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American Journal of Physics

Coherent states for the bouncing pendulum and the paddle ball

Mark Andrews

Published Feb 11, 2008• 1 citation
Publisher

Abstract

The coherent states of the simple harmonic oscillator with an impenetrable barrier at its center are studied. This half oscillator is the equivalent of a pendulum that bounces elastically off a vertical wall directly below the point of suspension with the angle of swing sufficiently small. The system can also be considered as a paddle ball, where the paddle is fixed and the ball is constrained by a spring attached to the paddle. The coherent states are almost the same as the familiar Gaussian coherent states of the full oscillator, except when they overlap the barrier. The solutions can be easily extended to two and three dimensions and gravity can be included if the impenetrable barrier is vertical. To better understand the form of the expectation values of the position and momentum, we investigate some general aspects of the effect of impenetrable barriers on the dynamics of wavepackets.

DOI
10.1119/1.2825388

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The coherent states of the simple harmonic oscillator with an impenetrable barrier at its center are studied. This half oscillator is the equivalent of a pendulum that bounces elastically off a vertical wall directly below the point of suspension with the angle of swing sufficiently small. The system can also be considered as a paddle ball, where the paddle is fixed and the ball is constrained by a spring attached to the paddle. The coherent states are almost the same as the familiar Gaussian coherent states of the full oscillator, except when they overlap the barrier. The solutions can be easily extended to two and three dimensions and gravity can be included if the impenetrable barrier is vertical. To better understand the form of the expectation values of the position and momentum, we investigate some general aspects of the effect of impenetrable barriers on the dynamics of wavepackets.
The usual 'quantum bouncer' is a quantum particle bouncing elastically under the influence of gravity. Here we consider a particle bouncing off an impenetrable wall, and bound to it by the harmonic potential instead of by the gravitational potential mgx . An analytic solution is possible, easily accessible to classes in intermediate quantum mechanics. The solution is related to Schrödinger's oscillating wavepacket (the original coherent state ). Although periodic, it changes its form during the oscillation, in contrast to Schrödinger's packet. The uncertainties in the position (Δ x ) and the momentum (Δ p ) oscillate also, as does the product Δ x Δ p .
We consider time-dependent Gaussian wave packet solutions of the Schrödinger equation, with arbitrary initial central position, x 0 , and momentum, p 0 , for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets . We show how difference or mirror solutions of the form ψ( x , t ) − ψ(− x , t ) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate ⟨ p 2 ⟩ t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, ⟨ x ⟩ t , and momentum, ⟨ p ⟩ t , and their relation to the impulsive force during the `collision' with the wall. Finally, the x 0 , p 0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.
We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate.
Gaussian Klauder coherent states are constructed for the harmonic oscillator, the planar rotor, and the particle in a box. The standard harmonic oscillator coherent states are given by expansions in the eigenstates of the Hamiltonian in terms of a complex parameter \ensuremath{\alpha}. When the complex modulus of \ensuremath{\alpha} is large, these states are identical in behavior with a particular choice of Gaussian Klauder coherent state. When the angular momentum of a planar rotor is large compared with Planck's constant, the angle distribution associated with a Gaussian Klauder coherent state for this case remains sharply localized for many rotations. Similarly, for the particle in a box, it is possible to choose parameters in the Gaussian Klauder coherent state so that a localized particle bounces back and forth at constant velocity between the walls of the box for many periods without significant delocalization. Buried in this behavior is the Fourier series for a triangle wave. These examples show how Gaussian Klauder coherent states are of utility in understanding quantum-classical correspondence.