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arXiv:quant-ph/0302076 · Physical Review A

Implications of Lorentz covariance for the guidance equation in two-slit quantum interference

Peter Holland, Chris Philippidis

Published Jun 20, 2003• 104 citations
arXivHTMLPDF

Abstract

It is known that Lorentz covariance fixes uniquely the current and the associated guidance law in the trajectory interpretation of quantum mechanics for spin-$\frac{1}{2}$ particles. In the nonrelativistic domain this implies a guidance law for the electron which differs by an additional spin-dependent term from that originally proposed by de Broglie and Bohm. In this paper, we explore some of the implications of the modified guidance law. We bring out a property of mutual dependence in the particle coordinates that arises in product states, and show that the quantum potential has scalar and vector components, which implies the particle is subject to a Lorentz-like force. The conditions for the classical limit and the limit of negligible spin are given, and the empirical sufficiency of the model is demonstrated. We then present a series of calculations of the trajectories based on two-dimensional Gaussian wave packets which illustrate how the additional spin-dependent term plays a significant role in structuring both the individual trajectories and the ensemble. The single packet corresponds to quantum inertial motion. The distinct features encountered when the wave function is a product or a superposition are explored, and the trajectories that model the two-slit experiment are given. The latter paths exhibit several new characteristics compared with the original de Broglie--Bohm ones, such as crossing of the axis of symmetry.

DOI
10.1103/physreva.67.062105

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It is known that Lorentz covariance fixes uniquely the current and the associated guidance law in the trajectory interpretation of quantum mechanics for spin-12\frac{1}{2}21​ particles. In the nonrelativistic domain this implies a guidance law for the electron which differs by an additional spin-dependent term from that originally proposed by de Broglie and Bohm. In this paper, we explore some of the implications of the modified guidance law. We bring out a property of mutual dependence in the particle coordinates that arises in product states, and show that the quantum potential has scalar and vector components, which implies the particle is subject to a Lorentz-like force. The conditions for the classical limit and the limit of negligible spin are given, and the empirical sufficiency of the model is demonstrated. We then present a series of calculations of the trajectories based on two-dimensional Gaussian wave packets which illustrate how the additional spin-dependent term plays a significant role in structuring both the individual trajectories and the ensemble. The single packet corresponds to quantum inertial motion. The distinct features encountered when the wave function is a product or a superposition are explored, and the trajectories that model the two-slit experiment are given. The latter paths exhibit several new characteristics compared with the original de Broglie--Bohm ones, such as crossing of the axis of symmetry.
The constraints imposed by relativistic covariance on the law of motion of fermions in the trajectory interpretation of quantum mechanics are examined. It is found that the quantum current satisfies a uniqueness property that fixes the guidance law. This in turn fixes the nonrelativistic equation which generally differs from that originally given by de Broglie and Bohm for electrons. These results extend to Bohm's noncovariant many-body theory. The demonstration does not exclude other guidance equations and this is shown by the explicit construction of an alternative Lorentz covariant many-body theory possessing future-causal space-time paths.
We present a novel covariant relativistic guidance formula à la de Broglie-Bohm derived from conserved quantities associated with the Lagrangian of the Dirac equation in a 1+1-dimensional spacetime. In line with the standard guidance formula of Bohmian mechanics, this generalized guidance formula incorporates space-time derivatives of the Dirac spinor phases and consistently reproduces relativistic features such as mass-dependent trajectories and zitterbewegung. We formally show that this formulation explicitly reveals how the relativistic quantum potential affects the trajectories. Using numerical simulations, we compare these trajectories with those presented in the usual approaches. Finally, we show how the guidance formula naturally transitions to the standard de Broglie-Bohm theory in the non-relativistic limit and gives additional contributions in the expansions giving new insight into the zitterbewegung in pilot-wave theories.
The Bohm-de Broglie quantum mechanics has made possible to calculate the trajectories of electrons in a typical double-slit experiment[C. Philippidis et al., Il Nuovo Cimento, 52 B, 15-28 (1979)]. The trajectories do not correspond to an uniform movement but to an accelerated one. The acceleration is caused by the quantum potential. From the quantum theoretical point of view, the accelerated electrons should, with a certain probabil-ity, emit photons during its movement from the slits to the screen. According to the Copenhagen interpretation we found that this quantity is strictly zero because the electron moves as a free particle along its path after it leaves the slit and before reach the screen. Then, there is no emission of photons. On the other hand, by using the Bohm-de Broglie (BdB) approach we calculate the emission power, giving a general formula and which results, for a concrete real experiment, in a very tiny but not a vanishing value. We give an idea of the type of spectrum that could be measured. Therefore we have shown that the theoretical predictions of the usual quan-tum mechanics and that of Bohm-de Broglie’s interpretation are different. An experiment could determine the correct prediction.
Abstract The 2011 experiment by Kocsis et al., using the weak measurement protocol, confirmed the distinctive particle trajectory pattern given by the Bohm-de Broglie causal interpretation model of the quantum version of Young's two-slit experiment. The model uses Gaussian wave-packets. Given that the original (classical) Young's two-slit experiment was modelled by Fresnel in terms of Huygens' spherical waves, we thought it would be interesting to test whether a spherical wave model of the quantum Young's two-slit experiment would yield a similar description, in particular, whether or not the distinctive particle trajectory pattern confirmed by experiment would be reproduced. PACS NO. 03.65.-w
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.
Abstract In Copenhagen’s interpretation, a self-interference mechanism is assumed for a wave function to split and to pass through the slits simultaneously, then recombine and collapse to generate a dot signal on a detector. Its use of a single wave function to describe a whole ensemble without accounting for a particle’s individuality is incomplete. To better understand the single-electron double-slit interference, we use a wave-packet trajectory approach to track each electron’s dynamics and calculate the time-dependent interference intensity. We analyze the self-interference and cross-interference mechanisms to illustrate how an electron’s coherence length affects the self-interference and how it dictates the wave packet overlaps between adjacent electrons for the cross-interference mechanism to be effective. To circumvent a conflicting dilemma between the electron’s indivisibility and its wave-packet split while passing through the slits, we consider an alternate mechanism involving slit-induced quantized momentum. This mechanism preserves the electron’s indivisibility and avoids the illusive split, recombination and mysterious collapse of a single particle’s wave function.