Why bouncing droplets are a pretty good modelof quantum mechanics

Robert Brady and Ross AndersonUniversity of Cambridge Computer Laboratory

JJ Thomson Avenue, Cambridge CB3 0FD, United Kingdom

robert.brady,ross.anderson@cl.cam.ac.uk

January 20, 2014

Abstract

In 2005, Couder, Protiere, Fort and Badouad showed that oil droplets bouncing on a vibrat-ing tray of oil can display nonlocal interactions reminiscent of the particle-wave associations inquantum mechanics; in particular they can move, attract, repel and orbit each other. Subse-quent experimental work by Couder, Fort, Protiere, Eddi, Sultan, Moukhtar, Rossi, Molacek,Bush and Sbitnev has established that bouncing drops exhibit single-slit and double-slit diffrac-tion, tunnelling, quantised energy levels, Anderson localisation and the creation/annihilation ofdroplet/bubble pairs.

In this paper we explain why. We show first that the surface waves guiding the dropletsare Lorentz covariant with the characteristic speed c of the surface waves; second, that pairs ofbouncing droplets experience an inverse-square force of attraction or repulsion according to theirrelative phase, and an analogue of the magnetic force; third, that bouncing droplets are governedby an analogue of Schrodinger’s equation where Planck’s constant is replaced by an appropriateconstant of the motion; and fourth, that orbiting droplet pairs exhibit spin-half symmetry andalign antisymmetrically as in the Pauli exclusion principle. Our analysis explains the similaritiesbetween bouncing-droplet experiments and the behaviour of quantum-mechanical particles. Italso enables us to highlight some differences, and to predict some surprising phenomena thatcan be tested in feasible experiments.

1 Introduction

In 1978 Walker reported that a droplet of soapywater could bounce for several minutes on avibrating dish of the same fluid [1]. In 2005Couder, Protiere, Fort and Badouad startedthe systematic study of this phenomenon usingdroplets of silicone oil; the droplets can be madeto bounce indefinitely on an oil surface that is vi-brated vertically, and with the right amplitudeand frequency of vibration, droplets can movelaterally or ‘walk’ [2]. A thin film of air betweenthe droplet and the surface prevents coalescence.

Figure 1 illustrates the apparatus, figure 2 hassix photographs of a bouncing droplet, and fig-ure 3 illustrates the vertical motion as a functionof time. Here the droplet touches down everyother cycle; at lower driving amplitudes thereare less interesting modes of bouncing, where thedroplet grazes off the peak near f in the figureor touches down every cycle.

These experiments have since been repro-duced in laboratories around the world, such asby Molacek and Bush [3], and have appearedon TV [4]. They are of interest because thedroplets exhibit much of the behaviour that had

1

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previously been thought unique to quantum-mechanical particles.

Vibration exciter

Oil

Droplet of same oil

Figure 1: The experimental apparatus. Theshape of the container reduces unwanted wavesfrom the edge.

2 Basic mechanism

Figure 2: A droplet of silicone oil bouncingon the surface of the same liquid which isvibrated vertically. (courtesy Suzie Protiere,Arezki Boudaoud and Yves Couder [5])

To first order the height of the surface h abovethe ambient level obeys the wave equation

1

c2∂2h

∂t2− ∂2h

∂x2− ∂2h

∂y2= 0 (1)

where c is the wave speed. Figures 2 indi-cates that the waves are predominantly standing

c d e f a b

Figure 3: The height of the droplet (red) andthe surface (blue) as it oscillates vertically withtime. Surface waves are neglected. The labelscdefab refer to the photograph in figure 2.

waves rather than propagating ones (note the re-duced amplitude in photographs b and e). Therelevant circularly symmetric solution is

h = − ho cos(ωot) Jo(ωo r/c) (2)

where ho is the maximum height and Jo is aBessel function of the first kind.

2.1 Parametric reinforcement

The standing waves in (2) are reinforced para-metrically. See Figure 4. When the oil tray is atits greatest height it is accelerating downwards,typically at 3.3 – 4.3 g. This reverses the effec-tive force of gravity, lifting the wave crests andenlarging the waves.

Figure 4: Parametric reinforcement of the stand-ing waves in (2).

This can also be understood in terms of propa-gating waves as shown in figure 5. Like a pebblethrown into a pond, a droplet creates outgoingpropagating waves when it lands. They are rein-forced and reflected back when the oil tray is atits greatest height. The combination of outgoingand incoming waves forms the standing waves.

2

time

Sound cone

(a) (b)

ct

time

Oil tray at its greatest height

Droplet lands

Next landing

Figure 5: Bragg reflection. (a) When a dropletlands it creates a trough in the surface whichpropagates outwards (marked ‘sound cone’). (b)When the oil tray is at its greatest height it accel-erates downwards. This reverses the effective di-rection of gravity, reinforcing the wave and form-ing an inward-directed trough that reaches thecentre when the droplet next lands.

When the parametric driving is large enough,each bounce of the droplet influences the nextthrough this mechanism. The system behavesas if it had a memory of previous bounces; thisis called the ‘high memory’ regime.

2.2 Wave speed

The speed c in (1) depends on frequency. We donot pursue this complication since the frequencywas not varied in the experiments.

Consider an isolated propagating wave. If itoscillates in-phase with the forcing acceleration(similar to figure 4) at one position, it will havethe opposite phase a quarter of a wavelengthaway. Thus, any effect on the wave speed willapproximately cancel and the phase velocity islargely unaffected.

But the standing wave (2) near the dropletis always in-phase with the forcing acceleration.The restoring force on the wave is proportionalto hg − am cos(2ωot) where am is the maxi-mum vertical acceleration, so the net change ofmomentum on a half-cycle is proportional to∫ π

2

−π2

cos(ωot)g − am cos(2ωot)dt ∝ g − am3

Since am > 3g in all the relevant experiments,the parametric driving outweighs the net restor-

ing force of gravity and reverses it, leaving onlysurface tension to restore the waves. This sub-stantially reduces the speed c. Surface tensionis less effective in restoring larger waves, whichare slowed the most. These standing waves arethe ones that interact with the droplet and de-termine its motion, so we will need a reducedvalue of c in the equations that follow.

