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Quantum pendulum

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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.

Contents

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 ϕ ) 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 , U = m g l ( 1 − cos ⁡ ϕ ) .

This results in the Hamiltonian

H ^ = p ^ 2 2 m l 2 + m g l ( 1 − cos ⁡ ϕ ) .

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 .

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 , Ψ = ψ e − i E t / ℏ , E ψ = − ℏ 2 2 m l 2 d 2 ψ d η 2 + m g l ( 1 + cos ⁡ η ) ψ .

This is simply Mathieu's 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 ,

where the solutions are Mathieu functions.

Energies

Given q , for countably many special values of a , called characteristic values, the Mathieu equation admits solutions that are periodic with period 2 π . 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 C E ( n , q , x ) , S E ( n , q , x ) respectively, although they are traditionally given a different normalization (namely, that their L2 norm equal π ).

The boundary conditions in the quantum pendulum imply that a n ( q ) , b n ( q ) are as follows for a given 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 , a n ( q ) , b n ( q ) = 2 m E l 2 ℏ 2 − 2 m 2 g l 3 ℏ 2 .

The energies of the system, E = m g l + ℏ 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 = m 2 g l 3 ℏ 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 π .

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 ) , S ( b n ( q ) , q , x ) = S E ( n , q , x ) S E ′ ( n , q , 0 ) .

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

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

References

Quantum pendulum Wikipedia
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