We show that the Schrödinger equation describes the ensemble mean dynamics of solitons in a Galilean invariant field theory where we interpret solitons as particles. On a zero background, solitons move classically, following Newton's second law; however, on a nonzero amplitude chaotic background, their momentum and position fluctuate fulfilling an exact uncertainty relation, which gives rise to the emergence of quantum phenomena. The Schrödinger equation for the ensemble of solitons is obtained from this exact uncertainty relation, and the amplitude of the background fluctuations is what corresponds to the value of ℏ . We confirm our analytical results running simulations of solitons moving against a potential barrier and comparing the ensemble probabilities with the predictions of the time-dependent Schrödinger equation, providing a deterministic version of the quantum tunneling effect. We conclude with a discussion of how our theory does not present statistical independence between measurement and experiment outcomes.
We show that the Schrödinger equation describes the ensemble mean dynamics of solitons in a Galilean invariant field theory where we interpret solitons as particles. On a zero background, solitons move classically, following Newton's second law; however, on a nonzero amplitude chaotic background, their momentum and position fluctuate fulfilling an exact uncertainty relation, which gives rise to the emergence of quantum phenomena. The Schrödinger equation for the ensemble of solitons is obtained from this exact uncertainty relation, and the amplitude of the background fluctuations is what corresponds to the value of ℏ . We confirm our analytical results running simulations of solitons moving against a potential barrier and comparing the ensemble probabilities with the predictions of the time-dependent Schrödinger equation, providing a deterministic version of the quantum tunneling effect. We conclude with a discussion of how our theory does not present statistical independence between measurement and experiment outcomes.
We present a modified version of Nelson's seminal paper on the derivation of the time-dependent Schrödinger equation which draws on the equation of motion of a particle that moves under the influence of a classical force field and additional stochastic forces. The emphasis of our elaboration is focused on the implication of allowing stochastic forces to occur, viz. that the energy E of the particle is no longer conserved on its trajectory in a conservative force field. We correlate this departure Δ E from its classical energy with the energy/time uncertainty relation where Δ t is the average time for Δ E to persist. The stability of atoms, the zero-point energy of oscillators, the tunneling effect and the diffraction at slits are shown to be directly connected with the occurrence of such energy fluctuations. We discuss and rederive Nelson's theory entirely from this point of view and generalize his approach to systems of N particles which interact via pair forces. Achieving reversibility in a description of particle motion that is akin to Brownian motion, represents a salient point of the derivation. We demonstrate that certain objections raised against Nelson's theory are without substance. We also try to put the particular worldview of this version of stochastic quantum mechanics into perspective with regard to the established Copenhagen interpretation.
The nature of the problem The quantum world is inexplicable in classical terms. The predictions pertaining to the interaction of matter and light embodied in Newton's laws for the motion of particles and Maxwell's equations governing the propagation of electromagnetic fields are in flat contradiction with the experimental facts at the microscopic scale. A key feature of quantum effects is their apparent indeterminism, that individual atomic events are unpredictable, uncontrollable and literally seem to have no cause. Regularities emerge only when one considers a large ensemble of such events. This indeed is generally considered to constitute the heart of the conceptual problem posed by quantum phenomena, necessitating a fundamental revision of the deterministic classical world view. Some of the principal phenomena, among those discussed in this book, are as follows: ( a ) Self-interference: a beam of electrons sent one at a time through a barrier containing two apertures builds up through a series of localized detection events on a screen an interference pattern characteristic of waves (cf. the frontispiece and §5.1). ( b ) Tunnelling: an α-particle trapped in a nucleus can pass through a potential barrier in a manner forbidden to a classical particle (§5.3). ( c ) The stability of matter: atoms and molecules are found to exist only in certain discrete, or ‘stationary’, energy states. For this reason they do not ‘collapse’, the result predicted by classical electrodynamics. During transitions between stationary states (quantum jumps) an atom exchanges a discrete quantity of energy with the electromagnetic field (§§4.5, 7.6). […]
The Schrödinger free particle equation in 1+1 dimension describes second-order effects in ensembles of lattice random walks, in addition to its role in quantum mechanics, and its solutions represent the continuous limit of a property of ensembles of Brownian particles. In the present paper, the classical Schrödinger and Dirac equations have been derived from the Brownian motions of a particle, and it has been shown that the classical Schrödinger equation can be transformed into the usual Schrödinger quantum equation on applying the Heisenberg uncertainty principle between position and momentum, while the Dirac quantum equation follows from its classical counterpart on applying the Heisenberg uncertainty principle between energy and time, without applying any analytical continuation.
