N -slit interference: Path integrals, Bohmian trajectories. Valeriy I. Sbitnev ∗ B.P.Konstantinov St.-Petersburg Nuclear Physics Institute, Russ. Ac. Sci., Gatchina, Leningrad district, 188350, Russia. (Dated: April 1, 2019) Path integrals give a possibility to compute in details routes of particles from particle sources through slit gratings and further to detectors. The path integral for a particle passing through the Gaussian slit results in the Gaussian wavepacket. The wavepackets prepared on N slits and superposed together give rise to interference pattern in the near-field zone. It transforms to diffraction in the far-field zone represented by divergent principal rays, at that all rays are partitioned from each other by ( N − 2) subsidiary rays. The Bohmian trajectories in the near-field zone of N -slit gratings show wavy behavior. And they become straight in the far-field zone. The trajectories show zigzag behavior on the interference Talbot carpet (ratio of particle wavelength to a distance between slits are much smaller than 1 and N ≫ 1). Interference from the the N -slit gratings is simulated by scattering monochromatic neutrons (wavelength=0.5 nm). Also we have considered simulation of interference fringes arising at scattering on an N -slit grating of fullerene molecules (according to the real experiment stated in e-print 1001.0468). PACS numbers: 03.75.-b, 03.75.Dg, 42.25.Fx, 42.25.Hz, 45.20.Jj, 47.10.Df, 61.05.fm I. INTRODUCTION. Classical mechanics operates with point parti- cles behavior of which is found unambiguously from its variational principles [1]. Initially, we begin with the action integral: momentum. It is worth mentioning the optical- mechanical analogy [1] of particular solutions of the HJ-equation. It brings to light on deep parallels between mechanical trajectories, opti- cal rays, and even fluid streams fall under these parallels. Here L ( − → q , ˙ − → q ; t ) is a Lagrangian function equal to difference of kinetic energy and potential en- ergy of the particle. Dynamical variables − → q and ˙ − → q are generalized coordinate and velocity of the particle. Ones proclaim, that the action S re- mains constant along an optimal path of the movement particle. It is the least action prin- ciple. According to this principle, finding of the optimal path adds up to solution of the ex- tremum problem δS = 0. This solution leads to Hamilton-Jacobi equation (HJ-equation): Here H ( − → q , − → p ; t ) = ( − → p ˙ − → q ) − L ( − → q , ˙ − → q ; t ) is the Hamilton function and − → p = ∇ S is a particle Please note, that the gradient of the function S , that is, ∇ S , is directed normally to its sur- face S = const. Consider two nearby surfaces S = C and S = C + ε , see Fig. 1. Let us trace a normal from an arbitrary point P of the first surface up to its intersection with the second sur- face at point P $^{′}$. Hereupon, make another shift $^{∗}$Electronic address: valery.sbitnev@gmail.com P P’ S=C S=C+ ε FIG. 1: Shift of a surface S = C to a new position S = C + ε on a value δ ≪ 1 [1]. Possible trajectories shown by dotted blue curves intersect the surfaces perpendicularly. S = t$_{1}$ ∫ t$_{0}$ L ( − → q , ˙ − → q ; t ) dt (1) − ∂S ∂ t = H ( − → q , − → p ; t ) . (2) arXiv:1001.0661v2 [quant-ph] 25 Jan 2010