Journal of Physics Conference Series

From Free Motion on a 3-Sphere to the Zernike System of Wavefronts Inside a Circular Pupil

Kurt Bernardo Wolf

Published Apr 1, 2020• 3 citations

Abstract

Abstract Classical or quantum systems that stem from a basic symmetry are seen to be special in having several important properties. The harmonic oscillator and the Bohr system are such. Recent research into the Zernike system provides reasons to include it in this privileged class. Here we show that free motion on the 3-sphere can be projected down to produce classical orbits or complete and orthogonal bases for wavefronts in a circular pupil. This line of inquiry has been pursued in company with N.M. Atakishiyev, G.S. Pogosyan, C. Salto-Alegre, and A. Yakhno.

DOI: 10.1088/1742-6596/1540/1/012011

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Journal of Physics Conference Series

From Free Motion on a 3-Sphere to the Zernike System of Wavefronts Inside a Circular Pupil

Kurt Bernardo Wolf

Published Apr 1, 2020• 3 citations

Publisher

Abstract

DOI: 10.1088/1742-6596/1540/1/012011

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G. F. Torres del Castillo

Revista Mexicana de FísicaMay 1, 20251 citation

We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with

T^*{\mathbb R}^{2}={\mathbb C}^2

as classical phase space. We show that the eigenfunctions

ψ_n

of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space

{\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2

. They describe

{\mathbb Z}_n

-invariant motion of particle along a circle

S^1

in lens space

S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n

, where

{\mathbb Z}_n

is the cyclic group of rotation by an angle

2π/n

on the circle

S^1

n=1,2,...\,

. Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states

ψ_n

that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.