Description of Classical and Quantum Interference in View of the Concept of Flow Line

Abstract

Bohmian mechanics, a hydrodynamic formulation of quantum mechanics, relies on the concept of trajectory, which evolves in time in compliance with dynamical information conveyed by the wave function. Here this appealing idea is considered to analyze both classical and quantum interference, thus providing an alternative and more intuitive framework to understand the time-evolution of waves, either in terms of the flow of energy (for mechanical waves, sound waves, electromagnetic waves, for instance) or, analogously, the flow of probability (quantum waves), respectively. Furthermore, this procedure also supplies a more robust explanation of interference phenomena, which currently is only based on the superposition principle. That is, while this principle only describes how different waves combine and what effects these combinations may lead to, flow lines provide a more precise explanation on how the energy or probability propagate in space before, during and after the combination of such waves, without dealing with them separately (i.e., the combination or superposition is taken as a whole). In this sense, concepts such as constructive and destructive interference, typically associated with the superposition principle, physically correspond to more or less dense swarms of (energy or probability) flow lines, respectively. A direct consequence of this description is that, when considering the distribution of electromagnetic energy flow lines behind two slits, each one covered by a differently oriented polarizer, it is naturally found that external observers' information on the slit crossed by single photons (understood as energy parcels) is totally irrelevant for the existence of interference fringes, in striking contrast with what is commonly stated and taught.

Bohm’s approach to quantum mechanics: Alternative theory or practical picture?

“…As a result of the global nature of the quantum phase field and, through it, the associated velocity field, the trajectories or streamlines that the Bohmian picture renders have nothing to do with those one typically consider in classical mechanics. Actually, this nonlocal or global behavior is typical of ensembles of streamlines mapping any kind of wave, regardless of its nature [91,98]. Specifically, in the case here considered, the trajectories tend to move to regions with lower values of the modulus of the velocity field (more stable, dynamically speaking), where they display what could be regarded as a nearly classical-like uniform motion (v ≈ constant), as can be seen in Fig.…”

“…For example, in Optics we find models based on the Poynting vector and the Maxwell equations already in the 1950s [83,84], and only much later the approaches of Bohm and Madelung were considered [85][86][87][88][89][90][91][92]. In Acoustics the use of streamlines was also suggested in the 1980s as a visualization and analysis tool [93][94][95][96][97], although there was not a direct link with Madelung or Bohm in this direction until recently [98]. Finally, we also find different models aimed at treating dissipation in quantum systems, such as the one proposed by Kostin [99], which is based on the idea of adding nonlinear contributions to the Schrödinger equation, which, interestingly, are given in terms of the phase of the wave function, and therefore the Bohmian momentum (which is identified with the dissipative term that appears in the classical Newton equations of motion with friction).…”

Interfering Quantum Trajectories Without Which-Way Information

“…Such simultaneous, well-defined values for position and velocity, as can be evaluated using equation ( 4), are itself against the world view of standard quantum mechanics. One can see that this single-valuedness leads to the non-crossing property [10,26,27,28,29] of dBB trajectories. The consequent which-way information is an inescapable conclusion in the dBB representation.…”

“…Those trajectories whose starting points are near the upper slit (z > 0) cannot go to the region below some point with z = 0 on the screen and vice-versa. Thus there is a kind of fictitious barrier between the two regions, so that the two families of trajectories appear to repel each other [10,26,27,28,29,30]. Conversely, by knowing the point at which the particle reaches the screen, one can identify the slit through which it has emanated.…”

Louis de Broglie Double Solution Theory Confirms the Wave-Particle Duality Principle

“…As a result, formulas ( 25) and ( 27) are sufficient for calculations of the Bohmian trajectories when the action function is strictly given. Computations of the Bohmian trajectories are given in many articles devoted to the de Broglie-Bohm theory [31,32,[60][61][62][63]. Here we show the Bohmian trajectories calculated for the heavy fullerene molecules [16,46] passing from slits of the interference grating, Figure 7.…”