A Quantization Scheme for the Experiments with ''Walking Droplets

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A Quantization Scheme for the Experiments with ''Walking Droplets'' Javier Montes, Fabio Revuelta, and Florentino Borondo J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b12043 • Publication Date (Web): 01 Feb 2019 Downloaded from http://pubs.acs.org on February 5, 2019

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The Journal of Physical Chemistry

A quantization scheme for the experiments with walking droplets J. Montes,† F. Revuelta,∗,†,‡ and F. Borondo∗,‡,¶

Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Agroambiental y de Biosistemas, Universidad Politécnica de Madrid, 28040Madrid, Spain, Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049Madrid, Spain, and Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049Madrid, Spain E-mail: [email protected]; [email protected]

Abstract

how this task should be performed, or even which was the value of the elementary quan3 4 5 tum. Ehrenfest and Einstein, among others,

In this paper we explore the interest and feasibility of quantizing the macroscopic surface droplets on a vertically vibrated liquid surface

were main actors at that time by recognizing 6 the relevance of the action in this issue. Also 7 Maslov played an important role by detecting

in the limit of high memory of the droplet tra-

a necessary topological contribution to the ac-

jectory, where an astonishing similarity with

tion, the now called Maslov index, which is not 8 always easy to compute.

wave generated in the dynamics of walking

the quantum behavior has been experimentally observed.

In the late 1970's and 80's, when the computational power was still rather reduced, semi-

1 Introduction

classical quantization of the action became an attractive tool to compute tional energy levels.

Quantization is a key issue in many wave phe-

quantum vibra-

A wide variety of dier-

ent strategies, all based on dierent principles,

nomena at every scale, ranging from the macro-

were introduced to perform this task. Just to

scale in musical instruments, to the quantum 1 micro-world.

name a few, we have the following methods. Probably, the most straightforward method 9 who obtained

Indeed, it played a fundamental role in the

is due to Noid and Marcus,

early development of the quantum theory, af-

the eigenenergies by imposing quantization con-

ter the introduction of the wave-particle du2 alism by de Broglie. Then At that time it

ditions on numerically constructed invariant 10 tori. Chapman, Garret and Miller obtained

was not clear what was the physical magni-

the generator for the transformation to good

tude that should be primarily quantized, nor

action-angle variable by iteratively solving the

To whom correspondence should be addressed Grupo de Sistemas Complejos, Escuela Técnica Superior de Ingeniería Agronómica, Agroambiental y de Biosistemas, Universidad Politécnica de Madrid, 28040 Madrid, Spain ‡ Instituto de Ciencias Matemáticas (ICMAT), Cantoblanco, 28049Madrid, Spain ¶ Departamento de Química, Universidad Autónoma de Madrid, Cantoblanco, 28049Madrid, Spain ∗

corresponding Hamilton-Jacobi equation using

†

Fourier series. Other authors used classical perturbation theory to construct

a normal form

approximation to the Hamiltonian, which was subsequently quantized. An accelerated conver11 gence was obtained by using Van Vleck and 12,13 Lie transformations in Refs. Other quan-

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Page 2 of 14

tization methods based on Fourier analysis of

quantum pattern is obtained from the angular

the trajectories

distribution of droplet trajectories, was one of

14

were also proposed . Finally,

we will mention the method developed by Rein-

them. Tunneling, 25 where a droplet overcomes

hardt et al.

based on the adiabatic conserva-

the barrier formed by dierent depths in the liq-

tion of the action on perturbed Hamiltonian,

uid container, has also been observed in WD.

also called 'adiabatic switching'.

Similarly, the existence of orbit quantization-

15

In connection to this it should be mentioned

like conditions, 26,27 where the uid container is

that some one decade ago Fort, Couder and

subjected to a constant velocity rotating motion

coworkers in Paris, 16 followed by the group of 17

and then only droplet orbits corresponding to

carried out experiments in a

specic values of the radius are stable, has also

macroscopic system showing properties highly

be reported. Other quantum eects have been

Bush at MIT,

reminiscent of the quantum behavior. 18 In these

reproduced, such as, Zeeman-like level split-

experiments, a millimetric oil droplet, bounc-

ting 28,29 in experimental situations in which a

ing on a liquid bath vertically vibrated close

harmonic potential is implemented in the hy-

(but below) the Faraday wave threshold,

was

drodynamical system either by means of a fer-

able to self-propel due to the interaction with

romagnetic material added to the droplet and

the wave generated at the bounces (see video

an external magnetic eld, or by using a ro-

at Ref.

