Influence of the Relative Phase in the de Broglie-Bohm Theory

Does the possibility exist for motion of a quantum particle to occur when the particle density, given by the square of the Schrödinger wavefunction, is not time-dependent? This question is answered in this Demonstration. A system with two degrees of freedom assigned by a superposition of two stationary eigenfunctions that has commensurate energy eigenvalues can exhibit motion in the associated de Broglie–Bohm theory, provided that the constant phases and are not zero or integer multiples of . The origin of the motion lies in the relative phase of the total wavefunction, which has no classical analogue in particle mechanics. As an example a wavefunction for a quantum isotropic harmonic oscillator is chosen: , where is the Hermite polynomial and , are arbitrary constants. In this case the squared wavefunction (particle density ) is not time-dependent:
.
The corresponding autonomous differential equation system (velocity field) derived from the phase of the total wavefunction is: (in the direction) and (in the direction), which can be integrated numerically with respect to time to yield the motion in the (, ) configuration space. If or (where , are integers), the velocity field is zero and the particles do not show motion in the configuration space. All other cases lead to a motion in the (, ) configuration space. The velocity field (streamlines) shows that saddle points exist, where the trajectories of the quantum particles become unstable and bifurcation of the trajectories occurs. The particles circle with different velocities around the minima of the squared wavefunction. The wavefunction and the trajectories are axis symmetrical for .

For and , there is an analytic solution for the motion with integration constants and : and .
The initial positions are estimated by solving the equations and numerically. These initial positions determine the frequencies of the circulating particles. Decreasing the distance to the minima leads to increasing the particles' velocity.
So the answer to the initial question is: there is a motion for single quantum particles where the total particle density is time-independent if the superposition of the stationary eigenfunctions is assigned with constant phases. The de Broglie–Bohm picture of the system differs completely from the Schrödinger picture. The graphic shows the squared wavefunction, the particles, the trajectories (black) and the velocity field (red).

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