Pilot-Wave Quantum Theory with a Single Bohm’s Trajectory

Abstract

The representation of a quantum system as the spatial configuration of its constituents evolving in time as a trajectory under the action of the wave-function, is the main objective of the de Broglie–Bohm theory (or pilot wave theory). However, its standard formulation is referred to the statistical ensemble of its possible trajectories. The statistical ensemble is introduced in order to establish the exact correspondence (the Born’s rule) between the probability density on the spatial configurations and the quantum distribution, that is the squared modulus of the wave-function. In this work we explore the possibility of using the pilot wave theory at the level of a single Bohm’s trajectory, that is a single realization of the time dependent configuration which should be representative of a single realization of the quantum system. The pilot wave theory allows a formally self-consistent representation of quantum systems as a single Bohm’s trajectory, but in this case there is no room for the Born’s rule at least in its standard form. We will show that a correspondence exists between the statistical distribution of configurations along the single Bohm’s trajectory and the quantum distribution for a subsystem interacting with the environment in a multicomponent system. To this aim, we present the numerical results of the single Bohm’s trajectory description of the model system of six confined planar rotors with random interactions. We find a rather close correspondence between the coordinate distribution of one rotor, the others representing the environment, along its trajectory and the time averaged marginal quantum distribution for the same rotor. This might be considered as the counterpart of the standard Born’s rule when the pilot wave theory is applied at the level of single Bohm’s trajectory. Furthermore a strongly fluctuating behavior with a fast loss of correlation is found for the evolution of each rotor coordinate. This suggests that a Markov process might well approximate the evolution of the Bohm’s coordinate of a single rotor (the subsystem) and, under this condition, it is shown that the correspondence between coordinate distribution and quantum distribution of the rotor is exactly verified

Quantum Stochastic Trajectories: The Fokker–Planck–Bohm Equation Driven by the Reduced Density Matrix

“…Correspondingly the single Bohm trajectory Q(t), in principle to be evaluated for a given time dependent pure state |Ψ(t)⟩ and initial configuration Q(0), represents the main objective of our analysis. 40 As shown in ref 7, the correspondence with the quantum description provided by the wave function is recovered from the distribution on the configuration space generated by a single Bohm trajectory Q(t) during its evolution.…”

“…This equivalence is exploited in numerical procedures for the calculation of the time dependent wave function Ψ­( q , t ) from a suitable ensemble of trajectories. , Our methodological point of view, however, is rather different: the Bohm coordinates are employed to characterize the quantum molecular trajectory as long as they allow the assignment of well-defined positions to the particles of the system, the nuclei of a molecule for instance. Correspondingly the single Bohm trajectory Q ( t ), in principle to be evaluated for a given time dependent pure state |Ψ­( t )⟩ and initial configuration Q (0), represents the main objective of our analysis . As shown in ref , the correspondence with the quantum description provided by the wave function is recovered from the distribution on the configuration space generated by a single Bohm trajectory Q ( t ) during its evolution.…”

“…The set q = ( q 1 , q 2 , ..., q n ) includes the coordinates of all the oscillators. Because of the similarities between this model system and that analyzed in refs and , we just summarize its main features.…”

Quantum Statistical Ensemble Resilient to Thermalization

“…In the past, we analyzed in detail the uniform statistical distribution of populations in an active space without the lower boundary and the energy eigenvalues in the range E k < E max , the so-called random pure state ensemble (RPSE). ,− , Such a procedure leads to a thermal state specified by a single parameter, ζ = E max , which can be identified with the internal energy in the large size limit of the system . For the general case of an active space with both boundaries, ζ = ( E min , E max ), one can verify in the macroscopic limit the equivalence of E max with the internal energy, and that the thermodynamic parameters are independent of the lower energy boundary E min .…”

Quantum Molecular Trajectory and Its Statistical Properties

“…The same argument can be applied to Bohmian mechanics and this is corroborated by calculations on specific systems. For instance, in ref , we have analyzed the system of six interacting planar rotors in the presence of a confining potential and we have verified numerically the loss of correlation for the rotor coordinates. As an example of correlation function, in Figure we have reported that for the coordinate of the first rotor B ( q ) = q 1 .…”