If the forcing acceleration is increased further,eventually it overcomes the restoring force ofsurface tension and c approaches zero, resultingin a Faraday instability [6].

3 Lorentz symmetry

The droplet in figure 2 moves to the right asit bounces. Its speed depends on the verticaldriving acceleration as shown in Figure 6.

3 3.5 4 4.50

2

4

6

8

10

12

maximum driving acceleration (g)

speed (mm/s)

Figure 6: The speed of a walker depends on themaximum vertical driving acceleration, graphedas a multiple of the acceleration g due to gravity.(Data courtesy Antonin Eddi, published in [7])

As we can see in figure 7, at low speed thedroplet merely appears to be displaced slightlyto the right, but as the parametric driving andthe droplet speed increase, the wave field sup-porting the droplet becomes more complex. SeeOza, Rosales and Bush for a detailed treat-ment [8].

3

Figure 7: A droplet moving to the right. Atlow speed (left) it has been displaced from thecentre; at higher speeds the wave field becomesmore complex. (Courtesy Antonin Eddi, EricSultan, Julien Moukhtar, Emmanual Fort, Mau-rice Rossi and Yves Couder [7])

3.1 Simplified model

We now derive a simplified model of this mo-tion using the symmetries of the wave equation,and test its predictions against the experimentaldata.

If h(x, y, t) obeys the wave equation (1), thenso does h(x′, y′, t′) where

x′ = γ (x− vt)y′ = y

t′ = γ(t− vx

c2

)γ =

1√1− v2

c2

(3)

This is the Lorentz transformation familiarfrom electromagnetism; it also applies to so-lutions of the wave equation in fluids, and isused in aerodynamics and acoustics (e.g. [9]).Applying it to (2) for the waves near a sta-tionary droplet gives cos(ωot

′)Jo(ωor′/c) where

r′2 = x′2 + y′2, which is illustrated in Figure 8.In the moving solution, all lengths in the di-

rection of travel have contracted by the factor1/γ (substitute t = 0 into (3)), while all timeperiods have dilated by the factor γ (substitutex = vt).

The driving frequency is fixed, so the wavefield of the droplet must adapt to compensate.If h(x′, y′, t′) obeys the wave equation then sodoes h(αx′, αy′, αt′) where α is a scale factor.

(a) (b)

v

Figure 8: (a) The waves near a station-ary droplet cos(ωot)Jo(ωor/c) (b) The Lorentz-boosted wave field cos(ωot

′)Jo(ωor′/c) moves at

speed v.

Choosing α = γ gives

x′′ = γ2(x− vt)y′′ = γ y

t′′ = γ2(t− vx

c2

)Applying this coordinate transformation to (2)gives

h = −ho cos

(ωot−

γ2ωov

c2∆x

)Jo

(ωocr′′)

(4)

where ∆x = x − vt. This may be a reasonableapproximation to the waves near a walker be-cause it obeys the wave equation, it moves atspeed v, and it bounces at constant frequency.

As the vertical acceleration increases, thedroplet is thrown higher and lands later inthe cycle (see figure 3). Suppose it lands att = nτ + T where ωoτ = 2π. From (4) with(∆x, y, t) = (0, 0, T ),

∂h

∂x= −ho

γ2 ωoc

(vc

sin(ωoT ))

∂2h

∂x2= ho

γ4 ω2o

c2

(v2

c2cos(ωoT ) +

1

2

)(5)

The sloping surface displaces the droplet fromthe centre, as we can see in figure 7. It will settlenear ∂h/∂x = 0, where, from (5),

γ2ωo

(v2

c2+

1

2

)∆x = vT

where we have approximated sin(ωoT ) = ωoTand cos(ωoT ) = 1. Subsequent waves will always

4

be generated at this displaced position, with thenet result that the wave pattern moves at speedv ∝ ∆x to first order, and

γ2(v2

c2+

1

2

)∝ T (6)

3.2 Comparison with experiment

The test for such a simplified model can onlycome from the experimental data, and we seein Figure 9 that the linear relationship we de-rive between γ2(v2/c2 + 1

2) and T/τ is remark-

ably accurate out to an acoustic Lorentz factorof γ = 2.6. The landing time T was obtainedfrom the intersection between the path of thedroplet, for which h = −g, and the vertical os-cillation of the oil tray (see figure 3 where themaximum acceleration was 3.5g). The charac-teristic speed c of the standing waves near thedroplet was taken to be 11.95 mm/s, which is8% larger than the maximum speed measured.

T/τ

Figure 9: The experimental data in figure 6 plot-ted on new axes. T is the landing time. Compareequation 6.

There is further information in detailed ve-locimetry studies, reported by Eddi, Sultan,Moukhtar, Fort and Couder in [7]. Figure 10is the wave field measured near a walker at highparametric driving. Successive bounces of thedroplet can be seen in the peaks marked A,Band C. We can obtain an approximation to thewave field by treating these three peaks as thecentres of three wave fields given by (4). This

C B A

Figure 10: The wave field near a walker at largedriving amplitude. Courtesy Antonin Eddi, EricSultan, Julien Moukhtar, Emmanual Fort, Mau-rice Rossi and Yves Couder [7]

is approximate because it neglects viscosity andnonlinearities such as those introduced by theparametric driving.

The three waves reinforce nearly perpendicu-lar to the direction of motion, as can be seenfrom the taller waves there. They interfere de-structively at an angle behind them, producingthe wake-like lines with nearly zero amplitude.

Taller waves have a reduced wave speed, dueto the parametric driving, which will compressthe wave pattern perpendicular to the directionof motion. At the same time, the wave patternis elongated parallel to the direction of travelbecause the source (A,B,C) is elongated. How-ever, there is a counteracting effect. The Lorentzcontraction in (4) compresses the wave patternin the direction of motion. As we can see in fig-ure 11, the resulting waves are roughly circular.See [7, 10] for further studies of the wave field.

Figure 11: Contours of the waves in figure 10.

5

We conclude that bouncing droplets can beconsidered to be Lorentz covariant, to a goodapproximation, up to a Lorentz factor of 2.6.

