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

: $T = \frac{1}{2} m l^2 \dot{\phi}^2,$

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

This results in the Hamiltonian

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

The time-dependent Schrödinger equation for the system is

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

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:

: $\eta = \phi + \pi,$

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

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

This is simply Mathieu's differential equation

: $\frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0,$

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 $q$ :

: $\frac{d^2 \psi}{d \eta^2} + \left(\frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2} \cos\eta\right) \psi = 0,$

: $a_n(q), b_n(q) = \frac{2 m E l^2}{\hbar^2} - \frac{2 m^2 g l^3}{\hbar^2}.$

The energies of the system, $E = m g l + \frac{\hbar^2 a_n(q), b_n(q)}{2 m l^2}$ 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

: $q = \frac{m^2 g l^3}{\hbar^2}.$

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:

: $C(a_n(q), q, x) = \frac{CE(n, q, x)}{CE(n, q, 0)},$

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

Here are the first few periodic Mathieu cosine functions for $q = 1$ .

center

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

Bibliography

{{cite book | last1=Bransden | first1=B. H. | last2 = Joachain | first2 = C. J. | title = Quantum mechanics | edition = 2nd | publisher = Pearson Education|location=Essex| year = 2000|isbn=0-582-35691-1}}
{{cite book | last=Davies|first= John H.|title=The Physics of Low-Dimensional Semiconductors: An Introduction | publisher=Cambridge University Press|year=2006|isbn=0-521-48491-X|edition=6th reprint}}
{{cite book | last=Griffiths|first= David J.|title=Introduction to Quantum Mechanics |edition=2nd | publisher=Prentice Hall |year=2004 |isbn=0-13-111892-7}}
Muhammad Ayub, Atom Optics Quantum Pendulum, 2011, Islamabad, Pakistan., https://arxiv.org/abs/1012.6011

Category:Quantum models

Category:Pendulums