Quantum pendulum explained

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 beT = (1)/(2) m l ^{2} • \phi ^{2,} U = mgl ( 1 - \cos\phi).

This results in the Hamiltonian

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

The time-dependent Schrödinger equation for the system is

i \hbar (d\Psi)/(dt) =- (\hbar ^{2})/(2 m l ^{2}) (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:

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

This is simply Mathieu's differential equation

(d ^{2} \psi)/(d η ^{2}) + \left( (2 m E l ^{2})/(\hbar ^{2}) - (2 m ^{2} g l ^{3})/(\hbar ^{2}) - (2 m ^{2} g l ^{3})/(\hbar ^{2}) \cos η \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 :(d ^{2} \psi)/(d η ^{2}) + \left( (2 m E l ^{2})/(\hbar ^{2}) - (2 m ^{2} g l ^{3})/(\hbar ^{2}) - (2 m ^{2} g l ^{3})/(\hbar ^{2}) \cos η \right) \psi = 0 , a _{n(q),} b _{n(q)} = (2 m E l ^{2})/(\hbar ^{2}) - (2 m ^{2} g l ^{3})/(\hbar ^{2}) .

The energies of the system,

E = m g l + (\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 = (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) = (CE(n, q, x))/(CE(n, q, 0 )) , S(b _{n(q),} q, x) = (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.

See also

• Quantum harmonic oscillator

Bibliography

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