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of the surface on the value of 2 ε , thereupon on 3 ε , and so forth. Until all the space will be filled with such secants. Normals drawn from P to P $^{′}$, thereupon from P ′ to P $^{′′}$, and so forth, dis- close possible trajectories of a mechanical sys- tem, since |∇ S | = ε/δ represents a value of the gradient of S . This relation can be expressed in the vector form 

So far as the momentum − → p = m − → v has a direc- tion tangent to the trajectory, then the following statement is true [1]: trajectory of a moving point is perpendicular to the surface S = const. Fig. 1 shows possible trajectories by dotted curves in- tersecting the surfaces S perpendicularly. 

It is appropriate to mention here the Liouville theorem, that adds to the conservation law of energy one more a conservation law. Meaning of the law is that a trajectory density is con- served independently of deformations of the sur- face that encloses these trajectories. Mathemat- ically, this law is expressed in a form of the con- tinuity equation 

Here ρ is the density of moving mechanical points with the velocity − → v = − → p /m . 

Thus, two equations, the HJ-equation and the continuity equation, give mathematical descrip- tion of ensemble of moving points undergoing no noise. Draw attention here, that the conti- nuity equation depends on solutions of the HJ- equation via the term − → v = ∇ S/m . Whereas, the HJ-equation does not undergo dependence on solutions of the continuity equation. This is essential moment at description of moving en- semble of the classical objects in contrast to quantum ones, as we will see farther. 

In 1933 P. A. M. Dirac drew attention to a special role of the action in quantum mechan- ics [2] that can exhibit itself in expressions by means of a term exp { i S/ ¯ h } . It is appropriate to notice the following observation: the action here plays a role of a phase shift. And according to the principle of stationary action, the phase shift has to be constant along an optimal path of the 

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particle. In 1945 Paul Dirac emphasized once again, that the classical and quantum mechan- ics have many general points of crossing [3]. In particular, he had wrote ”We can use the formal probability to set up a quantum picture rather close to the classical picture in which the coor- dinates q of a dynamical system have definite values at any time. We take a number of times t$_{1}$, t$_{2}$, t$_{3}$, · · · 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 be- ing improbable and others being likely.” [3]. 

Dirac’s observations had influence to R. Feyn- man’s searching acceptable language for descrip- tion of moving quantum objects, where deci- sive role has a term exp { i Lδt/ ¯ h } [4]. Idea is that the above term proposes mapping a wave function from one state to another divided by a small time interval δt . Feynman’s genius in- sight has resulted in understanding that the in- tegral kernel (propagator) of the time-evolution operator can be expressed as a sum over all pos- sible paths (not just the classical one) connect- ing the points q$_{a}$ and q$_{b}$ with the weight factor exp { i S ( q$_{a}$, q$_{b}$ ; t ) / ¯ h } [5, 6]: 

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where A is an normalization constant. 

As a result we have now a powerful mathemat- ical apparatus, the path integral technique [7]. Briefly, according to this idea, there are many possible trajectories, that can be traced from a source to a detector. But only one trajectory, submitting to the principle of stationary action, is real. The others cancels each other because of interference effects. Such an interpretation is ex- tremely productive at generating intuitive imag- ination for more perfect understanding quantum mechanics. 

The contents of the article is as follows. In Sect. II being based on the path integral tech- nique we study migration of a particle across a 

