1. Introduction
One of the crises of contemporary mathematics belongs in part to the subject of the infinite and infinitesimals [1]. It originates from the barest necessity to develop a rigorous language for description of observable physical phenomena. It was a time when foundations of integral and differential calculi were developed. A theoretical foundation for facilitation of understanding of classical mechanics is provided by the concepts of absolute time and space originally formulated by Sir Isaac Newton [2]. Space is distinct from body. And time passes uniformly without regard to whether anything happens in the world. For this reason Newton spoke of absolute space and absolute time as of a "container" for all possible objects and events. The space-time container is absolutely empty until prescribed metric and a reference frame are introduced. Infinitesimals are main tools of differential calculus [3, 4] within chosen reference frames.
Infinitesimal increment being a cornerstone of theoretical physics has one receptee default belief, that increment
In light of classical views Newton maintained the theory that light was made up of tiny particles, corpuscles. They spread through space in accordance with law of classical mechanics. Christian Huygens (a contemporary of Newton), believed that light was made up of waves vibrating up and down perpendicularly to direction of the light propagation. It comes into contradiction with Newtonian idea about corpuscular nature of light. Huygens was a proclaimer of wave mechanics as opposite to classical mechanics [5].
We abstain from allusion to physical vacuum but expand Huygensian idea to its logical completion. Let us imagine that all Newtonian absolute space is not empty but is populated everywhere densely by Huygensian vibrators. The vibrators are silent at absence of wave propagating through. But as soon as a wave front reaches some surface all vibrators on this surface begin to radiate at a frequency resonant with that of incident wave. From here it follows, that in each point of the space there are vibrators with different frequencies ranging from zero frequency up to infinite. All are silent in absence of an external wave perturbance. Thus, the infinitesimal volume
Let us return to our days. One believes that besides matter and physical fields there is nothing more in the universe. Elementary particles are only a building material of "eternal and indestructible" substance of the cosmos. However we should avow that all observed matter and physical fields, are not the basis of the world, but they are only a small part of the total quantum reality. Physical vacuum in this picture constitutes a basic part of this reality. In particular, modern conception of the physical vacuum covers Huygens's idea perfectly. All space is fully populated by virtual particle-antiparticle pairs situated on a ground level. Such a particle-antiparticle pair has zero mass, zero charge, and zero magnetic moment. Famous Dirac’ sea (Dirac postulated that the entire universe is entirely filled by particles with negative energy) is boundless space of electron-positron pairs populated everywhere densely - each quantum state possessing a positive energy is accompanied by a corresponding state with negative energy. Electron has positive mass and positron has the same mass but negative; electron has negative charge and positron has the same positive charge; when electron and positron dance in pair theirs magnetic moments have opposite orientations, so magnetic moment of the pair is zero.
Let electron and positron of a virtual pair rotate about mass center of this pair. Rotation of the pairs happens on a Bohr orbit. Energy level of the first Bohr orbit, for example,
A short outline given above is a basis for understanding of interference effects to be described below.
