Connecting the spherical harmonics with the periodic orbits: application to optical microsphere cavities

The connection between the spherical harmonics and the stationary orbital states is theoretically established using the rotational transformation and the SU(2) algebra. Through the inverse quantum Fourier transform, the spherical harmonics can be decomposed into the superposition of the stationary orbital states. Using the partial sums of the decompositions, the gradual change of superpositions during the generation of spherical harmonics can be visualized. Furthermore, the expansion coefficients of hyperbolic sine functions are utilized to analytically derive the time-varying wave packet state corresponding to the inclined circular orbit. The orthogonality of Euler functions is further used to show that a periodic integration of the wave packet state can exactly generate the stationary orbital states. These derived formulas are expected to be used to explore the resonant modes of optical microsphere cavities.