Bohm Trajectories for a Particle in an Infinite 3D Box

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Quantum motion occurs in configuration-space particle trajectories associated with the de Broglie–Bohm causal interpretation of quantum mechanics. Previous Demonstrations (see related links) showed that 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. In addition to [1], the conditions for chaotic behavior can occur in a system with three degrees of freedom for a time independent wavefunction.


This Demonstration considers a three-dimensional cubic box with infinite potential walls in a degenerate stationary state with a constant phase shift. A free particle is contained between impenetrable and perfectly reflecting walls, separated by a distance . A system with three degrees of freedom assigned by a superposition of three stationary eigenfunctions that has commensurate energy eigenvalues and with a relative constant phase shift can exhibit chaotic motion in the associated de Broglie–Bohm picture, provided that the constant phase is not zero or an integer multiple of . The origin of the motion lies in the relative phase of the total wavefunction, which has no analog in classical particle mechanics. The dynamic structure is quite complex. Some of the curves are closed and periodic, while others are quasi-periodic. In the region of nodal points of the wavefunction the trajectories apparently become chaotic. The parameters have to be chosen carefully, because of the singularities of the velocities and the large oscillations that can lead to very unstable trajectories. Further investigation to capture the full dynamics of the system is necessary. The graphics show three-dimensional contour plots of the squared wavefunction and two initially neighboring trajectories. Black points mark the actual positions of the two quantum particles.


Contributed by: Klaus von Bloh (August 2013)
Open content licensed under CC BY-NC-SA



A degenerate, unnormalized wavefunction for the three-dimensional box can be expressed by , where and are eigenfunctions and permuted 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.

In the case of the three-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 , , , and . Adopting , and , the equations turn into the desired form. The corresponding autonomous differential equation system (velocity field) is derived from the phase function of the total wavefunction.

In the program, increase PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps for greater accuracy.


[1] R. H. Parmenter and R. W. Valentine, "Deterministic Chaos and the Causal Interpretation of Quantum Mechanics," Physics Letters A, 201, 1995 pp. 1–8. doi:10.1016/0375-9601(96)00096-5.

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