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.

Schrödinger equation
Using Lagrangian mechanics from classical mechanics, one can develop a Hamiltonian for the system. A simple pendulum has one generalized coordinate (the angular displacement $$\phi$$) 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


 * $$T = \frac{1}{2} m l^2 \dot{\phi}^2,$$
 * $$U = mgl (1 - \cos\phi).$$

This results in the Hamiltonian


 * $$\hat{H} = \frac{\hat{p}^2}{2 m l^2} + mgl (1 - \cos\phi).$$

The time-dependent Schrödinger equation for the system is


 * $$i \hbar \frac{d\Psi}{dt} = -\frac{\hbar^2}{2 m l^2} \frac{d^2 \Psi}{d \phi^2} + mgl (1 - \cos\phi) \Psi.$$

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:
 * $$\eta = \phi + \pi,$$
 * $$\Psi = \psi e^{-iEt/\hbar},$$
 * $$E \psi = -\frac{\hbar^2}{2 m l^2} \frac{d^2 \psi}{d \eta^2} + mgl (1 + \cos\eta) \psi.$$

This is simply Mathieu's differential equation


 * $$\frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0,$$

whose solutions are Mathieu functions.

Energies
Given $$q$$, for countably many special values of $$a$$, called characteristic values, the Mathieu equation admits solutions that are periodic with period $$2\pi$$. The characteristic values of the Mathieu cosine, sine functions respectively are written $$a_n(q), b_n(q)$$, where $$n$$ is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written $$CE(n,q,x), SE(n,q,x)$$ respectively, although they are traditionally given a different normalization (namely, that their $$L^2$$norm equals $$\pi$$).

The boundary conditions in the quantum pendulum imply that $$a_n(q), b_n(q)$$ are as follows for a given $$q$$:


 * $$ \frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0,$$


 * $$a_n(q), b_n(q) = \frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2}.$$

The energies of the system, $$E = m g l + \frac{\hbar^2 a_n(q), b_n(q)}{2 m l^2}$$ for even/odd solutions respectively, are quantized based on the characteristic values found by solving the Mathieu equation.

The effective potential depth can be defined as


 * $$q = \frac{m^2 g l^3}{\hbar^2}.$$

A deep potential yields the dynamics of a particle in an independent potential. In contrast, in a shallow potential, Bloch waves, as well as quantum tunneling, become of importance.

General solution
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 $$a_n(q), b_n(q)$$, the Mathieu cosine and sine become periodic with a period of $$2\pi$$.

Eigenstates
For positive values of q, the following is true:
 * $$C(a_n(q), q, x) = \frac{CE(n, q, x)}{CE(n, q, 0)},$$
 * $$S(b_n(q), q, x) = \frac{SE(n, q, x)}{SE'(n, q, 0)}.$$

Here are the first few periodic Mathieu cosine functions for $$q = 1$$. Note that, for example, $$CE(1, 1, x)$$ (green) resembles a cosine function, but with flatter hills and shallower valleys.