How Travels a Bohmian Particle?

ABSTRACT

Bohm’s mechanics was built for explaining individual results in measurements, and mainly for getting rid of the enigmatic reduction postulate. Its main idea is that particles have at any time definite positions and velocities. An additional axiom is that particles follow continuous trajectories that admit the first derivative in time, the velocity. In the quantum theory, if the position of a quantum object is well-defined at some time, a Δt time later the object may be found anywhere in space, so, the velocity defined as Δx/Δt is completely undefined. This incompatibility is regarded in standard quantum theory as nature’s property. The disagreement between quantum and Bohm’s mechanics is particularly strong in wave-like phenomena, e.g. interference. For a particle traveling through an interference fringe, Bohm’s velocity formula shows a dependence of the time-of-flight on the fringe length. Such a dependence is not supported by the quantum theory. Thus, for deciding which prediction is correct one has to measure times-of-flight. But this is a problem. If one detects a particle at two positions and records the detection times, the time difference is meaningless, because the first position measurement disturbs the particle’s Bohm velocity (if exists). This text suggests a way around: instead of measuring positions and times, the particles are raised to an excited, unstable level, by passing them through a laser beam. The unstable level will decay in time, s.t. the density of probability of the excited atoms will indicate the time elapsed since excitation. For comparing the Bohmian and quantum predictions, this text proposes in continuation to send the beam of excited particle upon a mirror. Bohm’s velocity leads to anomalies in the reflected wave.

Bohm’s mechanics was built for explaining individual results in measurements, and mainly for getting rid of the enigmatic reduction postulate. Its main idea is that particles have at any time definite positions and velocities. An additional axiom is that particles follow continuous trajectories that admit the first derivative in time, the velocity. In the quantum theory, if the position of a quantum object is well-defined at some time, a Δt time later the object may be found anywhere in space, so, the velocity defined as Δx/Δt is completely undefined. This incompatibility is regarded in standard quantum theory as nature’s property. The disagreement between quantum and Bohm’s mechanics is particularly strong in wave-like phenomena, e.g. interference. For a particle traveling through an interference fringe, Bohm’s velocity formula shows a dependence of the time-of-flight on the fringe length. Such a dependence is not supported by the quantum theory. Thus, for deciding which prediction is correct one has to measure times-of-flight. But this is a problem. If one detects a particle at two positions and records the detection times, the time difference is meaningless, because the first position measurement disturbs the particle’s Bohm velocity (if exists). This text suggests a way around: instead of measuring positions and times, the particles are raised to an excited, unstable level, by passing them through a laser beam. The unstable level will decay in time, s.t. the density of probability of the excited atoms will indicate the time elapsed since excitation. For comparing the Bohmian and quantum predictions, this text proposes in continuation to send the beam of excited particle upon a mirror. Bohm’s velocity leads to anomalies in the reflected wave.

KEYWORDS

Bohmian Particle; Bohmian Velocity; Bohmian Trajectory; Group Velocity; Interference; Time-of-Flight

Bohmian Particle; Bohmian Velocity; Bohmian Trajectory; Group Velocity; Interference; Time-of-Flight

Cite this paper

S. Wechsler, "How Travels a Bohmian Particle?,"*Journal of Modern Physics*, Vol. 3 No. 12, 2012, pp. 1842-1848. doi: 10.4236/jmp.2012.312231.

S. Wechsler, "How Travels a Bohmian Particle?,"

References

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[1] D. Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables. I,” Physical Review, Vol. 85, No. 2, 1952, pp. 166-179. doi:10.1103/PhysRev.85.166

[2] D. Bohm, “A Suggested Interpretation of the Quantum Theory in Terms of ‘Hidden’ Variables. II,” Physical Review, Vol. 85, No. 2, 1952, pp. 180-193. doi:10.1103/PhysRev.85.180

[3] D. Durr, S. Goldstein, R. Tumulka and N. Zanghì, “Bohmian Mechanics,” quant-ph/0903.2601, p. 3.

[4] D. Durr and S. Teufel, “Bohmian Mechanics the Physics and Mathematics of Quantum Theory,” Springer, 2008.

[5] J. G. Muga, R. Sala and J. P. Palao, “The Time of Arrival Concept in Quantum Mechanics,” quant-ph/9801043v1.

[6] C. Anastopoulos and N. Savvidou, “Time-of-Arrival Probabilities for General Particle Detectors,” quant-ph/1205.27 81v3.

[7] J. G. Muga, A. D. Baute, J. A. Damborenea and I. L. Egusquiza, “Model for the Arrival-Time Distribution in Fluorescence Time-of-Flight Experiments,” quant-ph/00 09111v1.

[8] J. A. Damborenea, I. L. Egusquiza, G. C. Hegerfeldt and J. G. Muga, “Atomic Time-of-Arrival Measurements with a Laser of Finite Beam Width,” quant-ph/0302201v1.

[9] C. Anastopoulos, “Time-of-Arrival Probabilities and Quantum Measurements: III Decay of Unstable States,” quant- ph/0706.2496v2.

[10] M. Kohl, T. W. Hansch and T. Esslinger, “Measuring the Temporal Coherence of an Atom Laser Beam,” condmat/0104384v2.

[11] I. Bloch, M. Kohl, M. Greiner, T. W. Hansch and T. Esslinger, “Optics with an Atom Laser Beam,” Physical Review Letters, Vol. 87, No. 3, 2001, Article ID: 030401. doi:10.1103/PhysRevLett.87.030401

[12] I. Bloch, T. W. Hansch and T. Esslinger, “Atom Laser with a cw Output Coupler,” cond-mat/9812.258v1.

[13] L. D. Landau and E. M. Lifshitz, “Quantum Mechanics,” Pergamon Press, Oxford, 1958, p. 24.

[14] G. Baym, “Lectures on Quantum Mechanics,” 3rd Edition, W. A. Benjamin, Inc., Massachusetts, 1974, p. 64.