20

19

). At each impact on the surface, the

tating container. Also, the emergence of a co-

droplet creates a circular capillary wave, that

herent statistics patterns due to interference of

in turn excites standing Faraday waves. These

the orbits 'bouncing' on the boundaries of a

waves propagate, damping away with a time

closed container, 30 which are the macroscopic

that is proportional to the extent to which the

analogue of the electronic quantum density ex-

amplitude of the forcing vibration approaches

isting in nanometric corrals on metallic sur-

faces, 31 or entangled bound states consisting of

the Faraday instability threshold. In this way, one can dene a memory parameter

M

to ac-

a number of droplets simultaneously performing

count for the damping over the time interval

complicated choreographies, 32 has been experi-

between successive rebounds. Accordingly, the

mentally observed in this macroscopic system.

wave reshapes from the last

M

Recent advances in the theory and experi-

the surface which results only impacts. When

M >> 1,

ments of WD can be found in Ref. 33

the

Due to

droplet suers a small horizontal force, making

the close resemblance between the

it to start 'walking' on the liquid surface, rea-

WD phenomenology and the quantum world, it

son why this system is usually known as

seems interesting to study quantization in this

droplets (WD).

walking

context, as done in this work.

Moreover, in this system there is a strong cou-

The organization of the paper is as follows.

pling between wave and particle, since the for-

In Sect. 2 we describe the theoretical frame-

mer impulses the latter, and the bounces of the

work . First, we discuss the dynamics of the

latter sustains the former.

WD in its sojourns

The amazing fact

on the liquid surface and

is that for high values of the memory, the sys-

the corresponding equations of motion. Second,

tem shows properties

we briey describe the main characteristics of

paradigmatic of

very similar to those

quantum systems . 18 In this 21

the chaotic potential used in our calculations.

to the quan-

In Sect. 3 we present our numerical results con-

tum pilot-wave theories proposed in the early

cerning WD orbits, generated waves, and a way

days of quantum mechanics by Louis de Broglie

to introduce quantization on them is proposed.

in his double solution conception,

Finally, we sum up the paper

sense, the uid can be compared

elaborated by Bohm. 23

22

and further

the conclusions of our work.

Indeed, several characteristic quantum phenomena have been emulated in WD experiments.

For example, diraction by single

and double slits interference, 24 where the usual


in Sect. 4 with

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2 Theory

where

2.1 Walking droplets

term accounts for the time decay of the ripple.

WD

Notice that Eq. (2) assumes that the bounces

In a typical WD experiment a cylindrical con-

of

tainer lled with silicon oil is bolted to a vi-

ω = 2πf

wavenumber,

A0 . Then, the eective gravity is 2 g + A0 ω sin ωt = g + γ sin ωt, γ being a param-

amplitude

γF , the

uid surface, also introducing at the same time a coupling between the corpuscular droplet and

are able to bounce indenitely without coalesc-

the wave on the surface that it creates.

ing on the surface. In this way, long-lived waves

theory.

the verti-

To study the motion of the walkers we scale

cal and horizontal motions of the droplet can

Eq. (1) in order to make it adimensional, thus

be assumed to be uncoupled, and the trajec-

obtaining

tory of the WD can be studied theoretically in

κ¨ x + x˙ = −∇V (x) − β∇ψ(x),

a stroboscopic fashion by following only the position at the bounces. The corresponding two-

Z

q = (X, Y ),

in which

where now

motion is averaged out, can be

V the β is a mem-

is the scaled mass,

lesser degree, the eect of the dispersing waves

created at the previous bounces in the wave

(1)

D

t, κ

ory parameter, which modulates, to a greater or

¨ + Dq(t) ˙ = −∇U (q(t)) − mg∇ψ(q(t)), mq(t) is the mass of the particle and

(3)

is the adimensional position at

corresponding scaled potential, and

this case reads

m

x(t)

the scaled time

followed using Newton's second law, which in

where

No-

tice the similarity with de Broglie's pilot wave

are created whose ripples are able to impel the

the vertical

which plays

tory of previous bounces on the shape of the liq-

regime, small droplets of the same oil

dimensional dynamics for

(or alternatively the Fara-

λF = 2π/kF ),

in Eq. (2) accounts for the inuence of the his-

However, in the

droplet, making of it a WD. Then,

J0 .

here the same role as the de Broglie's wave21 length in the quantum theory. The integral

oil surface is at rest, despite the fact that it is

γ γF

kF ,

day's wavelength

eter that is easily controlled in the experiment.

being excited by the shaker.

a Bessel function of the rst kind,

This function is characterized by the Faraday's

and an

Below the so-called Faraday's threshold,

the droplet deform the surface with the

shape of

brating electromagnetic shaker, which oscillates vertically with a frequency

qp (s) represents the vector position of the trajectory at time s, and the exponential

eld

is a

ψ

at time

t.

