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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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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
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The Journal of Physical Chemistry
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
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quan-
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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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The Journal of Physical Chemistry
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.
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β.
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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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The Journal of Physical Chemistry
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
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the memory parameter
β
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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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The Journal of Physical Chemistry
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)].
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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
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Acknowledgements
The Journal of Physical Chemistry
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)
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