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 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 ) 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:
This results in the Hamiltonian:
The time-dependent Schrödinger equation for the system is as follows:
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:
This is simply Mathieu's equation where the solutions are Mathieu functions
Solutions
Energies
Given , for countably many special values of , called characteristic values, the Mathieu equation admits solutions which are periodic with period . The characteristic values of the Mathieu cosine, sine functions respectively are written , where n is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written respectively, although they are traditionally given a different normalization (namely, that their L2 norm equal ).
The boundary conditions in the quantum pendulum imply that are as follows for a given q:
The energies of the system, 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:
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 , the Mathieu cosine and sine become periodic with a period of .
Eigenstates
For positive values of q, the following is true:
Here are the first few periodic Mathieu cosine functions for q=1:
Note that, for example, (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