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 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]: 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 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 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. where A is an normalization constant. 2 − → p = ∇ S. (3) ∂ρ ∂ t + ( − → v ∇ ρ ) = 0 . (4) K ( q$_{b}$, q$_{a}$ ) = ∑ all paths A exp { i S ( q$_{a}$, q$_{b}$ ; T ) / ¯ h } , (5)