Ones suppose that random fluctuations of electron-positron pairs take place always. What is more, these fluctuations are induced by other pairs and by particles traveling through this random conglomerate. Edward Nelson has described mathematical models [6, 7] representing the above random fluctuations as Brownian motions of particles that are subjected by random impacts from particles populating aether (Nelson's title of a lower environment). The model is viewed as a Markov process
loaded by a Wiener term
Nelson' vision that aether fluctuations look as Brownian movements of subparticles with
The article consists of five sections. Sec. 2 introduces a general conception of the path integral that describes transitions along paths both of classical and quantum particles. Here we fulfill expansions in the Taylor series of terms presented in the path integral. Depending on type of presented parameters we disclose either the Schrödinger equation or diffusion-drift equation containing extra term, osmotic diffusion. In the end of the section we compute passing nanoparticles through
2. Generalized path integral
Let many classical particles occupy a volume
The problem is to find transition probabilities that describe transition of the particle ensemble from one statistical state to another. These transient probabilities can be found through solution of the integral Einstein-Smoluchowski equation [9]. This equation in mathematical physics is known as Chapman-Kolmogorov [22-24] equation.. This equation looks as follows
This equation describes a Markovian process without memory. That is, only previous state bears information for the next state. Integration here is fulfilled over all a working scene R, encompassing finite space volume. Infinitesimal volume
Next we shall slightly modify approach to this problem. Essential difference from the classical probability theory is that instead of the probabilities we shall deal with probability amplitudes. The transition amplitudes can contain also imaginary terms. They bear information about phase shifts accumulated along paths. In that way, a transition from an initial state
Here function
where denominator Γ under exponent is a complex-valued quantity, i.e., Γ = β+
Factor 2 at the first term is conditioned by the fact that the kernel
Next let us imagine that particles pass through a path length one by one. That is, they do not collide with each other along the path length. The particles are complex objects, however. They are nanoparticles. Fullerene molecule, for example, contains 60 carbon atoms, Fig. 1. Conditionally we can think that the atoms are connected with each other by springs simulating elastic vibrations. In this view the Lagrangian
Here
Here we admit that
Let us consider the second row in Eq. (7). First of all we note that the term
The path integral (3) contains functions depending only on coordinates
2.1. Expansion of the path integral
The next step is to expand terms, ingoing into the integral (3), into Taylor series. The wave function written on the left is expanded up to the first term
As for the terms under the integral, here we preliminarily make some transformations. We define a small increment
The Lagrangian (7) is rewritten, in such a case, in the following view
Here
The potential energy
Taking into account the expressions (8)-(11) and substituting theirs into Eq. (3) we get
First, we consider terms enveloped by braces (a) and (b): (a) here displacement
In the light of the above observation we rewrite Eq. (12) as follows
Here we have expanded preliminarily exponents to the Taylor series up to the first term of the expansion. Exception relates to the term exp{ -
To derive outcomes of integration of terms containing factors
In the light of this observation let us now solve integral (13) accurate to terms containing
We note that the term
The parameter
is seen to be as a complex-valued diffusion coefficient consisting of real and imaginary parts.
2.1.1. Temperature is zero
Let
It is the Schrödinger equation.
2.1.2. Temperature is not zero
Let
Here
is the diffusion coefficient. The coefficient has dimensionality of [length2 time-1]. It is a factor of proportionality representing amount of substance diffusing across a unit area in a unit time - concentration gradient in unit time.
We can see that Eq. (20) deals with the amplitude function
Extra term
is an osmotic velocity required of the particle to counteract osmotic effects [6]. Namely, imagine a suspension of many Brownian particles within a physical volume acted on by an external, virtual in general, force. This force is balanced by an osmotic pressure force of the suspension [6]:
From here it is seen, that the osmotic pressure force arises always when density difference exists and especially when the density tends to zero. And vice-versa, the force disappears in extra-dense media with spatially homogeneous distribution of particles. As states the second law of thermodynamics spontaneous processes happen with increasing entropy. The osmosis evolves spontaneously because it leads to increase of disorder, i.e., with increase of entropy. When the entropy gradient becomes zero the system achieves equilibrium, osmotic forces vanish.
