Triple-Slit Experiment in the Causal Interpretation

This Demonstration simulates the triple-slit experiment numerically according to the causal interpretation of David Bohm and Louis de Broglie. The slits generate three Gaussian profiles in space, positioned at 0 and . The initial form of the unnormalized waves are , where are the wave numbers in the direction, are the widths of the wave packets at , ξi are the constant phase shifts and ), are the real-valued constants with , , , and where with . In this case the total wavefunction is the superposition of the three waves , where the time evolution is to be calculated from the free Schrödinger equation (): 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 three slits and the wavefunction collapses when a measurement is made at the detector screen.
The Bohmian interpretation is completely different. The quantum particle passes through one of the three slits. Which slit the particles pass through depends on the initial position only. 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. 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 second and third slits affect the motion of the particle, which evolves from the first slit immediately. So the information of the whole triple 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.
The trajectories in - space do not display in 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: .
On the right, the graphic shows the squared wavefunction and the trajectories. The left shows the particles' position, the squared wavefunction (blue), the quantum potential (red), and the velocity (green). The velocity and the quantum potential are scaled to fit.


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In the program, if you increase PlotPoints, AccuracyGoal, PrecisionGoal, and MaxSteps, the results will be more accurate. The starting positions of the particles are linearly distributed around the peaks of the wave density at . Furthermore, the number of the particles depends on the initial squared amplitude of each unnormalized wave, which is proportional to , , and . If the constant phase shift of the wave from the second slit is changed, the interference pattern and therefore the motion of the particles change significantly. The particle positions are plotted against their downscaled kinetic energy along the vertical axis.
The trajectories for the double slit were first numerically calculated in [1]. The effects of empty waves are discussed in [2]. For more detailed information about Bohmian mechanics, see [3].
[1] C. Philippidis, C. Dewdney, and B. J. Hiley, "Quantum Interference and the Quantum Potential," Il Nuovo Cimento B Series 11, 52(1), 1979 pp. 15–28. doi:10.1007/BF02743566.
[2] M. Schmidt and F. Selleri, "Empty-Wave Effects on Particle Trajectories in Triple-Slit Experiments," Foundations of Physics Letters, 4(1), 1991 pp. 1–17. doi:10.1007/BF00666413.
[3] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Mar 4, 2013) plato.stanford.edu/entries/qm-bohm.
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