R $^{3}$. And these all paths have to be evaluated jointly. Such a description goes back to the in- teger Chapman-Kolmogorov equation [27] Here U ( ⃗ q$_{x}$ ) is a potential energy of the particle localized at the point ⃗ q$_{x}$ ∈ R $^{3}$. And ( ⃗ q$_{1}$ − ⃗ q$_{x}$ ) /δt is a velocity ˙ ⃗ q$_{x}$ attached to the same point ⃗ q$_{x}$ and oriented in the direction of the point ⃗ q$_{1}$ . The next step is to expand terms, ingoing into the integral (36), into Taylor series. For example the wave function written at the left is expanded up to the first term which gives transitional probability densities of a Markov sequence. Essential difference from the classical proba- bility theory is that instead of the probabilities quantum mechanics deals with amplitudes con- taining imaginary terms, that bear phase infor- mation. The amplitude function charged with a phase term is a complex-valued function called the wave function. In this key, a transition from an initial state ⃗ q$_{0}$ to a final state ⃗ q$_{1}$ through all intermediate instances ⃗ q$_{x}$ given on a conditional set R $^{3}$, see Fig. 9, is represented by the following path integral where the the potential energy U ( ⃗ q$_{1}$ − ⃗ ξ ) is sub- jected to expansion into the Taylor series by the small parameter ⃗ ξ . The under integral wave function ψ ( ⃗ q$_{x}$, ⃗ q$_{0}$ ; t ) = ψ ( ⃗ q$_{1}$ − ⃗ ξ, ⃗ q$_{0}$ ; t ) is subjected also to expansion into the Taylor series up to the second terms of the expansion It is believed, that a main contribution in the integral is only given by intermediate sources from the set R 3 situated near the point ⃗ q$_{1}$ , i.e., corresponding to small ⃗ ξ . In this case, the La- grangian (38) can be rewritten as As for the terms under the integral, here we pre- liminarily make some transformations. Let us determine a small increment in the limits δt → 0 and ⃗ q$_{1}$ → ⃗ q$_{x}$ . Here symbol D $^{3}$q$_{x}$ represents a differential element of volume in the set R $^{3}$. Integral kernel, i.e., propagator, K ( ⃗ q$_{1}$, ⃗ q$_{x}$ ; t + δt, t ) we suppose has a standard form Standard Lagrangian in this place has a form [7] Here the exponent, containing the potential term U ( ⃗ q$_{1}$ − ⃗ ξ ) δt , was preliminarily expanded into the Taylor series. The term in this expansion having Taking into account the expressions (39)-42) and substituting theirs into Eq. 36) we get FIG. 16: Computation of all possible paths passing from point q$_{0}$ to point q$_{1}$ through possible intermediate points q$_{x}$ ∈ R 3 represents a core of the path integral method. 14 p ( x, z ; t + s ) = ∫ R n p ( x, y ; t ) p ( y, z ; s ) dy (35) ψ ( ⃗ q$_{1}$, ⃗ q$_{0}$, t + δt ) ≈ ψ ( ⃗ q$_{1}$, ⃗ q$_{0}$, t ) + ∂ψ ∂ t δt. (39) ⃗ ξ = ⃗ q$_{1}$ − ⃗ q$_{x}$ ⇒ D $^{3}$q$_{x}$ = −D $^{3}$ξ. (40) L ( ⃗ q$_{x}$, ˙ ⃗ q$_{x}$ ) = m 2 ξ 2 δt 2 − U ( ⃗ q$_{1}$ − ⃗ ξ ) , (41) ψ ( ⃗ q$_{1}$, ⃗ q$_{0}$, t + δt ) = ∫ R 3 K ( ⃗ q$_{1}$, ⃗ q$_{x}$ ; t + δt, t ) ψ ( ⃗ q$_{x}$, ⃗ q$_{0}$ ; t ) D $^{3}$q$_{x}$ (36) ψ ( ⃗ q$_{1}$ − ⃗ ξ, ⃗ q$_{0}$ ; t ) ≈ ψ ( ⃗ q$_{1}$, ⃗ q$_{0}$, t ) − ( ∇ ψ · ⃗ ξ ) + ∇ $^{2}$ψ · ξ $^{2}$/ 2 . (42) ψ ( ⃗ q$_{1}$, ⃗ q$_{0}$, t ) + ∂ψ ∂ t δt = − ∫ R 3 1 A exp$^{{}$ i ¯ h m 2 ξ 2 δt$^{}}$ × ( 1 − i ¯ h$^{(}$ U ( ⃗ q$_{1}$ ) − ( ∇ U · ⃗ ξ ) + Δ U · ξ $^{2}$/ 2 $^{)}$δt$^{)}$ × $^{(}$ψ ( ⃗ q$_{1}$, ⃗ q$_{0}$, t ) − ( ∇ ψ · ⃗ ξ ) + Δ ψ · ξ $^{2}$/ 2 $^{)}$D $^{3}$ξ. (43) K ( ⃗ q$_{1}$, ⃗ q$_{x}$ ; t + δt, t ) = 1 A exp$^{{}$ i ¯ h L ( ⃗ q$_{x}$, ˙ ⃗ q$_{x}$ ) δt$^{}}$. (37) L ( ⃗ q$_{x}$, ˙ ⃗ q$_{x}$ ) = m 2 $^{(}$⃗ q$_{1}$ − ⃗ q$_{x}$ δt ) 2 − U ( ⃗ q$_{x}$ ) . (38) q$_{0}$ q $_{x}$O R 3 q$_{1}$