On the basis of the observational fact that a wave packet, describing the localized ensemble of micro-objects, spreads according to the diffusion law, the quantum equivalence principle is formulated, that the motion of the ensemble of quantum objects is equivalent to the diffusion of the ensemble of classical objects in a fluctuating background field (physical vacuum).The observations also confirm the validity of the principle of relativity for quantum phenomena, formulating as the principle of constancy of quantum fluctuations, that these fluctuations and the describing them diffusion occur identically in all inertial frames.It is shown that these two physical principles, together with the principles of mechanics, lead to the diffusion quantum mechanics (DQM), containing the formalism of quantum mechanics as a particular case.In DKM the relation between the velocity of the diffusion flow and the probability density is given by the diffusion law (Fick's law), while the invariance of diffusion leads to its conservatism (diffusion without friction) and the constancy of the diffusion coefficient, fixed from correspondence with quantum mechanics.In DQM the Hamiltonian includes the kinetic energies of the drift, diffusion flux, and interaction potentials, while the probability density and the drift action function of particles S form a canonical pair.The canonical equations for them then lead to the continuity equation and to the Hamilton-Jacobi-Madelung equation.The probability density in them enters nonlinearly, but in the case of a canonical transformation to the complex amplitude of the probabilities, they are linearized and pass into the Schrödinger equation.As a result, the amplitudes of probabilities add up for the alternatives, i.e. their superposition takes place.DQM based on physical principles is a more general theory than quantum mechanics, since along with classical devices and particles it introduces a third participant of all processes -the background field, which is the source of quantum fluctuations of classical particles.In DQM, the quantum potential is the potential energy associated with localization, quantum statistics appears in the system of many classical particles in the fluctuating background, and composite particles of small size can have a small mass.DQM also explains the existence of rest energy, the constancy of the light velocity (velocity of quantum fluctuations) and predicts the existence and properties of gravitation as thermal diffusion in the inhomogeneous background field.
The relationship between chaos and quantum mechanics has been somewhat uneasy--even stormy, in the minds of some people. However, much of the confusion may stem from inappropriate comparisons using formal analyses. In contrast, our starting point here is that a complete dynamical description requires a full understanding of the evolution of measured systems, necessary to explain actual experimental results. This is of course true, both classically and quantum mechanically. Because the evolution of the physical state is now conditioned on measurement results, the dynamics of such systems is intrinsically nonlinear even at the level of distribution functions. Due to this feature, the physically more complete treatment reveals the existence of dynamical regimes--such as chaos--that have no direct counterpart in the linear (unobserved) case. Moreover, this treatment allows for understanding how an effective classical behavior can result from the dynamics of an observed quantum system, both at the level of trajectories as well as distribution functions. Finally, we have the striking prediction that time-series from measured quantum systems can be chaotic far from the classical regime, with Lyapunov exponents differing from their classical values. These predictions can be tested in next-generation experiments.
The motion of a ball through an appropriate lattice of round obstacles models the behavior of a Brownian particle and can be used to describe measurement on a macro system. On another hand, such motion is chaotic and a known conjecture asserts that the Hamiltonian of the corresponding quantum system must follow the random matrix statistics of an appropriate ensemble. We use the Hamiltonian represented by a random matrix in the Gaussian unitary ensemble to study the Schrödinger evolution of non-stationary states. For Gaussian states representing a classical system, the Brownian motion that describes the behavior of the system under measurement is obtained. For general quantum states, the Born rule for the probability of transition between states is derived. It is then shown that the Schrödinger evolution with such a Hamiltonian models measurement on macroscopic and microscopic systems, provides an explanation for the classical behavior of macroscopic bodies and for irreversibility of a measurement, and identifies the boundary between micro and macro worlds.