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Connecting the spherical harmonics with the periodic orbits: application to optical microsphere cavities
Optics Express
|June 11, 2026
Summary
This study theoretically links spherical harmonics to stationary orbital states using rotational transformations and SU(2) algebra. The findings enable visualization of spherical harmonic generation and offer new formulas for exploring optical microsphere cavity resonant modes.
Area of Science:
- Quantum mechanics
- Mathematical physics
- Optical physics
Background:
- Spherical harmonics are crucial for describing wave functions in systems with spherical symmetry.
- Stationary orbital states are fundamental in quantum mechanics, representing stable energy levels.
- Understanding the relationship between these concepts is key for advanced quantum and optical applications.
Purpose of the Study:
- To theoretically establish the connection between spherical harmonics and stationary orbital states.
- To develop methods for visualizing the generation of spherical harmonics.
- To derive new formulas for analyzing optical microsphere cavities.
Main Methods:
- Utilizing rotational transformation and SU(2) algebra to link spherical harmonics and orbital states.
- Applying the inverse quantum Fourier transform for decomposition.
- Employing partial sums for visualization.
- Deriving time-varying wave packet states using hyperbolic sine function coefficients.
- Leveraging Euler function orthogonality for state generation.
Main Results:
- A theoretical framework connecting spherical harmonics and stationary orbital states is established.
- Spherical harmonics can be decomposed into superpositions of stationary orbital states via inverse quantum Fourier transform.
- Visualization of superposition changes during spherical harmonic generation is achieved.
- Analytical derivation of time-varying wave packet states for inclined orbits.
- Demonstration that periodic integration of wave packet states generates stationary orbital states.
Conclusions:
- The established connection provides a novel perspective on quantum states.
- The derived formulas offer a pathway for exploring resonant modes in optical microsphere cavities.
- This work bridges fundamental mathematical concepts with practical optical applications.
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