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Physical Review A

Distributions of delay times and transmission times in Bohm’s causal interpretation of quantum mechanics

W. R. McKinnon, C. Richard Leavens

Published Apr 1, 1995• 68 citations
Publisher

Abstract

In Bohm's causal or trajectory interpretation of quantum mechanics, it is straightforward to determine, from a sufficiently large number of calculated particle trajectories, probability distributions for time delays caused by a potential barrier, or for transmission times through a barrier. We show that these distributions can be calculated directly and more efficiently from probability currents, without the calculation of Bohm trajectories as an intermediate step. The ideas are illustrated for Gaussian wave packets incident on a square potential barrier and used to explain why average causal delay times differ from the average delay times calculated in other approaches.

DOI
10.1103/physreva.51.2748

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In Bohm's causal or trajectory interpretation of quantum mechanics, it is straightforward to determine, from a sufficiently large number of calculated particle trajectories, probability distributions for time delays caused by a potential barrier, or for transmission times through a barrier. We show that these distributions can be calculated directly and more efficiently from probability currents, without the calculation of Bohm trajectories as an intermediate step. The ideas are illustrated for Gaussian wave packets incident on a square potential barrier and used to explain why average causal delay times differ from the average delay times calculated in other approaches.
Several practical implications of the noncrossing property of one-dimensional Bohm trajectories are examined. It is shown that the position of a Bohm particle, the average transmission, reflection and dwell times, and the probability distribution of these tunneling times, can all be obtained without actually calculating trajectories. On the other hand, the intuitive interpretation of the scattering of wave packets by potential barriers is discussed within the framework of Bohm's interpretation of quantum mechanics. In this regard, claims that Bohm's approach leads to counterintuitive results are shown to be subjective. \textcopyright{} 1996 The American Physical Society.
The de Broglie–Bohm interpretation of quantum mechanics with and without spin-dependence is used to determine electron trajectories, arrival-time distribution of electrons and the mean arrival time of spin-1/2 particles (including electrons) in the presence of a uniform field and a barrier separately. The difference for the mean arrival times, which are calculated with different guidance equations, is examined versus mass of the arriving particle and versus group velocity of the arriving electron wave packet and also versus the width of the barrier. Numerical calculations show that these differences are of the order of 10 −18 –10 −17 s. Another feature of the modified guidance law is that Bohmian trajectories cross each other, but this does not contradict the single-valuedness of the wavefunction.
The transmission of wave packets through barriers by tunneling is studied in detail by the method of quantum molecular dynamics. The distribution of the arrival times of a tunneling packet in front of and behind a barrier and the momentum distribution function of the packet are calculated. The average position and average momentum of the packet and their spread are investigated. It is found that below the barrier a part of the packet is reflected, and a Gaussian barrier increases the average momentum of the transmitted packet and its spread in momentum space.
Usually tunneling is established after imposing some matching conditions on the (time-independent) wavefunction and its first derivative at the boundaries of a barrier. Here an alternative scheme is proposed to determine tunneling and estimate transmission probabilities in time-dependent problems, which takes advantage of the trajectory picture provided by Bohmian mechanics. From this theory, a general functional expression for the transmission probability in terms of the system initial state can be reached. This expression is used here to analyze tunneling properties and estimate transmissions in the case of initial Gaussian wave packets colliding with ramp-like barriers.
Following de Broglie, Bohm, and others, it is assumed that quantum mechanics may be interpreted causally and that the $\ensuremath{\psi}$ function plays the role of a generating function for particle trajectories. By arguing that the $\ensuremath{\psi}$ function should not be interpreted as a probability amplitude, a new method for generating particle trajectories is postulated. The four-momentum of a scalar particle is assumed to be given as the gradient of an unspecified function $F(\ensuremath{\psi})$, where $\ensuremath{\psi}$ is a pure real solution of the Klein-Gordon equation. Since the location of a particle is determined solely by its trajectory, the probability distribution differs from $\ensuremath{\psi}{\ensuremath{\psi}}^{*}$; and therefore, ordinary experimental results differing from the traditional theory may, in principle, be predicted. Particle motion and trajectories are discussed for three examples: a free particle, a particle in a box, and the double slit.