Due to appearance of the term
Reduction to PDEs, Eq. (19) and Eq. (22), was done with aim to show that the both quantum and classical realms adjoin with each other much more closely, than it could seem with the first glance. Further we shall return to the integral path paradigm [8] and calculate patterns arising after passing particles through gratings. We shall combine quantum and classical realms by introducing the complex-valued parameter Γ = 2
2.2. Paths through N -slit grating
Computation of a passing particle through a grating is based on the path integral technique [8]. We begin with writing the path integral that describes passing the particle through a slit made in an opaque screen that is situated perpendicularly to axis
We believe, that before the screen and after it, the particle (fullerene molecule, for example) moves as a free particle. Its Lagrangian, rewritten from Eq. (17), describes its deflection from a straight path in the following form
The first term relates to movement of the center of mass of the molecule. So that
The particles flying to the grating slit along a ray α, Fig. 4, pass through the slit within a range from
Integral kernel (propagator) for the particle freely flying is as follows [8, 26]
Here
2.2.1. Passing of a particle through slit
By substituting the kernel (28) into the integral (27) we obtain the following detailed form
The integral is computed within a finite interval [-
Here parameter
We do not write parameters
2.2.2. Definition of new working parameters
First we replace the flight times
where
There is, however, one more parameter of time that is represented in definition of the coefficient Γ= 2
Emergence of the term 2
Let us divide the parameter Γ by {
Here
The length
3. Wave function behind the grating
Wave function from one slit after integration over
Here argument of ψ-function contains apart
In order to simplify records here we omit subscript 1 at
Let us consider that an opaque screen contains
Probability density in the vicinity of the observation point (
Calculation of the wave function (37) is fulfilled for the grating containing
which is a convenient natural length at representation of interference patterns. Since we restrict themselves by finite
Evaluation of sizes of the interference pattern is given by ratio of the Talbot length to a length of the slit grating. In our case the Talbot length is
Zigzag curves, drawn in the upper part of Fig. 5 by dark blue color, show Bohmian trajectories that start from the slit No. 15. One can see that particles prefer to move between nodes having positive interference effect and avoid empty lacunas. However the above we noted, that the ratio of the Talbot length to the length of the grating is about 6250 >> 1. It means that really the Bohmian trajectories look almost as straight lines slightly divergent apart. Zigzag-like behavior of the trajectories is almost invisible. Such an almost feebly marked zigzag-like behavior may be induced by fluctuations of virtual particles escorting the real particle.
As soon as we add the term
We may think that the technical vacuum can be not perfect. It causes additional scattering of particles on residual gases. Because of this additional scattering the interference pattern can be destroyed entirely, as shown in Fig. 8.
Now let us draw dependence of the probability density
Fig. 9 shows three characteristic patterns of the interference fringes. In Fig. 9(a) almost ideal interference fringes are shown obtained at
Disappearance of interference fringes is numerically evaluated by calculating a characteristic called visibility [14, 34]. The fringe visibility [27] is represented as a ratio of difference between maximal and minimal intensities of the fringes to their sum:
Evaluation of
4. Bohmian trajectories and variance of momenta and positions along paths
Here we repeat computations of David Bohm [35] which lead to the Hamiltoton-Jacobi equation loaded by the quantum potential and, as consequence, to finding Bohmian trajectories. But instead of the Schrödinger equation we choose Eq. (17) that contains complex-valued parameter Γ=β+i
Here
Further we apply polar representation of the wave function,
Firstly, we can see that at
Terms enveloped by braces (a), (b), and (c) are the kinetic energy of the particle, the quantum potential, and the right part is a kernel of the continuity equation (45), respectively. In particular, the term My attention to the Madelung' article was drawn by Prof. M. Berry.
Momentum of the particle reads
where
positions of the particle in each current time beginning from the grating' slits up to a detector is calculated by the following formula
Here
In case of
Here we take into consideration that in the first equation we may replace
Here
4.1. Dispersion of trajectories and the uncertainty principle
As for the Bohmian trajectories there is a problem concerning their possible existence. As follows from Eqs. (46) and (48) in each moment of time there are definite values of the momentum and the coordinate of a particle moving along the Bohmian trajectory. This statement enters in conflict with the uncertainty principle.
Here we try to retrace emergence of the uncertainty principle stemming from standard probability-theoretical computations of expectation value and variance of a particle momentum. We adopt a wave function in the polar representation
where
The velocity V g is seen to be complex-valued. Here
It is instructive to compare this velocity with the classical osmotic velocity given in Eq. (23). As can see the osmotic velocity stems from gradient of entropy that evaluates degree of order and disorder on a quantum level, likely of vacuum fluctuations.
