Double-slit experiment with single wave-driven particles and its relation to quantum mechanics

In a thought-provoking paper, Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)] describe a version of the famous double-slit experiment performed with droplets bouncing on a vertically vibrated fluid surface. In the experiment, an interference pattern in the single-particle statistics is found even though it is possible to determine unambiguously which slit the walking droplet passes. Here we argue, however, that the single-particle statistics in such an experiment will be fundamentally different from the single-particle statistics of quantum mechanics. Quantum mechanical interference takes place between different classical paths with precise amplitude and phase relations. In the double-slit experiment with walking droplets, these relations are lost since one of the paths is singled out by the droplet. To support our conclusions, we have carried out our own double-slit experiment, and our results, in particular the long and variable slit passage times of the droplets, cast strong doubt on the feasibility of the interference claimed by Couder and Fort. To understand theoretically the limitations of wave-driven particle systems as analogs to quantum mechanics, we introduce a Schrödinger equation with a source term originating from a localized particle that generates a wave while being simultaneously guided by it. We show that the ensuing particle-wave dynamics can capture some characteristics of quantum mechanics such as orbital quantization. However, the particle-wave dynamics can not reproduce quantum mechanics in general, and we show that the single-particle statistics for our model in a double-slit experiment with an additional splitter plate differs qualitatively from that of quantum mechanics.

Figure 1

Schematic illustrations of the experimental cell geometries seen from above. (a) Reference experiment with no barrier, (b) single slit, and (c) double slit. The outer rim (dark) encloses the shallow layer (light gray) and the deep layer (white) where the Faraday waves can be excited and the walkers move. The droplet trajectory deflection angles were determined from the points at which the droplets crossed the dashed circle segments with radius $3.0\text{cm}$.

Figure 2

Snapshots of a walker in the double-slit experiment. (a), (b) The droplet is propelled by the pilot wave towards the subsurface barrier with two slits. (c) The droplet follows a well-defined trajectory through one of the slits, whereas the pilot wave is free to pass through both slits. (d) After passage, the droplet moves along a deflected path.

Figure 3

Droplet trajectories. (a) Reference experiment without a barrier, 78 trajectories, (b) single-slit experiment, 169 trajectories, and (c) double-slit experiment, 301 trajectories. In (a) we only show the trajectories of the droplets that would have passed through a virtual single slit with geometry similar to our single-slit arrangement, and in (b) and (c) we only show the trajectories of the droplets that passed the slit arrangement at first impact.

Figure 4

Histograms with the number of counts $N$ showing the distributions of the droplet impact parameter $y$ at the slit entrances at $x=-0.5\text{cm}$. (a) Single-slit experiment, 169 trajectories and (b) double-slit experiment, 301 trajectories.

Figure 5

Histograms with the number of counts $N$ showing the single-particle statistics based on the deflection angles $\alpha$. (a) Reference experiment without a barrier, 78 trajectories, (b) single-slit experiment, 169 trajectories, and (c) double-slit experiment, 301 trajectories.

Figure 6

Trajectories and droplet speeds in the double-slit experiment. (a) Five selected trajectories during passage of the slit and (b) the corresponding droplet speeds. The slit passage times of the droplets from $x=-0.5$ to $0.0\text{cm}$ were $0.46\text{s}$ (dotted-dashed line, blue), $0.67\text{s}$ (dashed line, green), $0.50\text{s}$ (solid line, red), $0.63\text{s}$ (dotted line, black), and $0.58\text{s}$ (thick solid line, magenta).

Figure 7

Correlation between droplet impact parameter $y$ and trajectory deflection angle $\alpha$ after slit passage in the double-slit experiment with 301 trajectories. (a) Droplet impact parameter before the double slit at $x=-3.0\text{cm}$ and (b) at the entrance to the slits at $x=-0.5\text{cm}$.

Figure 8

Time series of a walking droplet and the associated wave in the double-slit experiment. The red dot highlights the position of the droplet, and the double slit is accentuated in white.

Figure 9

Time evolution of the probability density of a wave packet in the standard quantum mechanical double-slit experiment.

Figure 10

Time evolution of the probability density of a wave packet in the quantum mechanical double-slit experiment with a central splitter plate.

Figure 11

Time evolution of a quantum mechanical wave packet, showing the contours of the probability density as it goes through a double slit without a central splitter plate (a)–(d), and with a splitter plate (e)–(h).

Figure 12

Accumulated probability density at the cross section at 15.83 after the slit arrangement in the standard quantum mechanical double-slit experiment (dashed line, red) and with a central splitter plate (solid line, blue). The slit openings were 1.67 wide, the width of the central barrier was 5.00, and the length of the central splitter plate was 12.50.

Figure 13

Solutions $\mathrm{Re}\left(\mathrm{\Psi }\right)$ for a free particle in three dimensions moving on a straight line ${\mathbf{r}}_p\left(t\right)=vt\widehat{\mathbf{x}}$. (a) Non-Galilean invariant case and (b) Galilean invariant case. The $x$ and $y$ coordinates are made dimensionless using the de Broglie wavelength $h/\left(mv\right)$, we have set $z=0$ and $t=0$, and $\mathrm{Re}\left(\mathrm{\Psi }\right)$ is shown in arbitrary units.

Figure 14

Double-slit experiment with a splitter plate. A particle will have to move on one or the other side of the plate as shown by the two possible trajectories.

Figure 15

Time evolution of the wave field $\mathrm{Re}\left(\mathrm{\Psi }\right)$ without a particle at times (a) $t=0.6$, (b) $t=2$, and (c) $t=5$. The $x$ and $y$ coordinates are made dimensionless using the de Broglie wavelength $h/\left(mv\right)$, the time $t$ using the period $2h/\left(m{v}^2\right)$, we have set $z=0$, and $\mathrm{Re}\left(\mathrm{\Psi }\right)$ is shown in arbitrary units.