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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 non-linearity, 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 <math> \phi </math>) and two constraints (the length of the string is constant and there is no motion along the z axis). The kinetic energy and potential energy of the system can be found to be as follows:
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:<math>T=\frac{1}{2} m l^2 \dot{\phi}^2 </math>
:<math>U=m g l (1-\cos(\phi)) </math>
 
This results in the Hamiltonian:
:<math>\hat{H} = \frac{\hat{p}^2}{2 m l^2} + m g l (1-\cos(\phi))</math>
 
The time-dependent [[Schrödinger]] equation for the system is as follows:
 
:<math>i \hbar \frac{d\Psi}{dt} = - \frac{\hbar^2}{2 m l^2} \frac {\mathrm{d}^2 \Psi} {\mathrm{d} \phi^2}+m g l (1-\cos(\phi)) \Psi </math>
 
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:
:<math>\eta = \phi + \pi </math>
:<math>E \psi = - \frac{\hbar^2}{2 m l^2} \frac {\mathrm{d}^2 \psi} {\mathrm{d} \eta^2}+m g l (1+\cos(\eta)) \psi </math>
 
This is simply Mathieu's equation where the solutions are [[Mathieu functions]]
 
:<math>0 = \frac {\mathrm{d}^2 \psi} {\mathrm{d} \eta^2}+(\frac{2 m E^2} {\hbar^2}-\frac{2 m^2 g l^3} {\hbar^2}-\frac{2 m^2 g l^3} {\hbar^2} \cos(\eta)) \psi </math>
 
==Solutions==
 
===Energies===
Given <math>q</math>, for countably many special values of <math>a</math>, called ''characteristic values'', the Mathieu equation admits solutions which are periodic with period <math>2\pi</math>. The characteristic values of the Mathieu cosine, sine functions respectively are written <math>a_n(q), \, b_n(q)</math>, where n is a [[natural number]]. The periodic special cases of the Mathieu cosine and sine functions are often written <math>CE(n,q,x), \, SE(n,q,x)</math> respectively, although they are traditionally given a different normalization (namely, that their L<sup>2</sup> norm equal <math>\pi</math>). 
 
The boundary conditions in the quantum pendulum imply that <math>a_n(q), \, b_n(q)</math> are as follows for a given q:
 
:<math>0 = \frac {\mathrm{d}^2 \psi} {\mathrm{d} \eta^2}+(\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)) \psi </math>
 
:<math>a_n(q), \, b_n(q)=\frac{2 m E l^2} {\hbar^2}-\frac{2 m^2 g l^3} {\hbar^2} </math>
 
The energies of the system, <math>E=m g l+\frac{\hbar^2 a_n(q), \, b_n(q)}{2 m l^2}</math> 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 follows:
 
<math>q=\frac{m^2 g l^3} {\hbar^2} </math>
 
A depth potential depth yields the dynamics of a particle in an independent potential. In contrast, a shallow potential depth, [[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 <math> a_n(q), \, b_n(q) </math>, the Mathieu cosine and sine become periodic with a period of <math> 2 \pi </math>.
 
===Eigenstates===
 
For positive values of q, the following is true:
:<math>C \left( a_n(q),q,x \right) = \frac{CE(n,q,x)}{CE(n,q,0)}</math>
:<math>S \left( b_n(q),q,x \right) = \frac{SE(n,q,x)}{SE^\prime(n,q,0)}.</math>
Here are the first few periodic Mathieu cosine functions for q=1:
[[Image:MathieuCE.gif|center]]
Note that, for example, <math>CE(1,1,x)</math> (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., http://lanl.arxiv.org/PS_cache/arxiv/pdf/1012/1012.6011v1.pdf
 
==External links==
* [http://lanl.arxiv.org/PS_cache/arxiv/pdf/1012/1012.6011v1.pdf Atom Optics Quantum Pendulum]
 
{{DEFAULTSORT:Quantum Pendulum}}
[[Category:Quantum mechanics]]
[[Category:Quantum models]]

Revision as of 05:34, 26 February 2014


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