^{1,a)}

### Abstract

The de Broglie-Bohm “pilot-wave” theory replaces the paradoxical wave-particle duality of ordinary quantum theory with a more mundane and literal kind of duality: each individual photon or electron comprises a quantum wave (evolving in accordance with the usual quantum mechanical wave equation) *and* a particle that, under the influence of the wave, traces out a definite trajectory. The definite particle trajectory allows the theory to account for the results of experiments without the usual recourse to additional dynamical axioms about measurements. Instead, one need simply assume that particle detectors click when particles arrive at them. This alternative understanding of quantum phenomena is illustrated here for two elementary textbook examples of one-dimensional scattering and tunneling. We introduce a novel approach to reconcile standard textbook calculations (made using unphysical plane-wave states) with the need to treat such phenomena in terms of normalizable wave packets. This approach allows for a simple but illuminating analysis of the pilot-wave theory's particle trajectories and an explicit demonstration of the equivalence of the pilot-wave theory predictions with those of ordinary quantum theory.

Thanks to Shelly Goldstein, Doug Hemmick, and two anonymous referees for helpful comments on earlier drafts of the paper.

I. INTRODUCTION

II. SCHRÖDINGER WAVE SCATTERING AT A POTENTIAL STEP

III. PARTICLE TRAJECTORIES IN THE PILOT-WAVE THEORY

IV. TUNNELING THROUGH A RECTANGULAR BARRIER

V. DISCUSSION

### Key Topics

- Textbooks
- 17.0
- Wave functions
- 15.0
- Particle trajectory
- 13.0
- Quantum mechanics
- 13.0
- Particle velocity
- 11.0

## Figures

Visualization of a “plane-wave packet” interacting with the step potential shown in the top frame. The “ ” frame shows a plane-wave packet of length *L* and (central) wavelength incident from the left. (Note that, strictly speaking, a plane-wave packet by definition has ; the two length scales are inappropriately similar in the figure so that several other features will be more readily visible.) At *t* = 0 the leading edge of the incident packet arrives at the origin and leading edges for the reflected and transmitted packets are produced. At *t* = *T* the trailing edge of the incident packet arrives at the origin and trailing edges for the reflected and transmitted packets are produced. For the incident and reflected packets overlap in some (initially small, then bigger, then small again) region to the left of the origin. This is depicted in the “ ” frame. Note that the amplitudes of the reflected and transmitted waves are determined by the usual boundary-matching conditions imposed at *x* = 0. Finally, for the reflected and transmitted packets propagate away from the origin. Note that while the wavelength and packet length of the reflected wave matches those of the incident wave, the wavelength and packet length of the transmitted wave are respectively greater than and smaller than those of the incident wave, owing to the different value of the potential energy to the right of the origin.

Visualization of a “plane-wave packet” interacting with the step potential shown in the top frame. The “ ” frame shows a plane-wave packet of length *L* and (central) wavelength incident from the left. (Note that, strictly speaking, a plane-wave packet by definition has ; the two length scales are inappropriately similar in the figure so that several other features will be more readily visible.) At *t* = 0 the leading edge of the incident packet arrives at the origin and leading edges for the reflected and transmitted packets are produced. At *t* = *T* the trailing edge of the incident packet arrives at the origin and trailing edges for the reflected and transmitted packets are produced. For the incident and reflected packets overlap in some (initially small, then bigger, then small again) region to the left of the origin. This is depicted in the “ ” frame. Note that the amplitudes of the reflected and transmitted waves are determined by the usual boundary-matching conditions imposed at *x* = 0. Finally, for the reflected and transmitted packets propagate away from the origin. Note that while the wavelength and packet length of the reflected wave matches those of the incident wave, the wavelength and packet length of the transmitted wave are respectively greater than and smaller than those of the incident wave, owing to the different value of the potential energy to the right of the origin.

Space-time diagram showing a representative sample of possible particle trajectories for the case of a plane-wave packet incident from the left on a step potential at *x* = 0. The leading and trailing edges of the various packets are indicated by dashed grey lines while particle trajectories are shown in black. In general, the particle simply moves at the group velocity along with the packet that is guiding it. In the (triangular) overlap region, however, the particle moves more slowly; this gives rise to a bifurcation of the possible trajectories between those that arrive at the origin before being caught by the incident packet's trailing edge (and thus end up moving away with the transmitted packet), and those that are caught by the incident packet's trailing edge (and thus end up moving away with the reflected packet).

Space-time diagram showing a representative sample of possible particle trajectories for the case of a plane-wave packet incident from the left on a step potential at *x* = 0. The leading and trailing edges of the various packets are indicated by dashed grey lines while particle trajectories are shown in black. In general, the particle simply moves at the group velocity along with the packet that is guiding it. In the (triangular) overlap region, however, the particle moves more slowly; this gives rise to a bifurcation of the possible trajectories between those that arrive at the origin before being caught by the incident packet's trailing edge (and thus end up moving away with the transmitted packet), and those that are caught by the incident packet's trailing edge (and thus end up moving away with the reflected packet).

The critical trajectory, which arrives at the apex of the triangular overlap region on this space-time diagram, divides trajectories that transmit from those that reflect. The possible trajectories are distributed with uniform probability density throughout the incident packet, so the fraction of the total length *L* of the packet that is in front of the critical trajectory represents the transmission probability *P _{T} *. Equivalently, the critical trajectory is a distance behind the incident packet's leading edge. From

*t*= 0 exactly half this distance is covered before encountering the leading edge of the reflected packet; this occurs at time . In traversing the overlap region, the critical trajectory then moves through the remaining distance in a time , where is the time needed for the trailing edge of the incident packet to arrive at the origin. Equation (30) then follows by dividing this distance by this time.

The critical trajectory, which arrives at the apex of the triangular overlap region on this space-time diagram, divides trajectories that transmit from those that reflect. The possible trajectories are distributed with uniform probability density throughout the incident packet, so the fraction of the total length *L* of the packet that is in front of the critical trajectory represents the transmission probability *P _{T} *. Equivalently, the critical trajectory is a distance behind the incident packet's leading edge. From

*t*= 0 exactly half this distance is covered before encountering the leading edge of the reflected packet; this occurs at time . In traversing the overlap region, the critical trajectory then moves through the remaining distance in a time , where is the time needed for the trailing edge of the incident packet to arrive at the origin. Equation (30) then follows by dividing this distance by this time.

Space-time diagram showing a representative sample of possible particle trajectories for the case of a plane-wave packet incident from the left on a rectangular potential barrier. Particles beginning near the leading edge of the incident packet will tunnel through the barrier and emerge on the far side. The unusual accelerating character of the trajectories in the (gray shaded) classically forbidden region—indeed the mere presence of trajectories here—reflects the highly non-classical nature of the law of motion for the particle. (Note that the analysis in the main text assumes that the incident packet length *L* is very large compared to the barrier width *a*. This separation of length scales is not accurately depicted in this figure so that the qualitative nature of the trajectories in all relevant regions can be visualized simultaneously.)

Space-time diagram showing a representative sample of possible particle trajectories for the case of a plane-wave packet incident from the left on a rectangular potential barrier. Particles beginning near the leading edge of the incident packet will tunnel through the barrier and emerge on the far side. The unusual accelerating character of the trajectories in the (gray shaded) classically forbidden region—indeed the mere presence of trajectories here—reflects the highly non-classical nature of the law of motion for the particle. (Note that the analysis in the main text assumes that the incident packet length *L* is very large compared to the barrier width *a*. This separation of length scales is not accurately depicted in this figure so that the qualitative nature of the trajectories in all relevant regions can be visualized simultaneously.)

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