The initial wavefunction in the eikonal form,

,

obeys the free-particle Schrödinger equation in the

direction:

,

here with

. The initial wavefunction is a linear combination of two Gaussian packets with initial spatial half-widths

and group velocities

. The measured intensity

at any time

is given by

.

According to the theory proposed in [1], the measured intensity

takes the form

,

with

and

, where

is the coherence time, a measurement of how fast the system becomes classical. For simplicity,

is called the decoherence coupling factor. In this context, classical means that the probability density

becomes

.

If

(the quantum limit), there is full coherence with

and there is no difference between the FOA and SOA approaches. The trajectories run to the local maxima of the squared wavefunction and therefore correspond to the bright fringes of the diffraction pattern. The quantum particle passes through slit one or slit two and never crosses the axis of symmetry.

For unit mass

, the general equation of motion for the SOA is given by the acceleration term that is the second derivative of the position

with respect to time

:

,

where

is integrated numerically for various initial positions and velocities. The initial positions are linearly distributed around the peaks of the wave. The initial velocities are taken from the gradient of the phase function for

. The quantum potential

, from which the trajectories are calculated, is given by:

,

.

For

the coupling function

, the effect of the quantum potential

vanishes, and there is no dependence of the trajectories on the phase functions

and

. This leads to the crossing of trajectories.

The results become more accurate if you increase

AccuracyGoal,

PrecisionGoal and

MaxSteps.

For more detailed information about Bohmian mechanics, see [5, 6], and for an animated example describing a similar situation, see [7].

[3] A. S. Sanz and F. Borondo, "A Quantum Trajectory Description of Decoherence,"

*The European Physical Journal D*,

**44**(2), 2007 pp. 319–326.

doi:10.1140/epjd/e2007-00191-8.

[4] B.-G. Englert, M. O. Scully, G. Süssman and H. Walther, "Surrealistic Bohm Trajectories,"

*Zeitschrift für Naturforschung*,

**47**(12), 1992 pp. 1175–1186.

doi:10.1515/zna-1992-1201.