Full details of this adimen29

sionality procedure can be found in Ref.

parameter giving account of the friction. Ob-

We conclude this subsection by noting that

viously, the magnitude of these parameters are

although the actual values of

related to the characteristics of the uid.

lated for any specic liquid used in a given WD

In

q(t) is the position vector of the particle liquid surface at time t, U is the exter-

κ

and

β

are re-

Eq. (1),

experiment through the magnitude of the Fara-

on the

day's threshold,

nal potential acting on the particle (see Sec. 2.2 below), and

−mg∇ψ

γF ,

we will consider them in

the rest of the paper as independent parame-

is the term that accounts

ters in order to be able to explore the full range

for the force exerted by the wave formed on

of dynamical possibilities that they allow.

the liquid surface, as a result of the addition of the dierent dispersing ripples originated at

2.2 External potential

each bounce. The height of this wave at a given point of the surface,

q,

and a given time, t, can

In this work we use an external potential,

be modeled as

ψt (q) =

t −∞

V (q),

which ensures that in the absence of forces due to the vibrating liquid surface, the trajectories

J0 (kF |q(t) − qp (s)|) e−(t−s) ds,

of the walkers are very chaotic. Actually, the quantization of classical chaotic systems is an

(2)

area of much interest, usually known as


quan-


tum chaos , ena

35,36

34

and

many

interesting

phenom-

Page 4 of 14

a family of tiny islands of regularity.

fall within the scope of this area.

value of

A very popular choice for this type of poten-

α1

As the

is increased, the potential becomes

more and more regular.

α1 = 0

tials are the chaotic billiards, such as the Buni-

cases, such as

movitch stadium, which has been often used

completely regular.

or

Indeed, for especial

α 1 = 1,

the behavior is

both in theoretical and experimental studies. In our case, we prefer instead to use a coupled quartic oscillator potential in two dimensions

V (x, y) =

α1 2 2 α2 4 x y + (x + y 4 ) 2 4

for a number of reasons.

(4)

First, it is a contin-

uous, smooth potential easy to be treated by numerical procedures.

Second, this potential

is homogeneous, and then, in the Hamiltonian case, the trajectories exhibit mechanical similarity.

That is,

the classical motion at any

value of the energy can be obtained by scaling the results from

E = 1.

This avoids the

hassles derived from the dependence/evolution of the phase space structure with the energy. Third, using this potential avoids the complications (discontinuities) due to the bouncing of trajectories at hard walls (although in the case of the droplets paths, this is alleviated by the eect of the wave also bouncing at the walls). Fourth, the dynamical behavior of this system is extremely chaotic, and free of the marginally stable (perpendicular to the straight borders and whispering gallery) motions that complicate the dynamics of the Bunimovitch stadium.

Figure 1:

used in dierent studies of classical and quantum

chaos.

37

At

this

point

it

should

be

(a) Contour line corresponding to

E = 1 for the quartic potential (4) with α1 = 1 and α2 = 1/100. (b) C := {x = 0, Px > 0} Poincaré surface of

And fth, this potential has been extensively

re-

marked that, although creating such a poten-

section at the same energy, showing the very

tial in an actual WD experimental setup may

chaotic character of the corresponding Hamil-

not be easy, it is however not impossible either,

tonian dynamics.

since it can be obtained, at least to some extent, by adequately shaping the bottom of the uid

In Fig. 1 we present the quartic potential (4)

container or by using suitably shaped magnetic

for

elds and a ferromagnetic uid.

α1

and

α2 ,

can be obtained by a simple scaling), together

whose values determine the

with the

degree of the chaoticity of the particle dynamics.

