Bohm Trajectories for a Particle in an Infinite 3D Box HD

Programmed by: Klaus von Bloh 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 video considers a three-dimensional cubic box with infinite potential walls in a degenerate stationary state with a constant geometric phase shift (see reference [2]). A free particle is contained between impenetrable and perfectly reflecting walls, separated by a distance Pi. A system with three degrees of freedom assigned by a superposition of three permuted 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 Pi The origin of the motion lies in the relative phase of the total wavefunction, which leads to a Pancharatnam--Berry-phase like behavior. 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 value of the constant phase shift determines the length of the path for each trajectory. 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 eight trajectories. Colored points mark the actual positions of the quantum particles and yellows points indicate the end positions Sorry, some artifact effects occur during displaying the trajectories. References [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. [2] Klaus von Bloh,"Bohm Trajectories for a Particle in an Infinite 3D Box", http://demonstrations.wolfram.com/BohmTrajectoriesForAParticleInAnInfinite3DBox/ Wolfram Demonstrations Project, Published: August 5, 2013