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Bulletin of the American Physical Society

Quantum wave-mechanics of the simple pendulum via non-diffracting pendulum optical beams

Enrique J. Galvez, Jake M. Freedman, Joel Auccapuclla, Yingsi Qin, Kristina L. Wittler

Published Jun 3, 2020• 0 citations
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Parallels between the Helmholtz and Schrodinger equations can be exploited for using light beams to investigate quantum problems. We present the study of a type of non-diffracting beams known as pendulum beams, where the optical modes satisfy a form of the Helmholtz equation that is identical to the Schrodinger equation for the mechanical pendulum. We prepared optical beams in the corresponding eigenmodes and made measurements of their Fourier spectrum. We find remarkable quantitative agreement between the measured angular spectrum and the quantum mechanical probabilities.

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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.
Parallels between the Helmholtz and Schrodinger equations can be exploited for using light beams to investigate quantum problems. We present the study of a type of non-diffracting beams known as pendulum beams, where the optical modes satisfy a form of the Helmholtz equation that is identical to the Schrodinger equation for the mechanical pendulum. We prepared optical beams in the corresponding eigenmodes and made measurements of their Fourier spectrum. We find remarkable quantitative agreement between the measured angular spectrum and the quantum mechanical probabilities.