# 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.

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Contributed by: Klaus von Bloh (August 2013)

Open content licensed under CC BY-NC-SA

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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.

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Reference

[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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