# The Perturbed Double-Slit Experiment in Bohmian Mechanics

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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 [1].

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Contributed by: Klaus von Bloh (August 27)

After idea by: Vikram Athalye

Open content licensed under CC BY-NC-SA

## Details

,

with .

The propagator for , with the free parameter , is well known [1]:

with

,

,

and

.

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 [8] 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:

with

and

with .

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 [9].

References

[1] B. Hamprecht, "Exact Solutions of the Time Dependent Schrödinger Equation in One Space Dimension." arxiv.org/abs/quant-ph/0211040.

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

[3] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (May 9, 2023)plato.stanford.edu/entries/qm-bohm.

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

[5] 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.

[6] 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.

[7] 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.

[8] Wikipedia. "Equations of Motion." (May 9, 2023) en.wikipedia.org/wiki/Equations_of_motion.

[9] 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.

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