# The Double-Slit Experiment in a Uniform Potential

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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

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Contributed by: Klaus von Bloh (August 2022)

After idea by: Vikram Athalye

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

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 [5].

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 [1].

References

[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] P. Holland, *The Quantum Theory of Motion*, Cambridge: Cambridge University Press, 1993.

[3] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jun 22, 2022)plato.stanford.edu/entries/qm-bohm.

[4] D. Bohm and B. J. Hiley, *The Undivided Universe*, London and New York: Routledge, 1993. doi:10.4324/9780203980385.

[5] Wikipedia. "Equations of Motion." (Jun 22, 2022) en.wikipedia.org/wiki/Equations_of_motion.

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