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 theory 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 {\mathrm {d} ^{2}\Psi }{\mathrm {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 {\mathrm {d} ^{2}\psi }{\mathrm {d} \eta ^{2}}}+mgl(1+\cos \eta )\psi .

This is simply Mathieu's equation

{\frac {\mathrm {d} ^{2}\psi }{\mathrm {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,

where the 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 L² norm equal $\pi$ ).

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

{\frac {\mathrm {d} ^{2}\psi }{\mathrm {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.

Bibliography

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., http://lanl.arxiv.org/PS_cache/arxiv/pdf/1012/1012.6011v1.pdf

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