The Physical Pendulum in Quantum Mechanics

E. U. Condon

It is pointed out that the Mathieu functions of even order are the characteristic functions of the physical pendulum in the sense of Schr\"odinger's wave mechanics. The relation of various properties of the functions, as known from purely analytical investigations of them, to the pendulum problem is discussed.

Piotr Garbaczewski

J. J. Duistermaat

Gregory L. Baker, James A. Blackburn

Abstract The classical pendulum has a long and rich history in physics. On the other hand, the history of the quantum pendulum can only stretch back as far as the appearance of quantum mechanics itself in 1925–1926. Yet within three years of that exciting time, a paper on the quantum pendulum appeared in Physical Review authored by one of the pioneers of atomic physics, Edward Condon (Condon 1928). Condon’s paper laid out the basic problem and the means to its solution. Contributions to the study of the quantum pendulum continue to be made even in recent times. See, for example, (Pradhan and Khare 1973; Aldrovandi and Ferreira 1980; Cook and Zaidins 1986; Baker et al. 2002). In this tale we describe some of the novel and surprising features of the pendulum when it is analyzed according to the rules of quantum physics.

Giulio Casati

Enrique J. Galvez, Fabio J. Auccapuclla, Y Qin +2 more

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.