,

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.