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Journal of Physics A Mathematical and General

Special functions in phase space: Mathieu functions

Go. Torres-Vega, J. D. Morales‐Guzmán, Arturo Zúñiga-Segundo

Published Aug 7, 1998• 5 citations
Publisher

Abstract

We introduce functions which are solutions to a coherent-state representation of the Shrödinger equation for the pendulum potential. These functions are interpolation functions between the coordinate and momentum solutions for the quantum pendulum. We also introduce their classical analogues which are stationary solutions to the classical Liouville equation.

DOI
10.1088/0305-4470/31/31/017

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We introduce functions which are solutions to a coherent-state representation of the Shrödinger equation for the pendulum potential. These functions are interpolation functions between the coordinate and momentum solutions for the quantum pendulum. We also introduce their classical analogues which are stationary solutions to the classical Liouville equation.
Abstract The wave equation of electromagnetism, the Helmholtz equation, has the same form as the Schrödinger equation, and so optical waves can be used to study quantum mechanical problems. The electromagnetic wave solutions for non-diffracting beams lead to the two-dimensional Helmholtz equation. When expressed in elliptical coordinates the solution of the angular part is the same as the Schrödinger equation for the simple pendulum. The resulting optical eigenmodes, Mathieu modes, have an optical Fourier transform with a spatial intensity distribution that is proportional to the quantum mechanical probability for the pendulum. Comparison of Fourier intensities of eigenmodes are in excellent agreement with calculated quantum mechanical probabilities of pendulum stationary states. We further investigate wave-packet superpositions of a few modes and show that they mimic the libration and the nonlinear rotation of the classical pendulum, including revivals due to the quantized nature of superpositions. The ability to ‘dial a wavefunction’ with the optical modes allows the exploration of important aspects of quantum wave-mechanics and the pendulum that may not be possible with other physical systems.