Should particle trajectories comply with transverse momentum distribution?

1. Should particle trajectories comply with transverse momentum distribution? Milena Davidović, Faculty of Civil Engineering, Belgrade, Serbia, milena@grf.bg.ac.yu Dušan Arsenović and Mirjana Božić Institute of Physics, Belgrade, Serbia, arsenovic@phy.bg.ac.yu, bozic@phy.bg.ac.yu Angel S. Sanz and Salvador Miret-Artes Instituto de Matematicas y Fisica Fundamental, Madrid, Spain angel.sanz@uam.es, salvador@fam10.imaff.csic.es Central European Conference on Quantum Optics, Palermo, June 2007

2. Quantum interference experiments with beams of one per one particle have intensified theoretical search of the forms of particle trajectories behind an interference grating. • The aim of all approaches is to get consistency between quantum mechanical particle distribution and the distribution associated with particle's trajectories. • In this paper we compare the features of : • trajectories determined using momentum distribution (MD trajectories) associated with a particle wave function. • de Broglie-Bohm's (BB) trajectories

3. Bozic at al Proc.3rd Workshop on Mysteries, Puzzles and Paradoxes where is the Fourier transform of the initial wave function

4. Alternative expression for can be used The last expression is particularly useful when initial wave function consists of pieces where it takes zero value as in our case. We use following boundary conditions at the one- dimensional grating with n slits of equal widths at the openings outside the openings

5. Expression for screen arrival probability based on the idea of MD particle’s trajectories

6. Comparison of particles distributions in a far field evaluated using modulus squared of a wave function and expression based on particles trajectories. n=2

7. Comparison of modulus square of the wave function and the screen arrival probability based on the idea of particle’s trajectories Number of equal slits n = 40

8. Bohm’s trajectories Bohm's trajectories associated with one particle stateare determined by the differential relation: where is the phase of the wave function written in the form: It can be shown that

9. In our case we have The equation of motion along y- axis is simple: Equation for a motion along x-axis can be solved numerically We have found numerical solutions for the particle BB trajectories behind gratings with 5 and 30 slits.

10. Trajectories behind a grating with n=5 slits up to 1.25 Talbot distances. d=0.1e-6m; delta=0.05e-6m; m=1.19e-24kg; v=220m/s; lam=2.53e-12m

11. Trajectories behind a grating with n=5 slits up to 12.5 Talbot distances.

12. Trajectories behind the upper part of a Ronchi grating with n=30 slits d=0.2e-6m; delta=0.1e-6m; k=pi/8*1e12m-1; m=3.8189e-26kg; v=1084m/s;

13. Bohm’s trajectories behind two Gaussian slits. Gondran et al Am.J.Phys. 733, March 2005 Gondran et al have discussed a simulation of the double slit experiment, based on the solution of Schrodinger equation using Feynman path integral method.

14. Bohm’s trajectories behind five Gaussian and five quasi plane slits A. Sanz et al.. Particle diffraction studied using quantum trajectories, J. Phys C 12 (2002) 6109 The patterns defined by trajectories are similar for both transmission functions, but trajectories corresponding to quasi plane model have more wavy shape.

15. The intensity patterns obtained by means of the standard quantum mechanics and histogram obtained by Bohmian mechanics in far field show good agreement.

16. Bohm’s trajectories behind n=10 Gaussian slits A. S. Sanz and S. Miret-Artes, A causal look into the quantum Talbot effect, arXiv:quant-ph/0702224v1, Feb. 2007

17. We see that BB trajectories reproduce perfectly quantum mechanical distribution in the far field as well as in the near field. The consistency of the set of BB trajectories in the near field behind a multiple slit grating with Talbot effect is remarkable.

18. Comparison of the trajectories features Essential feature of BB deterministic trajectories is that particle passing through different slits may not reach the same point at the screen. MD TRAJECTORIES (trajectories determined using momentum distribution) MD trajectories from different slits may reach the same point on the screen. This property is a consequence of contextuality in addition of probabilities in QM.

19. MOMENTUM DISTRIBUTION It is evident that the probability amplitude of transverse momenta is an essential characteristics of a wave function and wave field behind a grating. So, the question arises : is the distribution of transverse momenta associated with Bohm’s trajectories identical or different from the distribution determined from the probability amplitude of transverse momenta. Long time ago Takabayasi (Prof. Theor. Phys. 1952) concluded that this two distributions are different functions. From our numerical evaluation and analytical analysis it follows : The distributions determined from the wave function and from Bohms’s trajectories are identical in the far field. In the near field the distribution of transverse momenta associated with BB trajectories change with the distance from a grating. It is different from the distribution determined by the wave function.

20. Momentum distribution at various distances from a grating of particles following Bohm’s trajectories for n=5 slits kx/ky

22. Red: Momentum distribution in the far field obtained from BB trajectories. Blue:

23. Analytical expression for the transverse momentum distribution in the far field

24. CONCLUSION Momenta of particles moving along MD trajectories are distributed in accordance to the momentum distribution determined by the particle wave function. MD trajectories reproduce well quantum mechanical space distribution in the far field. It seems that better agreement of MD trajectories in the near field could be obtained by combining peaces of various BB trajectories, understood as lines of a quantum mechanical current. .