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.