For

α1 = 1

and

α2 = 1/100

C := {x = 0, Px > 0}

chaotic character of the corresponding Hamiltonian dynamics, i.e. without considering the

Actually, for a long time it

inuence of the wave created by the WD.

was thought to be totally ergodic (in the limit

α2 → 0)

until Dahlquist and Russberg

38

Poincaré surface

of section at the same energy, showing the very

the corre-

sponding tra jectories present a very high degree of chaoticity.

α2 = 1/100 in the form of conE = 1 (recall that others contours

and

tour plot for

The quartic potential (4) includes two parameters,

α1 = 1

found


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3 Results In this section, we present some results corresponding to the dynamics of WDs when governed by the adimensional Eq. (3) including (unless otherwise stated) the external quartic potential (4). We also analyze the wave originated in the bouncing of the WD, when it sojourns over the liquid surface. All calculations presented here have been calculated for

κ=1

and 8.

3.1 Walking droplets orbits We begin by considering the dynamics of the orbits described by the WD. This dynamics is governed by Eq. (3), where the friction term dissipates WD energy. However, the WD can also gain energy from the vibrating surface at the bounces.

When these two terms balance,

the WD comes to a stable stationary situation with constant or alternatively time-periodic nal energy. This is indeed what happens to all trajectories in the regime dened by the set of parameter values considered in this paper. Some results for

κ = 1

are shown in Fig. 2,

where we rst consider the case in which the external potential is eliminated, i.e.

α1 = α2 = 0

in Eq. (4). In this case, the trajectories maintain the initial direction, and the velocity always converges to the same nal value, regardless of the value of the initial conditions. This is illustrated in panel (a) with three examples for

β = 100.

Figure 2: Dynamics of the walking droplet gov-

Panel (b) shows the dependence 29 of the nal velocity with β , which is given by

1/2 1 vf = √ −1 + 2β − 1 + 4β , 2 β 1/2 β → ∞.

which is proportional to ory limit, i.e. when

erned by Eq. (3) for potential [α1

and no external

in Eq. (4)].

tionary nal velocity for suciently long times,

v 9.7

in the high mem-

in this case, where

β = 100.

(b) Value of the stationary nal velocity as a function of the memory parameter

introducing a chaotic

force exerted on the droplet, is included in the dynamics.

κ = 1

(a) All initial conditions reach the same sta-

(5)

Let us discuss now what happens when the quartic potential (4),

= α2 = 0

Here, the dissipative character

of the dynamics, dictated by Eq. (3), makes that all trajectories are attracted to orbits either along the horizontal or the vertical axis, following after a certain time a straight line.


β.


Page 6 of 14

However, in this case the nal asymptotic portion of the trajectories exhibit a more complex time-dependence on the nal energy, which in our case is periodic. Furthermore, the nal mean total energy, E , approximately increases linearly with the memory parameter β , as E ∼ vf2 ∼ β . In Fig. 3, we illustrate these results using the same three initial conditions of Fig. 2(a). In panel (a), we present the phase space view (x, vx) of the three trajectories, which are quickly attracted to a gure approximately corresponding to a deformed tilted rectangle. In panel (b), we show the corresponding values of the energy, which are seen to exhibit a three-periodic asymptotic behavior. Finally, panel (c) shows the approximately linear dependence of the nal energy with the value of β . We close this section by summarizing two main conclusions, that are relevant for the rest of the discussion in this work. First, in the dynamical regime that we are considering, i.e. κ = 1, the asymptotic dynamics of the WD orbits are governed by an attractor, which always drives them to one axis. Second, in this dynamics the memory parameter β and the WD energy are both related in such a way that the latter increases with the former. The corresponding functional form depends on the characteristics of the external potential, as shown in the examples of Figs. 2 and 3. 3.2 Instant wave vs main wave

Let us now consider the second 'force' appearing in the equation of motion (3) for the WD, which comes from the ripples in the liquid surface induced by the dierent bounces of the WD on it. For this purpose, it is important to distinguish between two dierent surface waves. On the one hand, we have the instant wave at a given moment of time t, which results from the addition of all dispersing ripples originated by the dierent bounces of the WD on the surface. Assuming a Bessel form and in the adimensional version of the equation of motion, this

Figure 3: Same as Fig. 2 when the quartic potential (4) is added in the dynamics of Eq. (3). (a) Phase space view, and (b) time variation of their energies for the same trajectories in Fig. 2(a). (c) Value of the stationary nal energy as a function of the memory parameter β .

