very good agreement with the exact results for the Lipkin-Meshkov-Glick model at $J=15 \mathrm{and} 25$. The method readily extends to the $\mathrm{SU}(n)$ case.