We now turn to experiments conducted atconstant forcing frequency and amplitude, wherethe velocity of the droplet is constant and theperturbations inflicted by the parametric drivingon the Lorentz symmetry can largely be treatedas constant and neglected.

4 Force between droplets

When a walker approaches the edge of the con-tainer, it does not actually touch the edge but isdeflected away. The stroboscopic photograph infigure 12 shows a droplet travelling three timesround a rectangular dish. This experiment givesdeep insight into one type of interaction betweendroplets.

Figure 12: Stroboscopic photograph of a droplet’spath (dots) deflected near the walls of the con-tainer (solid). (Courtesy Suzie Protiere, ArezkiBadaoud and Yves Couder) [5]

4.1 Velocity normal to theboundary

The velocity normal to the boundary, V⊥, canbe measured from the photograph in figure 12,where an equal time passes between each strobo-scopic image. Figure 13 plots V 2

⊥ as a functionof the inverse distance 1

rfrom the boundary.

1/r (mm-1)

V

2 (m

m2

s-2)

Motion towards boundary

Motion away from boundary

Figure 13: The square of the velocity normal tothe boundary, V 2

⊥, as a function of inverse dis-tance from the boundary. The data are extractedfrom the stroboscopic images near the bottom offigure 12.

For each branch in figure 13, the data near theboundary (towards the right of the graph) fallvery nearly on a straight line, before deviatingat greater distances. This straight line can bewritten

V 2⊥ = V 2

o −B

r(7)

where the slope of the graph is −B and it de-pends on the branch. Extrapolating to 1/r = 0on the upper branch gives Vo ≈ 18mm/s, whichis the same as the speed of the droplet to the ac-curacy of measurement. The lower branch hasVo ≈ 14mm/s.

4.2 Inverse square force

These experimental results show there is an in-verse square force of repulsion near the bound-ary. The mechanism responsible for it is in factwell known [11]. It is used for removing un-wanted bubbles of gas from oils and other liq-uids using ultrasonic vibration, as in figure 14.Ultrasonic pressure waves cause nearby bubblesto expand and contract in phase with one an-other, inducing oscillatory radial flows in the liq-uid. Near the mirror plane equidistant from two

6

Figure 14: Degassing oil by applying ultrasonicvibration. The process takes about 5 seconds.(courtesy Hielscher Ultrasonics GmbH)

bubbles, the flows reinforce as illustrated in fig-ure 15. The increased velocity results in a re-duced Bernoulli pressure, giving rise to a forceof attraction between the bubbles which mergeand rise to the surface.

Figure 15: Schematic of the flow from twosources.

The exact same mechanism causes a bounc-ing droplet to avoid the boundary of the dish,which has the same effect as an imaginary imagedroplet on the other side, at the same distance,and bouncing antiphase. Each droplet drives ra-dial flows in the liquid, just like those near thebubbles in the degasser – except that the bounc-ing droplets are antiphase, so the force is one ofrepulsion rather than attraction.

4.3 Size of the bubble force

In Figure 16, a vacuum cleaner nozzle ingestsvolume Q1 of air per unit time. If the flow isspherically symmetric then the air speed at ra-dius r will be U = −Q1/(4πr

2). A second nozzle

at this radius, with volume Q2 per unit time, willingest momentum along with the air particles

dp

dt= ρoUQ2 = − ρo

Q1Q2

4πr2(8)

where ρo is the density. This is an inverse squareforce of attraction between the nozzles.

Q1

Q2

Figure 16: The flow near two vacuum cleanernozzles, neglecting exhaust.

The direction of flow, and hence of the force,will be reversed if one of them is set to blow;then the force will be reversed for both hoses, byconservation of momentum (we assume the flowremains spherically symmetric, which might beachieved using a baffle on the end of the hose).More generally, oscillatory motion results in anattractive force if it is in-phase, and a repulsionif it is antiphase. The magnitude of the force isgiven by (8) averaged over a cycle.

4.4 Size of the droplet force

In the droplet experiments, the force in (8) mustbe doubled due to the hemispherical geometrybut halved to average over a cycle, leaving itsmagnitude unchanged.

Suppose a droplet of volume V bounces atfrequency f . It will induce a flow fV directly,which will be enhanced by secondary flows dueto entrained fluid and the reinforced waves, giv-ing Q = βfV where β is a factor into which we

7

will also incorporate the effects of higher har-monics. Substituting into (8) and rememberingto invert the sign, the acceleration of a dropletis

a = − F

ρoV=

V β2f 2

4πr2

Using f = 25Hz and a droplet radius of 0.35mm(which was reported for a different run), the ac-celeration measured in the upper branch of figure13 gives β ≈ 5.

4.5 Bubble force in conventionalform

An inverse square force can always be written inthe form

F = αbc

r2(9)

where α is a dimensionless constant and b is aconstant with the dimensions of energy × time.

Suppose the radius of a bubble is given by

rb = ro(1 + A sinωt)

We will simplify the calculation by assumingthat A is small. The flow speed at the surface is

vs =drbdt

= A roω cos(ωt)

Multiplying by the area, the flow is Q = 4πr2ovs.Substituting into (8) gives

F = 4πρor3o.r

3oA

2ω2 cos2(ωt)1

r2

= 3md.r3oA

2ω2 1

2r2

wheremd is the displaced mass of the bubble andwe have replaced cos2(ωt) by its average value,12.This can be rearranged into the conventional

form (9) using the fact that the inertial mass ofthe bubble, due to the motion of the displacedfluid around it, is approximately m = 1

2md [11].

Thus

α = 3A2(roωc

)3b =

mc2

ω(10)

The dimensionless constant α depends onwhether the bubbles are resonant or not. Con-sider the resonant case. Neglecting geometricfactors (which are of order 1), the bubble ra-dius will vary from a small value to 2ro, givingA ≈ 1. The maximum surface speed roω will alsoincrease, but it cannot much exceed the speed ofsound in the fluid since the pressure would re-duce to zero due to the Bernoulli effect. There-fore, if the bubbles are resonant then both A andthe ratio roωo/c will be of order 1, and so α isalso of order 1.

