Wolfram Demonstrations Project

Quantum billiards are an important class of systems showing a large variety of dynamical behavior ranging from regular motion through quasiperiodic behavior to strongly chaotic behavior. Suppose a single quantum particle, an atom, is in a superposition of two energy eigenstates, in the absence of measurement. This could be achieved by exciting the atom with coherent laser pulses. For this system, a transformation from regular to quasiperiodic motion is shown in the Bohm trajectories for this "unobserved" system. If, before emitting a photon, a position or energy measurement is made, the atom will remain in the superposition state.

A quantum trajectory in an infinite spherical potential well of radius

could be described by the de Broglie–Bohm approach [1, 2], using spherical Bessel functions.

Due to the large oscillations of the superposed wavefunction in configuration space, the trajectory could become very unstable and it could leave the billiard boundary for certain time intervals. To ensure that the trajectories lie within the billiard boundary, the initial point is restricted to the region

In the graphic, you can see the wave density (if enabled); a possible orbit of a quantum particle, where the trajectory (blue) depends on the initial starting point (

); the initial starting point of the trajectory (shown as a small black sphere) and the position (shown as a small blue sphere).

Contributed by: Klaus von Bloh

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An unnormalized wavefunction

of the Schrödinger equation

with mass

is separable in spherical polar coordinates, such that

where

is a spherical harmonic and

a spherical Bessel function [3]. The boundary condition that

is fulfilled when

is the

zero of the spherical Bessel function

. The quantized energy levels are then given by

where

is the central potential energy with

In this Demonstration, the total wavefunction

is defined by a superposition of two stationary eigenstates:

with

For simplicity, set the mass

and

equal to 1 (atomic units [4]).

In this case, the wavefunction for the quantum particle in an infinite spherical well in spherical polar coordinates [1] reads

The velocity field

with the position

can be calculated from the current

or from the gradient of the total phase function

from the wavefunction

in the eikonal form (often called polar form)

From the total phase function

the gradient

, with the partial derivative

and so on, is given by:

and

For the initial point

the velocity becomes indeterminate, and for

the velocity field

becomes autonomous, with

or in Cartesian coordinates,

and the atom orbits the

axis along a circle of constant radius and with a constant angular speed depending on the azimuthal quantum number

(here,

, which is a multiple of

For greater accuracy, increase PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps.

Due to the computational limitations, here only two superposed states are investigated. For this case, there is no chaotic behavior in the quantum motion [5, 6].

References

[1] Bohmian-Mechanics.net. (Oct 18, 2019)

[2] S. Goldstein, "Bohmian Mechanics," The Stanford Encyclopedia of Philosophy, Summer 2017 Edition (E. N. Zalta, ed.), (Oct 24, 2019) plato.stanford.edu/archives/sum2017/entries/qm-bohm.

[3] O. F. de Alcantara Bonfim, J. Florencio and F. C. Sá Barreto, "Chaotic Bohm’s Trajectories in a Quantum Circular Billiard," Physics Letters A, 277(3), 2000 pp. 129–134. doi:10.1016/S0375-9601(00)00705-2.

[4] Wikipedia. "Hartree Atomic Units." (Oct 18, 2019) en.wikipedia.org/wiki/Atomic_units.

[5] R. H. Parmenter and R. W. Valentine, "Deterministic Chaos and the Causal Interpretation of Quantum Mechanics," Physics Letters A, 201(1), 1995 pp. 1–8. doi:10.1016/0375-9601(95)00190-E.

[6] R. H. Parmenter and R. W. Valentine, "Erratum: Deterministic Chaos and the Causal Interpretation of Quantum Mechanics (Physics Letters A 210 (1995) 1)," Physics Letters A, 213(5–6), 1996 p. 319. doi:10.1016/0375-9601(96)00096-5.