The Perturbed Double-Slit Experiment in Bohmian Mechanics
The Schrödinger equation for a harmonic oscillator with a linear time-dependent term can take into account the influence of a periodic time-dependent force on the system. This system has many applications in physics, including electromagnetic fields, nonlinear dynamics and other quantum phenomena .[more]
Here we consider the double-slit experiment under the influence of a harmonic potential and a periodic time-dependent uniform linear potential according to the causal interpretation of quantum mechanics (David Bohm and Louis de Broglie), often called Bohmian mechanics. The causal interpretation postulates the existence of a guiding wave or pilot wave that determines the behavior of particles via the quantum potential. In this interpretation, particles have well-defined positions and velocities. They are guided by the pilot wave, which is described by a wavefunction that satisfies the Schrödinger equation.
In the causal interpretation of quantum mechanics, interference effects can be explained by the interaction between the guiding wave of the two slits and the particle. When a particle is incident upon the double slit, the guiding wave passes through both slits and interferes with itself, creating the interference pattern displayed on the detector screen. Still, the particle passes through only one of the slits, but its trajectory is influenced by the quantum potential coming from the two slits. In the context of the harmonic oscillator with a periodic time-dependent term, the Bohmian approach can provide a more complete picture of the system's behavior. The guiding wave can modulate the position and velocity of the particle in a nontrivial way. This can lead to interesting effects, such as the appearance of nontrivial trajectories in configuration space, which is not the real space in the many-particle case. In practice, it is impossible to predict or control the quantum trajectories with complete precision [2–4].
Another interesting effect is the noncrossing theorem [5–7], which forbids the trajectories of particles to cross one another. The noncrossing theorem is a consequence of the fact that the wavefunction is a single-valued function: the wavefunction can only take on a single value at any given point in space and time.
Here, the trajectories in , space do not carry over to real space, but the structure in real (, space is the same. The motion (time evolution) of the particles is obtained by integrating the gradient of the real phase function from the total wavefunction in the eikonal representation: .
The graphic shows, on the right, the squared wavefunction and the trajectories. On the left, it shows the particle position (colored points), the squared wavefunction (blue), the quantum potential (red) and the velocity (green). The velocity and the quantum potential are scaled to fit. The starting positions of the particles are linearly distributed around the peaks of the wave.
The harmonic oscillator wavefunction is a solution of the Schrödinger equation (with mass ):
The two slits produce two Gaussian profiles in space centered at . The velocity field shows a complex behavior in the space. It is clearly seen that the trajectories of the particles do not cross each other during their motion.[less]
The propagator for , with the free parameter , is well known :
For further calculations, the parameters and are set to 1.
If the trigonometric time dependence is applied to the ground state of the harmonic oscillator, it will produce a wave packet centered about :
The velocity field is obtained by the gradient of the real phase function. In the one-slit case, the time evolution of a quantum particle analytic is easily calculated from the velocity field and it gives an analytic expression, because the velocity field depends on time only:
Therefore, the equation for the motion  could be calculated analytically, including the boundary condition :
For the double-slit preparation, a superposition of two unnormalized wave packets is considered:
The gradient of the total phase function (velocity) becomes:
More accurate results are obtained if you increase PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps. The trajectories for the double slit with free particles were first numerically calculated in .
 B. Hamprecht, "Exact Solutions of the Time Dependent Schrödinger Equation in One Space Dimension." arxiv.org/abs/quant-ph/0211040.
 P. Holland, The Quantum Theory of Motion, New York: Cambridge University Press, 1993.
 S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (May 9, 2023)plato.stanford.edu/entries/qm-bohm.
 D. Bohm and B. J. Hiley, The Undivided Universe, New York: Routledge, 1993. doi:10.4324/9780203980385.
 J. S. Bell, "De Broglie–Bohm, Delayed-Choice, Double-Slit Experiment, and Density Matrix," International Journal of Quantum Chemistry: Quantum Chemistry Symposium, 18(S14), 1980 pp. 155–159. doi:10.1002/qua.560180819.
 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.
 C. Dewdney, L. Hardy and E. J. Squires, "How Late Measurements of Quantum Trajectories Can Fool a Detector," Physics Letters A, 184(1), 1993 pp. 6–11. doi:10.1016/0375-9601(93)90337-Y.
 Wikipedia. "Equations of Motion." (May 9, 2023) en.wikipedia.org/wiki/Equations_of_motion.