4.6 Constant of the motion

If an unperturbed acoustic Lorentz transfor-mation (3) could be realized experimentally, bwould be a constant of the motion. The effectivemass m of a wave is proportional to its volume,so the Lorentz contraction multiplies it by 1/γ,whilst the angular frequency is multiplied by thesame factor, so b ∝ m/ω is independent of ve-locity. More formally, its dimensions, energy ×time, are Lorentz invariant.

The droplet’s speed was not varied during theexperiments. One way to achieve this might beto adjust the forcing amplitude and frequency(correcting for the perturbation to the wavespeed and height). Alternatively a droplet offerrofluid might be de-weighted magnetically soit lands later in the cycle and travels faster. Theforcing frequency might be adjusted to avoid theadditional factor (1 − v2/c2) from the scale en-largement in (4). The mass of the droplet it-self must be treated separately from that of thewaves.

4.7 Comparison to the force be-tween electrons

The electrostatic force between electrons is

F = α~cr2

α ≈ 1

137.036

~ =mc2

ω

8

where m is the mass of the electron, ω its angularfrequency, c is the speed of light, and ~ = h/2πwhere h is Planck’s constant.

Notice the analogy to (9) and (10), where b isdefined in the same way as ~.

From the value of α, it seems that the elec-trostatic force is about two orders of magni-tude weaker than the mechanical force betweenresonant bubbles. This suggests one limitationof the bouncing-droplet experiment as a modelof quantum mechanics, namely that spherically-symmetric resonant solutions are not a goodmodel for the electron. We will explore higher-order solutions that are not spherically symmet-ric below.

4.8 Maxwell’s equations

When the droplets or bubbles are stationary, wehave seen there is an inverse square force be-tween them (when averaged over a cycle). Thisobeys the same equations as the electrostaticfield near a charged particle, since both are in-verse square.

These equations can be extended to the case ofmoving droplets by noting that the solutions areacoustically Lorentz covariant to a reasonableapproximation. So we need equations that areLorentz covariant and that reduce to the equa-tions of electrostatics when stationary.

These conditions are met by Maxwell’s equa-tions with an acoustic value of c. In fact, theyare unique in that Maxwell’s equations (morestrictly, equations that are equivalent to themwhen they are averaged over a cycle) are the onlyones that satisfy them. Suppose the contrary,that there existed a different set of Lorentz co-variant equations that produce the same electricfield with the same boundary conditions. Theonly difference between the two solutions canbe in the magnetic field. But a Lorentz trans-formation turns a pure magnetic field into onewith an electrical component, and so the electri-cal fields differ in the new reference frame. Thisis a contradiction. Thus the interaction betweenthe droplets obeys Maxwell’s equations when itis averaged over a cycle.

This model predicts an interaction obeyingthe equations of magnetism, which is observedexperimentally as follows.

4.9 Magnetic interaction

We now turn to the lower branch of figure 13.

A walker’s speed is fixed by the driving ampli-tude as discussed above. As the walker’s velocitynormal to the boundary slows down and reversesin the experiment, it must accelerate parallel tothe wall to maintain constant speed. The ve-locity boost is observed in the experiment. Theresearchers estimated the angle of incidence (rel-ative to the normal to the boundary) at about38, and of reflection at about 53.

In addition to the force of repulsion betweenthe droplets, which obeys the same equations asthose of electrostatics, there is a force of attrac-tion when they are moving at a common velocityv parallel to the boundary. This obeys the equa-tions of magnetism. It is like the magnetic forceof attraction between two electrons moving at acommon velocity parallel to one another.

The magnetic force reduces the total force by afactor 1− v2/c2, which accounts for the reducedslope of the lower graph in figure 13. We willtake the approximation that the upper branch infigure 13 has no velocity perpendicular to the di-rection of travel, but by the time the droplet hasreached the lower branch it has been acceleratedto the full perpendicular speed, v = 18 mm s−1

by the tangential force. The force is proportionalto the slope of the graph, whose ratio is 14/18.Equating these two gives v = 0.47c, which sug-gests the droplets were moving at about half thewave speed.

4.10 Propagating waves

Maxwell’s equations tell us that if a source isaccelerated, propagating waves will be emitted.We are not aware of attempts to test for thesewaves experimentally with droplets. This mightbe possible by accelerating droplets of a fer-rofluid horizontally using magnetics. The period

9

of the acceleration should be longer than that ofthe droplets.

We predict that propagating waves will beobserved that are modulations of the stand-ing waves surrounding the source (not ordinarylongitudinal or transverse waves). They obeyMaxwell’s equations with an acoustic value of c.

The force obeying Maxwell’s equations is onlyone of the interactions between droplets. To seeanother we must turn to coherent motion.

5 Diffraction

We saw how a droplet is repelled from a barrier.When the barrier has one or more slits in it, somedroplets pass through as we can see in figure 17.

Figure 17: A droplet passing through an aperturein a submerged barrier.

When the Paris team measured which direc-tion they went, they found classical diffractionpatterns as you see for light waves, water wavesor quantum mechanical particles, as in figure 18.Diffraction is only observed in the high memoryregime. In the low memory regime the wavesfrom a droplet have little effect because theypropagate away and are lost to viscosity.

In these experiments the forcing frequencyand amplitude, and hence the walker velocity|v|, were kept constant. The variation was in vxand vy.

5.1 Wavelength

We saw (equation 2) that the surface height neara stationary droplet has two component factors

Figure 18: Histogram showing the number N ofdroplets (out of 125) that emerge at angle α tothe normal at large distance. The solid line isa single-slit diffraction pattern. Courtesy YvesCouder and Emmanuel Fort [12].

which we will now write ψ and χ where

h = ψ χ

ψ = cos(−ωot)χ = − ho J0(krr) (11)

We are interested in how the wave field of amoving droplet varies in the x direction, neglect-ing vy. An acoustic Lorentz transformation (3)gives the approximate solution

ψ = cos(−ωot′)χ = − ho J0(krr′)

where for simplicity we have neglected the scaleenlargement in (4), which is just a constant since|v| is constant. In this moving solution, the wavefield χ advances with the droplet at speed vx,and ψ has become a planar wave which can bewritten

ψ = cos(kx− ωt) (12)

The values of k and ω can be obtained bydefining S = −ωot′ and noting, from (3),

k =∂S

∂x=

∂S

∂t′∂t′

∂x=

γωoc2

vx

ω = −∂S∂t

= −∂S∂t′

∂t′

∂t= γωo (13)

10

The wavelength of ψ is λ = 2π/k, or

λ =2πc2

ωvx=

b

p(14)

where p = mvx is the momentum of the waveand b = 2πb where b = mc2/ω.

