content_image/410992.jpg
and Eqs. (32), where v$_{z}$ = h/λm$_{C}$$_{60}$ ≈ 110 m/s. Fig. 15 shows contour plot of the probability den- sity distribution p ( x, z ) projected on the plane ( x, z ). The quantum wave-nature of the fullerene molecular flow is drawn by color gray. The particle-nature presented by the Bohmian tra- jectories is shown in the upper part of the fig- ure. Predominantly, they are drawn by color violet. Some trajectories colored in blue are drawn against this background. Due to such a color representation flows of fullerene molecules through the near-field region and further are eas- ily visualized. Fullerene molecular flows are seen to have undulatory motions. 

Obviously, there are no possibilities to observe real movements of fullerene molecules through the slit grating and further [24]. Any perturba- tion of the molecule destroys its movement and, consequently, it destroys the interference pat- tern. In best case we can observe the fullerene molecular flows at crossing some kind of detect- ing plates (fixed at z$_{T}$/ 2 , z$_{T}$, 3 z$_{T}$/ 2 , · · · , for in- stance), as shown in Fig. 12(a). 

## VI. VARIATIONAL CALCULATIONS 

What could force the particle to carry out such undulatory and zigzag behaviors, as shown in the figures above? Possible answer can be as fol- lows: a set of ordered slits in the screen poses itself as a quantum object that provides a po- larization of the vacuum in the near-field region. The polarization, in turn, induces formation of a virtual particle escort around of a flying real par- ticle through. The escort ”informs” the particle about the environment. Such an insight consid- ering behavior of the particle with the point of view of Feynman path integrals, can be more productive, than fantastic ideas about splitting particles passing through the slit grating and their confluence as soon as the slits are left far behind. 

We need in this connection to continue discus- sion of the problems, relating to the virtual and real trajectories. Let us consider a number of variational procedures applied to the complex- valued function ψ ( ⃗ q, ⃗ p ; t ), wave function. It per- 

mits to see which variational scheme ends up the classical equations of movement and what scheme leads to the quantum mechanical equa- tions. The problem, in fact, is to retrace how the virtual trajectories relate to the real trajectories, Bohmian trajectories. 

Let the complex-valued function be as follows 

<img src='content_image/410992.jpg'>

Let us demand that this function would retain a constant value along a path from the initial time t = t$_{0}$ to final t = t$_{1}$ . In other words, varia- tion of this function along this time interval has to vanish, i.e., δψ = 0. Applying this varia- tion to the expression (34), we disclose, that the complex-valued function ends by two equations, separately for real and imaginary parts. Each equation should vanish. 

Real part of the above equation leads imme- diately to the continuity equation (4) for the probability density ρ . Imaginary part reduces to variation of the action S , see equation (1), along paths from the initial time t$_{0}$ to the final time t$_{1}$ . After a series of mathematical transfor- mations [1] we disclose, that the particle moves along an optimal path that is described by the Hamilton-Jacobi equation (2). 

We observe in the classical case, that the parti- cles move along the classical trajectories submit- ting to the principle of least action. By moving along the optimal paths, ensemble of the par- ticles resembles a ”cloud” having a density dis- tribution ρ . Evolution of the cloud obeys the continuity equation for the density distribution. 

In contrast to the previous searching of a single trajectory connecting the initial and final points, here all trajectories connecting these points are to be considered. They pass through all inter- mediate points belonging to a conditional set 

## A. Classical domain 

## B. Quantum domain 

