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
is the angular displacement from the vertical direction. The oscillation is presumed to occur between the limits
to avoid the transition to a spherical pendulum. The exact solution for this classical problem is known (see, for example, ) 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
, 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
. This has the form of Mathieu's differential equation, and its solutions are even and odd Mathieu functions of the form
. 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.