If we choose m to be the effective mass of thewave (as discussed in section 4.6), then the pa-rameter b that determines the wavelength of adroplet is the same as the constant in (10) thatdetermines its deflection from the boundary bythe inverse square force.

Equation (14) is the same as the de Brogliewavelength of a quantum mechanical particlewith b instead of Planck’s constant h. It canbe extended to arbitrary axes. If the velocity inthe direction of interest is v = (vx, vy) then ψ in(12) becomes

ψ = cos(k.x− ωt) (15)

wherep = bk (16)

and p = (px, py) is the momentum.

5.2 The diffraction pattern

We now show that the histogram in figure 18agrees with the diffraction pattern of ψ throughthe aperture.

The width of the aperture was 14.8 mm and,based on measuring the vertical distance fromthe barrier to the first node (measured near thecorner to the right of the aperture), the wave-length of ψ near the aperture was λ = 7.3 mm.When waves of wavelength λ diffract through asingle aperture of width L, the first minimumof amplitude is at angle θ where λ = L sin θ.The above measurements predict this will occurat θ = 30. The minimum in the experimentalhistogram occurs between 30 and 35.

So we observe that the minimum in the his-togram occurs where the waves of ψ interferedestructively. It is as if the droplet were re-pelled from these regions. This can be under-stood as follows. Bigger waves have deeper wavetroughs, so a droplet bouncing in them will be

physically lower than one bouncing elsewhere.Consequently it will be attracted towards themby the force of gravity. Thus droplets move awayfrom the regions of destructive interference of ψwhere the waves are smaller.

5.3 Double-slit diffraction

In another experiment, the droplet was made todiffract through two slits.

Figure 19: Droplet passing through a double slit

The histogram of the directions taken by thedroplets after they had passed through one orother of the slits is shown in figure 20.

The distance between the slits was 14.3mm.Using the above parameters we would expect thefirst diffraction minimum to be at approximately15, and this is what the researchers observed.

5.4 Classical approximation

Defining a quantity with the dimensions of en-ergy

E = bω

then, from (13),

ω2 − c2k2 = ω2o

E2 − p2c2 = m2oc

4

where we have used p2 = b 2k2 from (16) and thedefinition of b in (10). This will be recognised as

11

Figure 20: Histogram of the deflection angle for75 droplets that have passed through one of twoslits. The solid lines show a possible fit to adouble-slit diffraction pattern. Courtesy YvesCouder and Emmanuel Fort [12].

the relativistic equation of motion for a classicalparticle of rest mass mo and energy E. It has alow-velocity approximation

E = Eo

(1 +

p2c2

E2o

) 12

≈ Eo +p2

2mo

which is the Newtonian equation of motion.In order to include the inverse square force

discussed above, it suffices to add a term to theenergy, namely

bω = E = mc2 − V (17)

where V is the potential energy associated withthe interaction.

Revisiting the experimental constraints, inprinciple it might be possible to adjust the forc-ing frequency in accordance with (17) during theexperiment, but this was not done. The fixedforcing frequency constrained |v| to be constant.As we saw in figure 12, the velocity perpendic-ular to the direction of interest will adjust tocompensate. The experimenters observed an in-creased tangential speed when the droplets werenear the diffraction slits, but the perturbationto the expected diffraction patterns, if any, wassmall.

5.5 Klein-Gordon equation

From (11), the factor ψ for a stationary particleobeys the equation

∂2ψ

∂t2= − ω2

oψ (18)

In order to extend this to the case of a mov-ing particle, we need a Lorentz covariant equa-tion that reduces to (18) in the stationary case,namely

∂2ψ

∂t2− c2∇2ψ = ω2

oψ (19)

since the left hand side is Lorentz invariant.Equation (19) is the same as the Klein-Gordon

equation of quantum mechanics for a relativisticparticle. The only difference is that the charac-teristic speed in the experiment is the speed ofsurface waves in the oil rather than the speed oflight.

5.6 Schrodinger equation

If (11) receives a Lorentz boost with a smallvelocity v in the x direction then we get ψ =cos(−ωot′) = cos(vxωo/c

2 − ωot) where we haveapproximated γ = 1. Writing this in the form

ψ = R cos(θ − ωot) (20)

gives θ = vxωo/c2. Extending to arbitrary axes

gives the velocity of the droplet as determinedby the local waves

v =c2

ωo∇θ (21)

The function in (20) can be analytically con-tinued into the complex plane by defining

ψs = R eiθ (22)

so that ψ = <(e−iωotψs) where < means the realpart. Now, ψ obeys the Klein-Gordon equation(19), and we will seek a solution where both thereal and imaginary parts of e−iωotψs obey thissame equation, which is satisfied when

i∂ψs∂t

= − c2

2ωo∇2ψs (23)

12

where we have neglected the term in ∂2ψs/∂t2,

which is small when the velocity is small.Substituting (17) in the form bωo = moc

2−Vgives

i b∂ψs∂t

=

(− b 2

2mo

∇2 + V

)ψs (24)

This is the same as the Schrodinger equationfor the wavefunction of a quantum mechanicalparticle. The only difference is that the con-stant of motion in the experiment is b ratherthan Planck’s reduced constant ~ (although theyare defined in the same way and they are bothconstants of the motion).

5.7 Probability density

If the starting position of a droplet is not knownprecisely, and it is allowed to evolve over time,then there will be a range of final positions,which can be calculated probabilistically. Ourtreatment will follow the reasoning of DavidBohm [13], who solved this problem in 1952.

Substituting the definition ψs = Reiθ (equa-tion 22) back into (23), and taking the imaginarypart when θ = 0 gives

∂R

∂t= − c2

2ωo(R∇2θ − 2∇R∇θ)

which can be rearranged into

∂R2

∂t+ ∇(R2 v) = 0 (25)

where v is the velocity of the droplet in (21).This equation has a simple interpretation.

