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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