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 The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...

for the quantized system can be solved relatively easily.

Schrödinger equation

Using

Lagrangian mechanics In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...

from classical mechanics, one can develop a

Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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 m l^2 \dot^2,

U = mgl (1 - \cos\phi).

This results in the Hamiltonian :

\hat = \frac + mgl (1 - \cos\phi).

The time-dependent

for the system is :

i \hbar \frac = -\frac \frac + 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^,

E \psi = -\frac \frac + mgl (1 + \cos\eta) \psi.

This is simply Mathieu's differential equation :

\frac + \left(\frac - \frac - \frac \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 In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''cardinal ...

. 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 + \left(\frac - \frac - \frac \cos\eta\right) \psi = 0,

a_n(q), b_n(q) = \frac - \frac.

The energies of the system,

E = m g l + \frac

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.

A deep potential yields the dynamics of a particle in an independent potential. In contrast, in a shallow potential,

Bloch waves In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who di ...

, as well as

quantum tunneling In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...

, 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,

S(b_n(q), q, x) = \frac.

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.

Bibliography