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

@article{Wolf2020FromFM,
  title={From Free Motion on a 3-Sphere to the Zernike System of Wavefronts Inside a Circular Pupil},
  author={Kurt Bernardo Wolf},
  journal={Journal of Physics: Conference Series},
  year={2020},
  volume={1540}
}
  • K. Wolf
  • Published 1 April 2020
  • Physics
  • Journal of Physics: Conference Series
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… 
View on IOP Publishing
doi.org
2 Citations
Background Citations
1

2 Citations

Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by where q i and p i are generic canonical variables, γ n are arbitrary coefficients, and N∈N . For N = 2,…

On the generalization of classical Zernike system

We generalize the results obtained recently (Nonlinearity 36 (2023), 1143) by providing a very simple proof of the superintegrability of the Hamiltonian H = ~p 2 + F ( ~q · ~p ), ~q, ~p ∈ R 2 , for…

References

SHOWING 1-10 OF 18 REFERENCES

Superintegrable classical Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, as if it were a classical Hamiltonian with a non-standard potential. The…

Quantum superintegrable Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account, the quantum Zernike…

Elliptic basis for the Zernike system: Heun function solutions

The differential equation that defines the Zernike system, originally proposed to classify wavefront aberrations of the wavefields in the disk of a circular pupil, had been shown to separate in three…

New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion.

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the unit disk to classify wavefront aberrations in circular pupils is shown to have a set…

Quantum motion on the three-dimensional sphere: the ellipso-cylindrical bases

We study the free quantum motion on the three-dimensional sphere in ellipso-cylindrical coordinates, where we distinguish between prolate elliptic and oblate elliptic coordinates. The oblate and…

Spherical geometry, Zernike’s separability, and interbasis expansion coefficients

Free motion on a 3-sphere, properly projected on the 2-dimensional manifold of a disk, yields the Zernike system, which exhibits the fundamental properties of superintegrability. These include…

Interbasis expansions in the Zernike system

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I) serves to define a classical…

On the circle polynomials of Zernike and related orthogonal sets

  • A. BhatiaE. Wolf
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1954
ABSTRACT The paper is concerned with the construction of polynomials in two variables, which form a complete orthogonal set for the interior of the unit circle and which are ‘invariant in form’ with…

Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions

In this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems. We do this for…

Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space

Path integral formulations for the Smorodinsky-Winternitz potentials in twoand threedimensional Euclidean space are presented. We mention all coordinate systems which separate the…