Quantum Pendulum

For an idealized classical pendulum consisting of a point mass attached to a massless rigid rod of length attached to a stationary pivot, in the absence of friction and air resistance, the energy is given by
where is the angular displacement from the vertical direction. The oscillation is presumed to occur between the limits , where to avoid the transition to a spherical pendulum. The exact solution for this classical problem is known (see, for example, [1]) and turns out to be very close to the behavior of a linear oscillator, for which can be approximated by . The natural frequency of oscillation is given by the series
where , the limiting linear approximation for the natural frequency (a result of great historical significance).
The nonlinear pendulum can be formulated as a quantum-mechanical problem represented by the Schrödinger equation
where . This has the form of Mathieu's differential equation, and its solutions are even and odd Mathieu functions of the form and [2]. However, we describe a more transparent solution, which uses the Fourier series used to compute the Mathieu functions.
Accordingly, the solution of the Schrödinger equation is represented by a Fourier expansion
This can be put in a more compact form:
The matrix elements of the Hamiltonian are given by
in terms of a set of normalized basis functions
, .
The built-in Mathematica function Eigensystem is then applied to compute the eigenvalues and eigenfunctions for , which are then displayed in the graphic.


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[1] A. Beléndez, C. Pascual, D. I. Méndez, T. Beléndez and C. Neipp, "Exact Solution for the Nonlinear Pendulum," Revista Brasileira de Ensino de Física, 29(4), 2007 pp. 645–648. doi:10.1590/S1806-11172007000400024.
[2] T. Pradhan and A. V. Khare, "Plane Pendulum in Quantum Mechanics," American Journal of Physics, 41(1), 1973 pp. 59–66. doi:10.1119/1.1987121.
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