When the velocity v of a compressible fluid, suchas the air, varies with position, its density ρobeys the continuity equation ∂ρ

∂t+∇(ρ v) = 0,

which is the same as (25) with R2 replaced byρ. Since the velocity of the droplets is v, it fol-lows that the probability density for the posi-tion of the droplet, averaged over nearby trajec-tories, must be R2 = |ψs|2 (provided the initialvalue of |ψs|2 is appropriately calibrated, or ‘nor-malised’). This is confirmed by the experimen-tal results in figure 21. The graph shows that

the probability a droplet crosses the barrier (or‘tunnels’) reduces exponentially with its width.If you solve Schrodinger’s equation with a bar-rier, you get the same exponential decay of |ψ2|with the width of the barrier. The same prob-ability density |ψ2

s | is assumed as a postulate inthe Copenhagen interpretation of quantum me-chanics.

Figure 21: Droplets encounter a region of re-duced depth, which repels them. The x axis isthe width of the barrier, and the y axis is theprobability the droplet tunnels through the bar-rier, plotted on a logarithmic scale. CourtesyAntonin Eddi [14]

The foregoing calculation was performed in1952, more than 50 years before the first dropletexperiments. It led Bohm to hypothesise theexistence of a tiny particle which moves at thevelocity v in (21), guided by waves that obeySchrodinger’s equation (23), whose probabilitydensity is |ψ2

s |. These are exactly the equationsfor a bouncing droplet at low velocity. His in-sight is remarkable. For him, this was a purelyabstract exercise; he did not have the dropletmodel to inspire him to derive these relation-ships from Euler’s equation.

Based on these equations, Bohm showed theresulting mechanics to be indistinguishable fromthe Copenhagen interpretation of quantum me-chanics. He subsequently found that Louis deBroglie had suggested a similar idea in the 1920s;the model is now called the de Broglie-Bohm in-

13

terpretation of quantum mechanics.

5.8 Same equations, same solu-tions

Given that we have the same mathematics upto a constant factor, we expect the calculationsof quantum mechanics to carry over to otherdroplet experiments, and can start to under-stand why bouncing droplets are a pretty goodmodel of the quantum world.

Figure 22: A droplet in a rotating bath is at-tracted towards the centre, and exhibits quantizedorbits. Courtesy Emmanual Fort [15]

In figure 22, the experiment was conductedin a rotating bath, where the droplet was at-tracted towards the centre. The droplets exhib-ited quantized orbits.

Other experimentalists have discovered fur-ther evocative results. For example, ValeriySbitnev has recently reported that a droplet andan antidroplet (a bubble) can be created whentwo suitably-shaped soliton waves collide on thesurface of the fluid [16]. The droplet and an-tidroplet move apart, just as when a particle andantiparticle are created in quantum mechanics.However, before we approach the domain of fieldtheory, we first have to discuss spin.

6 Rotational motion

We have so far only considered waves with cir-cular and spherical symmetry. The photographsin figure 23 suggest we also have to think aboutsolutions to the wave equation which depend onangle.

Figure 23: The waves with two droplets. Theside drawings show a Bessel function J1, whichis the lowest rotating component of the standingwaves between the droplets. The bouncing is an-tiphase in (a) and (c) and in sympathy in (b)and (d). (Photograph courtesy Suzie Protiere,Arezki Boudaoud and Yves Couder [5])

In this experiment, the droplets orbit aroundone another with a period of approximately 20bouncing periods in (a), and longer in (b)-(d).Their velocity is approximately that of an ordi-nary walker driven with the same vertical accel-eration, and the period increases with radius.

6.1 Harmonic solutions

The motion in the photographs can be describedusing the solutions to the wave equation in cir-cular coordinates (r, θ), namely

hm = ho cos(ωot−mθ) Jm(krr) (26)

where Jm is a cylindrical Bessel function of thefirst kind, m is an integer whose sign is signif-icant, and ωo = c kr. The waves near the twodroplets contain components with various valuesof m, but the main experimental results can beunderstood from the lowest order rotating com-ponents, with m = ±1. We neglect higher har-monics as well as Jo in (b) and (d).

14

Figure 23 shows the Bessel function J1, whichgives the wave height of the lowest rotating com-ponent on a line joining the droplets. As wehave seen, the droplets prefer to land in the wavetroughs, so they are in free flight over the crests,as shown. The rotating wave pattern is

h = 12h0 [cos(ωot+ Ωt− θ) J1(k1r)

+ cos(ωot− Ωt+ θ) J1(k2r)]

≈ ho cos(ωot) cos(Ω t− θ) J1(krr) (27)

where ck1 = ωo + Ω and ck2 = ωo − Ω. Inthe second expression we have used the identitycos(A + B) + cos(A − B) = 2 cosA cosB, andhave approximated k1 ≈ k2 ≈ kr, which is validat small r and Ω. This can be regarded as astanding wave that rotates with the droplets atangular frequency Ω.

The factor cos(Ωt − θ) vanishes on the nodeline θ = Ωt± 1

2π. On either side of this line, its

sign reverses; we see in the photographs in fig-ure 23(b) – (d) that the crests turn into troughsand the troughs crests. The node line is nearlynormal to the line joining the droplets, indicat-ing that they are bouncing close to the anglewith the largest wave amplitude. The patternis not so evident in (a), due to the greater an-gular velocity and the presence of higher-ordercomponents.

6.2 Angular momentum

Figure 24 shows how the wave height h1 in (26)varies with angle. The wave propagates in the+θ direction. The flow velocity u is irrotational(∮

u.dl = 0) when the path of integration dl ison the submerged line A, but this does not meanthe wave has no angular momentum since it isnot irrotational at B. The elevations carry extrafluid around the centre.

At large radius, the net flow around the centreapproximates to that of a vortex when averagedover a period and a wavelength. The Bessel func-tion in (26) approximates to a standing wave inthe radial direction, whose amplitude reduces asA ∼ r−

12 . The flow speed is u ∝ A so the net

flow is proportional to uA ∼ r−1, which is thesame as a vortex.