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Figure 4: Squared instant (blue) wave for t > 1000 when the droplet is at x = 0 with vx > 0, and corresponding main wave (red) for dierent values of β : (a) 5.05, (b) 89.59, (c) 242.33, and (d) 370.58. are shown in Fig. 4 in blue and red, respectively. As can be seen, the β parameter plays the role here of the excitation energy in quantum systems as the number of nodes, n, i.e. the vibrational excitation, the main wave function (red) depends on it: For β = 5.05 we have that n = 2 (a); for β = 89.59, n = 6 (b); for β = 242.33, n = 8(c); and for β = 370.58, n = 9(d). Moreover, the shape of these main waves looks very similar to that appearing in the standard quantum mechanical harmonic oscillator, 39 being the 'probability' more highly localized at the positions of the two symmetrically located turning points, ±xT P , of the WD orbits. More interesting, it is that the instant wave (blue) tends to the main wave as the value of β increases. To gauge this result one should take into account that the instant wave is somehow only half of the main one, since it is computed at x = 0 with vx > 0, and then the intensity into the x > 0 region has not had a chance to develop. Likewise, as can be seen by examining the dierent panels in Fig. 4, we observe that as

wave simply reads [cf. Eq. (2)] ψt (x) =

t −∞

J0 (|x − xp (s)|) e−(t−s) ds.

(6)

Notice that this term, is modulated by the memory parameter β in the dynamical Eq. (3), and also implicitly depends on it through the term xp (s). On the other hand, we have the main (averaged) wave, which is given by

ψ(x) =

∞ −∞

J0 (|x − xp (s)|) ds,

(7)

where the exponential term responsible for the wave dispersion has been eliminated. Let us remark that the wave dened in this way needs to be properly normalized. Some results for these two waves, computed for a representative trajectory (recall that, as previously discussed, that our results are independent of the initial conditions) propagated for dierent values of the memory parameter β


7


the memory parameter

β

Page 8 of 14

increases, more and

more peaks of the main wave (red) are well reproduced by the ripples in the instant wave (blue).

In this way, while the agreement be-

tween both waves in panel (a) is very poor, in panel (b) both waves agree well in the vicinity of the left turning point at

xT P −20

(although

the intensities are still very dierent).

How-

ever, in panel (c) the agreement is much better around that point, also existing a good agreement at the maximum next to it. And nally, in panel (d) we observe good agreement (again leaving aside the intensity) with the ve leftmost maxima of the main wave. We could continue increasing the memory of the wave created along the WD trajectory, i.e.

β,

and get better

and better agreement between instant and the main waves. In this way, we can conclude that in the high memory limit the instant wave tends very accurately to reproduce the shape of the main wave, obviously putting it aside the fact that the latter is symmetric with respect to the origin

x = 0.

This limiting result is reasonable,

and can be easily understood if one takes into account that when the memory is high, the ripples originated at each consecutive bounce do not have the opportunity to decay, and thus

Figure 6:

they eectively contribute to the main wave.

(a) Squared main wave given by

Eq. (6) as a function of the position

This fact enables the experimental verication

x, for κ = 8 β:

and dierent values of the memory parameter

of the quantization phenomenon reported here.

250 (b), 160 (c), and 90 (c), respectively, playing here the role of the energy in a regular mechanical system. When the corresponding turn-

3.3 Quantization

ing points, whose position depends on

Despite the similarities of the main wave shapes

terference takes place, this indicating the posi-

obtained in the previous subsection and the true

tion of a quantization condition. (b)-(d) Three

quantum case, it cannot be forgotten that the

'quantized states' corresponding to

WD problem is macroscopic and then classical.

in panel (a).

tized, and then there is no immediate way to claim the existence of a physically based quantization rule in it. However, driven by the striking similarity described above, we will further investigate this issue in this part of the paper.

ansatz

that is , at

the same time, coherent with the nature of the problem, and able to satisfactorily explain our numerical ndings.

n = 8, 7,

and 6, also shown at the colored horizontal lines

Here the energy is continuous and it is not quan-

The idea is to make an

β , are ad-

equately separated, a maximal constructive in-

For this purpose, we will


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Figure 5: Quantization of the walking droplet main wave. It is shown here how a 'quantized' wave with a clear, regular nodal pattern is obtained [panel (b)] by adding two Bessel functions centered at the turning points, when they are adequately separated [panel (a)]. As can be easily ascertain this 'quantized state' corresponds to n = 8. The positions of the corresponding turning points, at x1,2 = ±13.32 are marked with vertical dashed lines. In panel (d) we show how a non-quantized situation takes place when we move the position of the turning points to x1,2 = ±11.00 [panel (c)].


