Quantum pendulum

The quantum pendulum is fundamental in understanding hindered internal rotations in chemistry, quantum features of scattering atoms, as well as numerous other quantum phenomena.

Though a pendulum not subject to the small-angle approximation has an inherent nonlinearity, the Schrödinger equation for the quantized system can be solved relatively easily.

Using Lagrangian mechanics from classical mechanics, one can develop a Hamiltonian for the system.

A simple pendulum has one generalized coordinate (the angular displacement

) and two constraints (the length of the string and the plane of motion).

The kinetic and potential energies of the system can be found to be This results in the Hamiltonian The time-dependent Schrödinger equation for the system is One must solve the time-independent Schrödinger equation to find the energy levels and corresponding eigenstates.

This is best accomplished by changing the independent variable as follows: This is simply Mathieu's differential equation whose solutions are Mathieu functions.

, for countably many special values of

, called characteristic values, the Mathieu equation admits solutions that are periodic with period

The characteristic values of the Mathieu cosine, sine functions respectively are written

The periodic special cases of the Mathieu cosine and sine functions are often written

The boundary conditions in the quantum pendulum imply that

for even/odd solutions respectively, are quantized based on the characteristic values found by solving the Mathieu equation.

In contrast, in a shallow potential, Bloch waves, as well as quantum tunneling, become of importance.

The general solution of the above differential equation for a given value of a and q is a set of linearly independent Mathieu cosines and Mathieu sines, which are even and odd solutions respectively.

In general, the Mathieu functions are aperiodic; however, for characteristic values of

For positive values of q, the following is true: Here are the first few periodic Mathieu cosine functions for

(green) resembles a cosine function, but with flatter hills and shallower valleys.