The Double-Slit Experiment in a Uniform Potential
This Demonstration considers the Fraunhofer diffraction experiment with a double slit under the influence of a uniform linear potential according to the causal interpretation of quantum mechanics (David Bohm and Louis de Broglie). This linear potential might, for example, be from gravity, with . The slits produce two Gaussian profiles in space centered at . The initial form of the unnormalized waves is given by[more]
with , where are the group velocities in the direction, are the widths of the wave packets at and is the mass. In this case, the total wavefunction is the superposition of the two waves , where the time evolution is calculated from the Schrödinger equation (here with ): (, and so on).
According to the Copenhagen interpretation of quantum theory, it cannot be decided which slit the particle goes through because of the self-interference effect. It has to be assumed that the particle goes through both slits, and the wavefunction collapses when a measurement is made at the detector screen.
The de Broglie–Bohm interpretation is completely different. The quantum particle passes through one of the two slits. Which slit the particles pass through depends on the initial position only [1, 2]. The quantum particles possess well-defined positions and velocities at all times, but these variables can never be measured simultaneously, because of the uncontrollable particle-apparatus coupling [2–4]. In the causal interpretation of quantum theory, there is a quantum force proportional to , where is called the quantum potential. The quantum potential leads to highly nonclassical motion of particles, because the quantum potential from the one slit affects the motion of the particle from the other slit, which evolves from the first slit immediately. So the information of the whole double-slit apparatus is contained in , which Bohm and Hiley later called active information . The trajectories run to the local maxima of the squared wavefunction and therefore correspond to the bright fringes of the diffraction pattern under the influence of the uniform potential.
In the classical mechanics, all bodies fall at an equal rate, but in the Bohmian approach, the acceleration depends on the mass and the initial width of the wave packet at .
The trajectories in , space do not carry over to real space, but the structure in real , space is the same. The motion 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, at right, the squared wavefunction and the trajectories. At 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.[less]
In general, the expectation value in quantum theory is the probabilistic expected value of the measurement. For example, the expectation value of the Fraunhofer diffraction pattern from a single slit under an influence of the gravity, with the normalized wave function , might be given analytically by
with the group velocity, the gravitational acceleration, the mass, the initial width of the wave packet and the initial position.
The time-dependent expectation value for single-slit Fraunhofer diffraction is given by:
where is the complex conjugate of .
For a very special case (, with and ), the expectation value is given by:
The values of are in full agreement with the corresponding uniformly accelerated motion in classical mechanics .
More accurate results are obtained if you increase PlotPoints, AccuracyGoal, PrecisionGoal and MaxSteps. The trajectories for the double slit without uniform potential were first numerically calculated in .
 P. Holland, The Quantum Theory of Motion, Cambridge: Cambridge University Press, 1993.
 S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jun 22, 2022)plato.stanford.edu/entries/qm-bohm.
 D. Bohm and B. J. Hiley, The Undivided Universe, London and New York: Routledge, 1993. doi:10.4324/9780203980385.
 Wikipedia. "Equations of Motion." (Jun 22, 2022) en.wikipedia.org/wiki/Equations_of_motion.