Real part of Eq. (54) gives the current velocity
Let us now calculate variance of the velocity V g. This computation reads
Terms over bracket (d) kill each other as follows from Eq. (54). It is reasonable in the perspective to multiply Var(V g) by
So, this expression has a dimensionality of energy. The first term to be computed represents the following result
Here the term enveloped by bracket (a) is a kinetic energy of the particle, the term enveloped by bracket (b) with negative sign added is the quantum potential
As for the second term in Eq. (57) we have
One can see that the variance consists of real and imaginary parts. Observe that the right side is represented through square of gradient of the complexified action [28], namely
The first term in this expression represents kinetic energy,
This frequency multiplied by
Let we have two Bohmian trajectories. Along one trajectory we have
One can suppose that emergence of the second trajectory was conditioned by a perturbation of the particle moving along the first trajectory. If it is so, then emergence of the second trajectory stems from an operation of measurement of some parameters of the particle. One can think that duration of the measurement is about
Now let us return to Eq. (48) and rewrite it in the following view
The initial Bohmian trajectory is marked here by subscript 1. Observe that
Here we take into account
5. Concluding remarks
Each nanoparticle incident on a slit grating passes only through a single slit. Its path runs along a Bohmian trajectory which is represented as an optimal path for the nanoparticle migrating from a source to a detector. Unfortunately, the Bohmian trajectory can not be observable since a serious obstacle for the observation comes from the uncertainty principle. In other words, an attempt to measure any attribute of the nanoparticle, be it position or orientation, i.e., the particle momentum, leads to destroying information relating to future history of the nanoparticle. What is more, any collision of the nanoparticle with a foreign particle destroys the Bohmian trajectory which could give a real contribution to the interference pattern. It relates closely with quality of vacuum. In the case of a bad vacuum such collisions will occur frequently. They lead to destruction of the Bohmian trajectories. Actually, they degenerate to Brownian trajectories.
Excellent article [21] of Couder & Fort with droplets gives, however, a clear hint of what happens when the nanoparticle passes through a single grating slit. In the light of this hint we may admit that the particle “bouncing at moving through vacuum” generates a wave at each bounce. So, a holistic quantum mechanical object is the particle + wave. Here the wave to be generated by the particle plays a role of the pilot-wave first formulated by Lui de Broglie and later developed by Bohm [37]. It is interesting to note in this context, that the pilot-waves have many common with Huygens waves [5].
A particle passing through vacuum generates waves with wavelength that is inversely proportional to its momentum (it follows from the de Broglie formula,
Now we may suppose that the subcritical Faraday ripples on the silicon oil surface simulate vacuum fluctuations. Consequently, the vacuum fluctuations change their own pattern near the slit grating depending on amount of slits and distance between them. We may imagine that the particle passing through vacuum (bouncing through, Fig. 11) initiates waves which interfere with the vacuum fluctuations. As a result of such an interference the particle moves along an optimal path - along the Bohmian trajectory. Mathematically the bounce is imitated by an exponential term exp{i
In conclusion it would be like to remember remarkable reflection of Paul Dirac. In 1933 Paul Dirac drew attention to a special role of the action S in quantum mechanics [38] - it can exhibit itself in expressions through exp{iS/ћ}. In 1945 he emphasized once again, that the classical and quantum mechanics have many general points of crossing [39]. In particular, he had written in this article: "We can use the formal probability to set up a quantum picture rather close to the classical picture in which the coordinates q of a dynamical system have definite values at any time. We take a number of times t1, t2, t3, … following closely one after another and set up the formal probability for the q's at each of these times lying within specified small ranges, this being permissible since the q's at any time all commutate. We then get a formal probability for the trajectory of the system in quantum mechanics lying within certain limits. This enables us to speak of some trajectories being improbable and others being likely."
Acknowledgement
Author thanks Miss Pipa (administrator of Quantum Portal) for preparing programs that permitted to calculate and prepare Figures 5 to 8. Author thanks also O. A. Bykovsky for taking my attention to a single-particle interference observed for macroscopic objects by Couder and Fort and V. Lozovskiy for some remarks relating to the article.
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Notes
- My attention to the Madelung' article was drawn by Prof. M. Berry.