Regularized canonical correlation analysis

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 $\varphi$) 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:

$T=\frac{1}{2}m{l}^2{\dot{\varphi }}^2$

$U=mgl(1-\cos (\varphi ))$

This results in the Hamiltonian:

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

The time-dependent Schrödinger equation for the system is as follows:

$i\hslash \frac{d\mathrm{\Psi }}{dt}=-\frac{{\hslash }^2}{2m{l}^2}\frac{{\mathrm{d}}^2\mathrm{\Psi }}{\mathrm{d}{\varphi }^2}+mgl(1-\cos (\varphi ))\mathrm{\Psi }$

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:

$\eta =\varphi +\pi$

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

This is simply Mathieu's equation where the solutions are Mathieu functions

$0=\frac{{\mathrm{d}}^2\psi }{\mathrm{d}{\eta }^2}+(\frac{2m{E}^2}{{\hslash }^2}-\frac{2m^{2}g{l}^3}{{\hslash }^2}-\frac{2m^{2}g{l}^3}{{\hslash }^2}\cos (\eta ))\psi$

Solutions

Energies

Given $q$, for countably many special values of $a$, called characteristic values, the Mathieu equation admits solutions which 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:

$0=\frac{{\mathrm{d}}^2\psi }{\mathrm{d}{\eta }^2}+(\frac{2mE{l}^2}{{\hslash }^2}-\frac{2m^{2}g{l}^3}{{\hslash }^2}-\frac{2m^{2}g{l}^3}{{\hslash }^2}\cos (\eta ))\psi$

$a_n(q),b_n(q)=\frac{2mE{l}^2}{{\hslash }^2}-\frac{2m^{2}g{l}^3}{{\hslash }^2}$

The energies of the system, $E=mgl+\frac{{\hslash }^2{a}_n(q),b_n(q)}{2m{l}^2}$ 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 follows:

$q=\frac{m^{2}g{l}^3}{{\hslash }^2}$

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 $a_n(q),b_n(q)$, the Mathieu cosine and sine become periodic with a period of $2\pi$.

Eigenstates

For positive values of q, the following is true:

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

$S(b_n(q),q,x)=\frac{SE(n,q,x)}{S{E}^{\prime }(n,q,0)}.$

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

File:MathieuCE.gif

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

Bibliography

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Muhammad Ayub, Atom Optics Quantum Pendulum, 2011, Islamabad, Pakistan., http://lanl.arxiv.org/PS_cache/arxiv/pdf/1012/1012.6011v1.pdf

External links

• Atom Optics Quantum Pendulum