K ( x$_{b}$, t$_{b}$ ; x$_{a}$, t$_{a}$ ) = $^{[}$2 π i ¯ h ( t$_{b}$ − t$_{a}$ ) m $_{]}$− 1 / 2 · exp$^{{}$ i m ( x$_{b}$ − x$_{a}$ ) 2 2¯ h ( t$_{b}$ − t$_{a}$ ) } . (9) ψ ( x$_{1}$, x$_{0}$ ) = b ∫ − b $^{(}$2 π i ¯ hτ m $_{)}$− 1 / 2 exp$^{{}$ i m (Δ x − ξ ) 2 2¯ hτ $^{}(}$2 π i ¯ hT m $_{)}$− 1 / 2 exp$^{{}$ i m ( x$_{0}$ + ξ ) 2 2¯ hT } dξ. (10) G ( ξ ) = exp {− ξ $^{2}$/ 2 b $^{2}$} . (11) Integration here is executed along the slit width a = 2 b , i.e., from − b to + b . Formally, ranges of the integration can be broadened from −∞ to + ∞ . But in this case, one should introduce step function G ( ξ ) equal to unit in the interval [ − b, + b ] and equal to zero outside this interval. Such a form-factor with sharp edges can be suitable, if wavelength of the particle is much more than interatomic distances at slit’s edges. If a material from which the slit grating is produced shows fuzzy edges of the slits, then such a form-factor is incorrect. Sim- plest form-factor, simulating such fuzzy edges of the slits, is the Gaussian form-factor Effective width of the curve is conditioned by a parameter b . For such a form-factor roughly two thirds of all its area is situated between the points − b and + b . If the particles would move in classical way, then we should wait, that after time τ they will be distributed along axis x like- The expression (8) is written in accordance with a rule of summing amplitudes for successive events in time. The first event is the moving par- ticle from the origin to the slit. The second event is the movement of the particle from the slit to the point x$_{1}$ . The slit has a finite width. And passing through the slit is conditioned by differ- ent alternative possibilities. Therefore, we need to integrate along all the slit width in order to get a right result. All particles, moving through the slit, are free particles and their correspond- ing kernels are given by the expression (9). By substituting this kernel to the integral (8) we get the following detailed form It describes a transition amplitude from x$_{a}$ to x$_{b}$ for a time interval ( t$_{b}$ − t$_{a}$ ) [7]. Consequently, the integral (8) computes the probability amplitude of transition from the origin x = 0 to the point x$_{1}$ through the all possible intermediate points ξ situated within the interval ( x$_{0}$ − b, x$_{0}$ + b ). Here the kernel reads 4 ψ ( x$_{1}$, x$_{0}$ ) = b ∫ − b K ( x$_{1}$, T + τ ; x$_{0}$ + ξ, T ) K ( x$_{0}$ + ξ, T ; 0 , 0) dξ. (8) FIG. 3: Trajectories of particles, passing through the Gaus- sian slit [7], form a ray with an angle α of the particle beam. 2b 2b$_{1}$ x$_{1}$ x$_{0}$ x t x T