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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 ϕ {\displaystyle \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 = 1 2 m l 2 ϕ ˙ 2 , {\displaystyle T={\frac {1}{2}}ml^{2}{\dot {\phi }}^{2},} U = m g l ( 1 − cos ⁡ ϕ ) . {\displaystyle U=mgl(1-\cos \phi ).}

This results in the Hamiltonian

H ^ = p ^ 2 2 m l 2 + m g l ( 1 − cos ⁡ ϕ ) . {\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2ml^{2}}}+mgl(1-\cos \phi ).}

The time-dependent Schrödinger equation for the system is

i ℏ d Ψ d t = − ℏ 2 2 m l 2 d 2 Ψ d ϕ 2 + m g l ( 1 − cos ⁡ ϕ ) Ψ . {\displaystyle i\hbar {\frac {d\Psi }{dt}}=-{\frac {\hbar ^{2}}{2ml^{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:

η = ϕ + π , {\displaystyle \eta =\phi +\pi ,} Ψ = ψ e − i E t / ℏ , {\displaystyle \Psi =\psi e^{-iEt/\hbar },} E ψ = − ℏ 2 2 m l 2 d 2 ψ d η 2 + m g l ( 1 + cos ⁡ η ) ψ . {\displaystyle E\psi =-{\frac {\hbar ^{2}}{2ml^{2}}}{\frac {d^{2}\psi }{d\eta ^{2}}}+mgl(1+\cos \eta )\psi .}

This is simply Mathieu's differential equation

d 2 ψ d η 2 + ( 2 m E l 2 ℏ 2 − 2 m 2 g l 3 ℏ 2 − 2 m 2 g l 3 ℏ 2 cos ⁡ η ) ψ = 0 , {\displaystyle {\frac {d^{2}\psi }{d\eta ^{2}}}+\left({\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}\cos \eta \right)\psi =0,}

whose solutions are Mathieu functions.

Solutions

Energies

Given q {\displaystyle q} , for countably many special values of a {\displaystyle a} , called characteristic values, the Mathieu equation admits solutions that are periodic with period 2 π {\displaystyle 2\pi } . The characteristic values of the Mathieu cosine, sine functions respectively are written a n ( q ) , b n ( q ) {\displaystyle a_{n}(q),b_{n}(q)} , where n {\displaystyle n} is a natural number. The periodic special cases of the Mathieu cosine and sine functions are often written C E ( n , q , x ) , S E ( n , q , x ) {\displaystyle CE(n,q,x),SE(n,q,x)} respectively, although they are traditionally given a different normalization (namely, that their L 2 {\displaystyle L^{2}} norm equals π {\displaystyle \pi } ).

The boundary conditions in the quantum pendulum imply that a n ( q ) , b n ( q ) {\displaystyle a_{n}(q),b_{n}(q)} are as follows for a given q {\displaystyle q} :

d 2 ψ d η 2 + ( 2 m E l 2 ℏ 2 − 2 m 2 g l 3 ℏ 2 − 2 m 2 g l 3 ℏ 2 cos ⁡ η ) ψ = 0 , {\displaystyle {\frac {d^{2}\psi }{d\eta ^{2}}}+\left({\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}\cos \eta \right)\psi =0,} a n ( q ) , b n ( q ) = 2 m E l 2 ℏ 2 − 2 m 2 g l 3 ℏ 2 . {\displaystyle a_{n}(q),b_{n}(q)={\frac {2mEl^{2}}{\hbar ^{2}}}-{\frac {2m^{2}gl^{3}}{\hbar ^{2}}}.}

The energies of the system, E = m g l + ℏ 2 a n ( q ) , b n ( q ) 2 m l 2 {\displaystyle E=mgl+{\frac {\hbar ^{2}a_{n}(q),b_{n}(q)}{2ml^{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 ℏ 2 . {\displaystyle q={\frac {m^{2}gl^{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 ) {\displaystyle a_{n}(q),b_{n}(q)} , the Mathieu cosine and sine become periodic with a period of 2 π {\displaystyle 2\pi } .

Eigenstates

For positive values of q, the following is true:

C ( a n ( q ) , q , x ) = C E ( n , q , x ) C E ( n , q , 0 ) , {\displaystyle C(a_{n}(q),q,x)={\frac {CE(n,q,x)}{CE(n,q,0)}},} S ( b n ( q ) , q , x ) = S E ( n , q , x ) S E ′ ( n , q , 0 ) . {\displaystyle 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 {\displaystyle q=1} .

Note that, for example, C E ( 1 , 1 , x ) {\displaystyle CE(1,1,x)} (green) resembles a cosine function, but with flatter hills and shallower valleys.

See also

• Quantum harmonic oscillator

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., https://arxiv.org/abs/1012.6011