A

0 2ππθ

B

Figure 24: The wave h1 in (26) at fixed radiusat the instant ωot = π/2. The flow speed (redarrows) is proportional to the wave height.

At small radius the flow diverges from that ofa vortex, and in particular there is no singularity.It is illustrated in figure 25.

6.3 Attraction to the boundary

Two vortices of opposite circulations are at-tracted towards one another because their flowsreinforce in the region between them, giving areduced Bernoulli pressure. The rotating pairsshould similarly be attracted towards their im-ages in the boundary, which rotate in the op-posite direction. A protrusion on the boundarymight be used to test for this.

The fine structure constant of the interactionis as follows. We saw that individual droplets arerepelled from the boundary by an inverse squareforce. This static force obeys the equations ofelectrostatics, with a fine structure constant ofα ∼ 0.3. We also saw evidence for a motion-dependent force which obeys the same equationsas magnetism, in which a droplet and its imagein the boundary are attracted towards one an-other when they both move in the same directionparallel to the boundary.

The static forces between a pair and its im-age in the boundary nearly cancel out, since ifone droplet of a pair is attracted to the imagethen the other, being antiphase, will be repelled.However, the motion-dependent force is alwaysattractive. For example, consider the interac-tions with the wave crest marked in red on the

15

+-

(a)

(b)

+-

Figure 25: (a) Schematic drawing of the rotatingdroplet pair photographed in figure 23(a), and itsimage in the boundary. At the instant drawn,the green circles are the wave troughs (where thedroplet lands) and the red circles are crests. (b)The fluid flow due to the rotational motion. Thetrough has negative volume and contributes tothe flow in the same Cartesian direction as thecrest.

left of figure 25. It is moving in the same di-rection as the (green) trough in the image, analignment which we saw produces a force of at-traction (section 4.9). The crest in the image(red) moves in the opposite direction and withthe opposite phase. Each of these individuallyreverses the direction of the interaction, so thecombination leaves the sign unchanged.

We saw that the ratio of the moving to thestatic forces is v2/c2 (the same as the ratio ofthe magnetic to the electrostatic forces for par-ticles moving at velocity v). The force on eachdroplet is doubled, since there are two images,but it must be averaged over a rotation, givinga factor of 1

2. The fine structure constant of the

interaction is thus

α2 =v2

c2α1 (28)

where α1 is the fine structure constant for thestatic force between individual droplets. Thestrength of the interaction depends on the ro-tational speed, which can be varied in the ex-periment. A typical value might be v = 1

4c and

α1 ∼ 0.3 giving α2 ∼ 1/50.Our model predicts that an orbiting droplet

pair will be attracted towards the boundarywith this reduced fine structure constant. Thisphenomenon has been noted by the experi-menters; orbiting pairs that approach a sub-merged boundary at a shallow angle can stick toit and then move along it, playing ‘hopscotch’ aseach droplet takes it in turn to leapfrog the onein front. However an experiment with precisemeasurements has not yet been performed.

6.4 The emergence of spin-halfbehaviour

The rotating waves in (26) can be treated asindependent because they are orthogonal in thesense that∫ 2π

0

hm hn dθ = 0 (m 6= n) (29)

as may be verified by direct substitution.We have seen that the angular momentum of

the wave h1 in (23) is in the +z direction (ver-tically upwards), and it is in the −z directionfor h−1. The photograph in figure 23 shows thecase where the waves have nearly equal ampli-tude. What if the amplitudes are not equal?

It simplifies the analysis to consider ‘degen-erate’ solutions, that is, solutions that have thesame energy. (Solutions of arbitrary energy canbe obtained by scaling the wave height.) Thedegenerate solutions are

h = cos(α) h1 + sin(α) h−1 (30)

where α is a real parameter. This is a so-lution to the wave equation because it is asum of solutions. Its energy is proportional tocos2 α + sin2 α, which is constant, so the wavesare degenerate. The angular momentum is

L = Lo(cos2 α− sin2 α)

= Lo cos(2α) (31)

where Lo is the angular momentum of h1. Theangular momentum and the wave pattern h varycontinuously with the parameter α, as shown inthe table below

16

α L/Lo h

0 1 h1π4

0 1√2(h−1 + h1)

π2

-1 h−13π4

0 1√2(h−1 − h1)

π 1 −h1

As we can see in the table, the wave fieldreverses sign after the direction of the angularmomentum has gone through a complete cycle.Two cycles are needed to return to the startingposition.

Fermions are like these waves, in that theirwavefunctions reverse sign if the direction oftheir angular momentum is rotated through360. It is commonly believed that this be-haviour cannot emerge from classical mechanics.However the rotating droplets show that this be-lief is wrong.

In fact, double symmetry is already known insystems that contain two harmonic sub-systems.Leroy, Bacri, Hocquet and Devaud provided an-other example in 2006 when they showed thattwo weakly coupled pendula with nearly thesame frequency also have this symmetry [17].

6.5 Bloch sphere

The elementary waves in (26) are the real partof

ξm = A e−i(ωot−mθ) Jm(krr) (32)

where A is the amplitude. This can be factoredas before into

ξ = ψ χ

ψ = e−iωot

χ = A eimθ Jm(krr) (33)

The last section showed that ψ obeysSchrodinger’s equation. Now let us examine thefactor χ.

When the wave height in (30) is extended intothe complex plane as in (33), we get a simple wayto provide an arbitrary origin of time for each ofthe two components

χ = eiS[cos(12β)χ1 + eiϕ sin

(12β)χ−1]

(34)

where S is an arbitrary overall phase, ϕ is therelative phase of the two components, and wehave defined β = 2α.

x

y

z

φ

β

Figure 26: A Bloch sphere.

The parameters in this equation can be rep-resented on a sphere as shown in figure 26. Theangular momentum normal to the surface (inthe z direction) is proportional to cos β. Whenβ = 1

2π, the angular momentum vanishes and

there are standing waves whose amplitude isgreatest at angle ϕ to the x axis. However,this diagram should not be over-interpreted. Itwould be wrong to conclude that the system isphysically oriented in the direction indicated inthe figure. The existence of a simple geometri-cal way to picture the parameters in (34) shouldnot blind us to the fact that we are describingordinary surface waves which cannot rotate outof the plane of the surface.

