The Causal Interpretation of a Particle in a Two-Dimensional Square Box
The de Broglie–Bohm interpretation of quantum theory contradicts the opinion that in the case of a macro system, the motion of the quantum system should approach the motion following from classical mechanics. Without measuring the momentum of the particle, there are some cases where the unobserved quantum particle is, in contradiction to the classical counterpart, at rest. In the case of motion, a quantum particle possesses highly nonclassical but well-defined trajectories. Furthermore, the motion of a quantum particle could be obtained when the corresponding particle density (given by the modulus of the Schrödinger wavefunction) is not time dependent. To study this effect, we consider a two-dimensional square box with infinite potential walls in a degenerated stationary state with a constant phase shift. A free particle is contained between impenetrable and perfectly reflecting walls, separated by a distance . In this case, the energy eigenvalues and eigenfunctions for the two-dimensional box can be derived from those of the infinite square-well solutions of the one-dimensional Schrödinger equation. This quantum system can exhibit motion in the associated de Broglie–Bohm theory. The origin of the motion lies in the relative phase of the total wavefunction, which has no classical analogue in particle mechanics.
The graphic shows the squared wavefunction, the particles, the trajectories (yellow), and the velocity field (red) for various constant phase factors. If the "quantum potential" button is active, you see the trajectories and the velocity field with the associated quantum potential.
A degenerate, unnormalized wavefunction for the two-dimensional box can be expressed by , where , , and are eigenfunctions and eigenenergies of the corresponding stationary one-dimensional Schrödinger equation with . By expressing the wavefunction in the eikonal form , the particle density and the velocity for this special superposition state become time independent. From the above definition it follows that the total amplitude and phase are:
(particle number density ) and
with , (amplitude functions), (phase function), and the complex conjugate . The corresponding autonomous differential equation system (velocity field in the configuration space) derived from the total phase of the wavefunction with mass is:
, ( component of the velocity),
( component of the velocity),
with , etc.. The quantum potential is defined as: . In the case of the two-dimensional box, the eigenfunctions and eigenenergies that obey the free stationary Schrödinger equation with Dirichlet boundary condition are:
, , with the wavenumbers , , and the total energy , where , . Adopting and , the equations turn into the desired form. For or with , the velocity becomes zero, because the quantum potential is precisely equal to the total kinetic energy . In both cases the particles are at rest. For all other cases, the quantum potential is very complicated. Definitively, the maximum velocity of a quantum particle depends on and the particle number density . The velocity field changes sign when and the modulus of the velocity reaches a maximum when with . Local singularities in the velocity field and do appear to exist for certain values (). In the program, if PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps are increased, the results will be more accurate.
 D. Bohm, "A Discussion of Certain Remarks by Einstein on Born's Probability Interpretation of the -function," Scientific Papers Presented to Max Born on His Retirement from the Tait Chair of Natural Philosophy in the University of Edinburgh, London: Oliver and Boyd, 1953 pp. 13–19.
 A. Einstein, "Elementare Überlegungen zur Interpretation der Grundlagen der Quanten-Mechanik," Scientific Papers Presented to Max Born on His Retirement from the Tait Chair of Natural Philosophy in the University of Edinburgh, London: Oliver and Boyd, 1953 pp. 33–40.