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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 ${\displaystyle \phi }$) 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

${\displaystyle T={\frac {1}{2}}ml^{2}{\dot {\phi }}^{2},}$

${\displaystyle U=mgl(1-\cos \phi ).}$

This results in the Hamiltonian

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

The time-dependent Schrödinger equation for the system is

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

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:

${\displaystyle \eta =\phi +\pi ,}$

${\displaystyle \Psi =\psi e^{-iEt/\hbar },}$

${\displaystyle E\psi =-{\frac {\hbar ^{2}}{2ml^{2}}}{\frac {d^{2}\psi }{d\eta ^{2}}}+mgl(1+\cos \eta )\psi .}$

This is simply Mathieu's differential equation

${\displaystyle {\frac {d^{2}\psi }{d\eta ^{2}}}+\left({\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}\cos \eta \right)\psi =0,}$

whose solutions are Mathieu functions.

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

The boundary conditions in the quantum pendulum imply that ${\displaystyle a_{n}(q),b_{n}(q)}$ are as follows for a given ${\displaystyle q}$:

${\displaystyle {\frac {d^{2}\psi }{d\eta ^{2}}}+\left({\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}\cos \eta \right)\psi =0,}$

${\displaystyle a_{n}(q),b_{n}(q)={\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}.}$

The energies of the system, ${\displaystyle E=mgl+{\frac {\hbar ^{2}a_{n}(q),b_{n}(q)}{2ml^{2}}}}$ 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

${\displaystyle q={\frac {m^{2}gl^{3}}{\hbar ^{2}}}.}$

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.

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 ${\displaystyle a_{n}(q),b_{n}(q)}$, the Mathieu cosine and sine become periodic with a period of ${\displaystyle 2\pi }$.

For positive values of q, the following is true:

${\displaystyle C(a_{n}(q),q,x)={\frac {CE(n,q,x)}{CE(n,q,0)}},}$

${\displaystyle S(b_{n}(q),q,x)={\frac {SE(n,q,x)}{SE'(n,q,0)}}.}$

Here are the first few periodic Mathieu cosine functions for ${\displaystyle q=1}$.

Note that, for example, ${\displaystyle CE(1,1,x)}$ (green) resembles a cosine function, but with flatter hills and shallower valleys.

• Quantum harmonic oscillator

• Bransden, B. H.; Joachain, C. J. (2000). Quantum mechanics (2nd ed.). Essex: Pearson Education. ISBN 0-582-35691-1.
• Davies, John H. (2006). The Physics of Low-Dimensional Semiconductors: An Introduction (6th reprint ed.). Cambridge University Press. ISBN 0-521-48491-X.
• Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-111892-7.
• Muhammad Ayub, Atom Optics Quantum Pendulum, 2011, Islamabad, Pakistan., https://arxiv.org/abs/1012.6011