Nonetheless, equation (34) is the same as thatof a spin-half particle whose wavefunction is χwhere χ1 is the spin-up state and χ−1 the spin-down state, which is usually represented on the‘Bloch sphere’ in figure 26. By inspection, thesign of χ reverses when β increases by 2π, whichis characteristic of spin-half systems.

6.6 Pauli spin matrices

The mapping of the wave height near a dropletonto the Bloch sphere can be shown more for-

17

mally by writing (34) as a dot product of twovectors a and χ

χ = a.χ = (a1, a2).(χ1, χ−1)

where the values of ai are obtained from (34).The angular momentum of the first componentis proportional to |a1|2, and that of the secondcomponent is proportional to −|a2|2, so the nor-malised total is

σz =|a1|2 − |a2|2

|a1|2 + |a2|2(35)

This can also be written

σz =a∗. σz a

a∗.a(36)

where σz = ( 1 00 −1 ) is the same as the Pauli spin

matrix for the z direction.As in quantum mechanics, we can extend this

as follows. The Pauli matrices areσx = ( 0 1

1 0 ), σy = ( 0 −ii 0 ), σz = ( 1 0

0 −1 ), and spinprojections σi are defined by

σi =a∗. σi a

a∗.a

where i can be x, y or z. The eigenvectors of σiare

β ϕ (a1, a2) (σx, σy, σz) Eigenvector of12π 0 1√

2(1, 1) (1, 0, 0) σx

12π

12π

1√2(−i, i) (0, 1, 0) σy

0 0 (1, 0) (0, 0, 1) σz

It will be noticed that (σx, σy, σz) correspondto the Cartesian coordinates of a unit vector atthe spherical angle (β, ϕ) in figure 26. This isthe basis of the Bloch sphere, which maps be-tween the two representations. The mapping isa double covering because χ reverses sign when βincreases by 2π. The same mathematics is usedto describe fermions in quantum mechanics.

6.7 Antisymmetry

When the driving amplitude is reduced, the rota-tion speed of the droplets photographed in figure

+- BA

+

-

x

y

Figure 27: Schematic of two droplet pairs neareach other. Elevations are marked red and de-pressions green.

23 slows to zero. Figure 27 is a schematic of twodroplet pairs near each other. A is a solution to(34) with (β, ϕ) = (1

2π, 0) and B has (1

2π, 1

2π).

The droplets in B have opposite phases, so onewill attract the other pair whilst the other repelsand the net forces cancel. However, there is stillan effect involving orientation. One droplet in Ais closer to B than the other, which will cause Bto rotate anticlockwise. This prediction mightbe tested experimentally.

After B has rotated into the preferred align-ment, the solutions will be oriented in the x di-rection and the wave height is the real part of

ξ = ξa(x, t) − ξa(x− d, t) (37)

where ξa is the wave due to A and d is the sep-aration of the pair, B − A.

Equation (37) is antisymmetric, and, in par-ticular, exchanging A and B reverses the signof the wave field. This may be compared tothe principle, formulated by Wolfgang Pauli in1925, that the total wave function for two iden-tical fermions is anti-symmetric with respect toexchange of the particles.

7 Discussion

In this paper we have explained why the bounc-ing droplet experiments of Couder, Fort and col-leagues are a pretty good model for quantummechanics.

We have derived from first principles thatbouncing droplets are, to a rather good approx-imation, Lorentz covariant, with c being the

18

speed of surface waves; that they obey an ana-logue of Schrodinger’s equation where Planck’sconstant is replaced by an appropriate constantof the motion; that the force between them obeysMaxwell’s equations, with an inverse-square at-traction and an analogue of the magnetic force;and finally that orbiting droplet pairs exhibitspin-half symmetry and align antisymmetricallyas in the Pauli exclusion principle.

These results explain why droplets undergosingle-slit and double-slit diffraction, tunnelling,Anderson localisation, and other behaviour nor-mally associated with quantum mechanical sys-tems. We make testable predictions for the be-haviour of droplets near boundary intrusions,and for an analogue of polarised light.

The mathematical model described here maybe useful as a teaching aid. Bouncing-dropletexperiments have become popular with under-graduates; we show here that they can be ex-plained with the mathematics routinely taughtin a first undergraduate course in fluid mechan-ics. Indeed they are already sometimes one ofthe examples used to motivate such courses.

For an introductory course in quantum me-chanics, droplet models might help studentsovercome the initial feeling of bewilderment atthe quantum-mechanical wavefunction ψ. Here,ψ emerges naturally from known physical prin-ciples. This should help explain how thewavefunction of several interacting particles canemerge as a function of their position, momen-tum and spin, yet still be defined as a single am-plitude and a single phase at each point in space– helping students to avoid confusion over suchconcepts as configuration space and quantum en-tanglement. A vivid experimental model with aclear mathematical explanation should also helpdemystify spin-half behaviour and antisymme-try.

Finally, one might ask whether it is possible toextend this model from two dimensions to three.In separate work we show a model of rotonsin liquid helium with similar properties to thedroplets described here. Second sound in heliumcan be modelled as waves in a gas of quasiparti-cles, the lambda point as its Kosterlitz-Thouless

transition, while transverse sound emerges as thepolarised-light analogue whose existence we pre-dict here for droplet experiments [18]. So theremay be room for further research on even morecomplex and realistic analogue models of quan-tum mechanics.

Acknowledgments

We are very grateful to Yves Couder, EmmanuelFort and Antonin Eddi not just for discussionsbut for access to their raw data and permissionto use it here. We are also grateful to Robin Ball,John Bush, Graziano Brady, Basil Hiley, KeithMoffatt, Valeriy Sbitnev and to seminar partici-pants at Warwick and Cambridge for comments,criticism and feedback.

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[3] J. Molacek and J. W. M. Bush. Dropsbouncing on a vibrating bath. Submittedto J. Fluid Mech., 2013.

[4] Y. Couder. television programme.https://www.youtube.com/watch?v=

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