УДК 530.145.65
MACROSCOPIC MODEL OF QUANTUM MECHANICS
Atknin I.I.
Language supervisor: I. V. Alekseenko
Siberian Federal University
We know, that the idea of the wave - particle duality was originated when Isaac
Newton and Christian Huygens suggested conflicting theories. The Newton applied that light
consist of corpuscles in contrast to the Huygens, who determined the wave theory of light.
Concepts of light theory have changed a lot since the last century. Today light particles are
known as photons. Reviewing the investigations of many scientists, we conclude, that current
scientific theory holds such microparticles as photons, electrons and even molecules, which
are able to be particles and waves at the same time.
According to this idea, de Broglie proposed the pilot wave theory and later it was
developed by David Bohm. Due to this theory, particles are guided by waves. The pilot wave theory restores determinism and realism in quantum mechanics, but the physical nature
of the guiding wave field remains unclear. So, at that time the theory of de Broglie - Bohm
was overtaken by the interpretation of probabilistic approach, because there was no
macroscopic pilot wave analog to draw upon. Now there is one. Yves Couder and colleagues
found a macroscopic pilot system that demonstrates properties, which previously were
peculiar to the microscopic world
When a bath with a fluid is vibrated in the periodic mode, we are able to find the
critical vibration frequency, above which the flat hydrostatic surface becomes unstable to a
regular pattern of standing millimetric Faraday waves (fig. 1). The critical value of
acceleration depends on the depth of liquid, surface tension and viscosity.
Fig. 1. Snapshot of Faraday waves on a the fluid surface induced
by driving the bath vertically in a periodic mode
Here is investigated the situation when a droplet of the same fluid is placed on
the surface (fig. 2) of the vertically oscillating (1) bath [1]. The diameter of the droplet is
about 1 mm.
g (t) = g m cos(wt)
where w = 2p f0 ; g m = -Aw2 - amplitude of acceleration.
(1)
Fig. 2. Small cell containing a liquid is subjected
to a vertical sinusoidal acceleration (1)
Liquid droplet usually disappears rapidly (a few tenth of a second) when it falls on
the same liquid surface. The forcing acceleration is made to exceed g (the gravitational
acceleration) in order to avoid this coalescence. The collide period of time becomes shorter
than the time, which is required to break the air film between the droplet and fluid surface.
Thus, before the distance becomes critical to merger by van der Waals forces, the droplet lifts
off again and can be maintained for an unlimited time in a kind of “levitation” on the surface
[2].
The experiment includes a droplet, placed on a bath of silicone oil in the
vibrating regime between the Faraday threshold and threshold of bouncing. Depending on the
droplet size, it is possible to achieve the result, when a droplet could remain bouncing
infinitely. Its bouncing emits damped surface waves (fig. 3).
Fig. 3. Three photographs demonstrate a droplet, bouncing on the
surface of a bath, when the same liquid oscillating vertically.
These waves are able to be a mediator among multiple bouncers on the surface.
When pair of neighbouring walkers stuck together, they may be locked into orbit around each
other (fig. 4 Left). This set of equal - sized droplets can form structure corresponding to
Archimedean tilings (fig 4 right) [3].
Fig 4. Left: A pair of walkers lock into collective orbit.
Right: Side-view snapshot of a hexagonal lattice
aggregate of bouncing droplets.
The waves, which are generated by the bouncer, create an irregularity of the surface
(waves). When the droplet lands on the sloping surface, it rebounds out of the vertical
bouncing state and slides along the horizontal plane [4]. As a result, the droplet is piloted by
its wave field (fig. 5). We should, also, notice that this motion depends on the depth of the
liquid, the bigger the height of these waves, which are generated, more readily the droplet
moves.
Fig. 5. The droplet, touching the sloping surface of the wave, generated by the
previous collisions.
If a metal strip is put on the bottom of the bath so that the liquid layer is above the
strip (subsurface barrier), creating the equivalent of the barrier for the moving bouncer
(walker). It replicates the single - particle single- and double-slit experiments in macroscopic
realm (fig. 6). The experiment demonstrates the way the droplet is launched into the direction
of the slit.
Fig. 6. A walker passes through the slit.
The complex pilot wave field when the droplet is in the slit region.
When the droplet gets closer to the slit region, the wave propagation is
distorted by interference between the reflected and direct waves, as depicted in fig. 6. The
statistics of the deviations is investigated in fig. 7 in terms of many independent realization,
where it duplicates the classical diffraction pattern of microscopic particle proved by Taylor
in 1909.
Fig. 7. The droplet trajectory passes through the slit and the histogram for 125
successive events.
In the double - slit experiment while the droplet passes through the first slit, the
generated wave has time to pass through the both slits, where the wave velocity is higher than
the velocity of the moving droplet .These waves interact after the slits, creating the
interference.
Quantum tunneling is a quantum mechanic phenomenon when a particle tunnels
through a barrier that it classically could not surmount. Exploring a droplet interaction with a
barrier, we can see a macroscopic model of quantum tunneling. In the macroscopic
experiment a bouncing droplet is confined by walls around, above which the droplet could
jump over, but not walk (fig. 8). However, the probability of tunneling decreases
exponentially with increasing width of the barrier [5].
Fig. 8. The recorded path of the droplet inside the square trap, e – barrier width (2,5
mm), P – probability of escape. V – velocity of motion.
a)
V=11,8 mm/s, P≈10% ; b) V=13,2 mm/s, P≈30%;
In conclusion, we should notice that scientists are fully confident, that quantum and
classical realms are totally different. Due to the experiment with vibrating droplet, which
demonstrates the similarity of the wave - particle duality between quantum and classical
scales. The system remains deterministic in contrast to quantum world. The system is two dimensional, but the system is received energy by the third dimension. Here may be the clue
for solving the problem of probabilistic understanding of current quantum theory. Probably
the solving is hidden in one more dimension whose nature remains unclear. This system
provides a great number of questions which are difficult or even impossible to answer.
REFERENCES
1. Particle–wave association on a fluid interface // J. Fluid Mech. (2006), vol. 554,
pp. 85–108.
2. Y. Couder, E. Fort, C.-H. Gautier, and A. Boudaoud // From Bouncing to
Floating: Noncoalescence of Drops on a Fluid Bath, PRL 94, 177801 (2005)
3. Eddi A, Decelle A, Fort E, Couder Y (2009) // Archimedean lattices in the bound
states of wave interacting particles. Europhys Lett 87:56002-p1–56002-p6.
4. Protiere S., Boudaoud A. and Couder Y., J. Fluid Mech., 554 (2006) 85.
5. A. Eddi, E. Fort, F. Moisy, and Y. Couder // Unpredictable Tunneling of a
Classical Wave-Particle Association.