9


Page 10 of 14

start by considering that since the trajectories

limit are presented in Fig. 6, where we show in

spend a great deal of time at the turning points,

panel (a) the location of the turning points, as a

where the velocity becomes zero in the change

function of the position

of sign, these regions will be the most important

ing (squared) main waves obtained for

ones in the main wave construction by the suc-

at dierent values of

cessive bounces of the droplet on the liquid sur-

the energy). As can be seen at specic value

face. In fact, as the problem is symmetric with

of

respect to the origin, two such waves should

to a maximal constructive interference. Three

exist, one at each of the symmetrical turning

of these cases have been marked with horizon-

points. Moreover, these two waves should have

tal cyan, pink, and green lines, respectively.

a constructive interference, in order to be able

Furthermore, the corresponding squared main

to construct a wave that looks like some sort

waves are also presented, separately for the sake

of 'quantum-like' wavefunction. Obviously this

of clarity, in the panels (d), (c), and (b), re-

situation will only happened for some particu-

spectively. In the three of them we see a clear,

lar values of the memory parameter

In this

regular nodal pattern corresponding to the con-

way, a reasonable condition for the quantization

secutive 'vibrational quantum numbers' (from

of

bottom to top)

β

β.

can be obtained.

β

β

x,

and the correspond-

κ = 8

(which plays the role of

clear quantized 'states' are obtained, due

n = 6, 7,

and 8, with a great

This argument is illustrated in the results pre-

accumulation of intensity/'probability' in the

sented in Fig. 5, where panels (a) and (c) repre-

turning points, similarly to what happens in the

sent two squared Bessel functions, J02 , centered

at each of the turning points,

xT P = x1,2

semiclassical high ics.

of

39

n limit of Quantum Mechan-

the WD trajectories associated to two dierent values of

β.

The corresponding normalized re-

4 Summary and discussion

sults obtained by adding these two functions are also given in panels (b) and (d), respec-

Summarizing, in this paper we have studied

tively. As can be seen, in the rst case, where

xT P = ±13.32,

the dynamics of a WD under the inuence of

[see panel (b)], the two par-

a two-dimensional coupled quartic potential in

tial wave are in phase, and then giving rise to

the high memory limit. We have demonstrated

a constructive interference, which results in a

the emergence of nodal patterns in the surface

nal wave which resembles a quantum state.

wave that it creates, as a result of a coherent

On the other hand, for the second value of

xT P = ±11.00

interference. The tuning parameter in this case

[see panel (d)], the interference

is destructive at intermediate values of

x,

is the memory parameter

this

not appearing a regular nodal pattern that can

simple

be assigned to any quantized situation.

to explain where the quantization

understand the similarities found between WD

showed the dependence of the nal energy of

experiments and the quantum theory, in such a

can be used to obtain suit-

way that the former can be an adequate play-

able quantization conditions for the main 'pilot'

ground to study certain aspects of the later. For

wave in the WD problem.

example, to deepen in the understanding of the

The idea is quite simple. Since the WD en-

unnished de Broglie's double solution theory,

ergy, and thus the separation between turning points, increases with the memory constant

ansatz

We believe that this work opens a door to

our results discussed in Subsect. 3.1, where we

β,

which plays the

condition of the wave eld is fullled.

We conclude our study by showing next how

the orbits with

β,

role of an energy. Likewise, we have proposed a

specially as revived by Bohm's work. A more

β,

quantitative analisys of the quantization con-

we can vary continuously this parameter to

dition reported in this paper will be reported

nd the dierent consecutive excited vibra-

elsewhere. 40

tional 'states' at the points where the constructive interference between left and right waves is maximum. The results for the high memory


Page 11 of 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Acknowledgements


This research has been partially funded by the Ministerio de Economía y Competitividad (Spain) under Contracts No. MTM201563914-P and ICMAT Severo Ochoa Contract No. SEV-2015-0554, and from the People Programme (Marie Curie Actions) of the European Union's Horizon 2020 research and innovation programme under Grant No. 734557. F.R. gratefully acknowledges the nancial support of the Programa Propio of the Universidad Politécnica de Madrid.

(9) (10)

(11)

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A Quantization Scheme for the Experiments with ''Walking Droplets

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