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single Gaussian slit. As a result we get a Gaus- sian wave packet equivalent to that used in the articles [8–10]. Sect. III introduces superposition of the wave packets emitting from N slits. Here we describe also interference from N -slit grating in the near-field zone and diffraction in the far- field zone. Sect. IV describes the quantum HJ- equation and the continuity equation, that lead to the guidance equation for finding Bohmian trajectories. Simulation of the fullerene molec- ular interference stated in [11] is considered in Sect. V. Sect. VI discusses virtual trajectories, virtual sources of spherical waves, and much more. Almost all virtual trajectories cancel each other. Remaining trajectories are Bohmian ones. Sect. VII, concluding section, discusses subtle problems of virtual trajectories emerging in vac- uum, due to which an optimal paths through N -slit grating are disclosed. 

Before we will analyze interference on the N - slit grating, let us consider a particle passing through a single slit. The problem has been con- sidered in detail in [7]. We will study migration of the free particle along axis x, see Fig. 2. Its Lagrangian is as follows 

Here m is mass of the particle and ˙ x is its ve- locity. By translating a particle’s position on a small value δx = ( x$_{b}$ − x$_{a}$ ) ≪ 1, being performed for a small time δt = ( t$_{b}$ − t$_{a}$ ) ≪ 1, we find that a weight factor of such a translation has the fol- lowing form 

Pay attention on the following situation: since argument of the exponent contains multiplica- tion of the Lagrangian L by δt , putting into operation the Lagrangian (6) we obtain result ( x$_{b}$ − x$_{a}$ ) $^{2}$/ ( t$_{b}$ − t$_{a}$ ). Further we will see, that the weight factor (7) plays an important role. Now, by means of such small increments let us trace 

## II. GAUSSIAN SLIT 

<img src='content_image/383628.jpg'>

<img src='content_image/383629.jpg'>

passing the particle from a source through the slit and farther, Fig. 2. 

<img src='content_image/383630.jpg'>

We suppose, that at the time t = 0 the particle leaves the origin of coordinates x = 0. Let we know, that after a time T the particle enters to the vicinity x$_{0}$ ± b of a point x$_{0}$ , see Fig. 2. The question is: what is the probability to disclose the particle after a time τ at a point x$_{1}$ remote from the point x$_{0}$ at a distance Δ x = ( x$_{1}$ − x$_{0}$ )? Let the particle outgoing from the point x = 0 at the time t = 0 passes a slit between the points x$_{0}$ − b and x$_{0}$ + b at the time t = T . Let us com- pute the probability of discovering the particle at some point x$_{1}$ after the time τ , i.e., at t = T + τ . Because of existence of an opaque screen, the problem can have no solution if we apply laws only for moving the free particles along direct paths. The particle should pass through the slit in order to reach the point x$_{1}$ . In this connec- tion, we partition the problem into two parts. Each part relates to movement of the free par- ticle. In the first part we consider the particle moving from the point x = 0 at the time t = 0 to a point x = x$_{0}$ + ξ , reaching it at time t = T , where | ξ | ≤ b . The second part deals with the particle passing through the point x = x$_{0}$ + ξ at the time t = T and moving to the point x$_{1}$ , reaching it at the time t = T + τ . A full proba- bility amplitude is equal to integral convolution of two kernels, each describing movement of the free particle: 

