# 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
    $T={\frac {1}{2}}ml^{2}{\dot {\phi }}^{2},$
    $U=mgl(1-\cos \phi ).$
This results in the Hamiltonian
    ${\hat {H}}={\frac {{\hat {p}}^{2}}{2ml^{2}}}+mgl(1-\cos \phi ).$
The time-dependent Schrödinger equation for the system is
    $i\hbar {\frac {d\Psi }{dt}}=-{\frac {\hbar ^{2}}{2ml^{2}}}{\frac {d^{2}\Psi }{d\phi ^{2}}}+mgl(1-\cos \phi )\Psi .$
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:
    $\eta =\phi +\pi ,$
    $\Psi =\psi e^{-iEt/\hbar },$
    $E\psi =-{\frac {\hbar ^{2}}{2ml^{2}}}{\frac {d^{2}\psi }{d\eta ^{2}}}+mgl(1+\cos \eta )\psi .$
This is simply Mathieu's differential equation
    ${\frac {d^{2}\psi }{d\eta ^{2}}}+\left({\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}\cos \eta \right)\psi =0,$
whose solutions are Mathieu functions.

## Solutions
### 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 $q$:
    ${\frac {d^{2}\psi }{d\eta ^{2}}}+\left({\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}\cos \eta \right)\psi =0,$
    $a_{n}(q),b_{n}(q)={\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}.$
The energies of the system, $E=mgl+{\frac {\hbar ^{2}a_{n}(q),b_{n}(q)}{2ml^{2}}}$ 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
    $q={\frac {m^{2}gl^{3}}{\hbar ^{2}}}.$
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.