On the thermodynamic origin of the quantum potential

In a new thermodynamic interpretation, the quantum potential is shown to result from the presence of a subtle thermal vacuum energy distributed across the whole domain of an experimental setup. Explicitly, its form is demonstrated to be exactly identical to the heat distribution derived from the defining equation for classical diffusion wave fields. For a single free particle path, this thermal energy does not significantly affect particle motion. However, in between different paths, or at interfaces, the accumulation-depletion law for diffusion waves provides an immediate new understanding of the quantum potential’s main features.     

Emergence and collapse of quantum mechanical superposition: Orthogonality of reversible dynamics and irreversible diffusion

Based on the modelling of quantum systems with the aid of (classical) non-equilibrium thermodynamics, both the emergence and the collapse of the superposition principle are understood within one and the same framework. Both are shown to depend in crucial ways on whether or not an average orthogonality is maintained between reversible Schrödinger dynamics and irreversible processes of diffusion. Moreover, the said orthogonality is already in full operation when dealing with a single free Gaussian wave packet. In an application, the quantum mechanical “decay of the wave packet” is shown to simply result from sub-quantum diffusion with a specific diffusivity varying in time due to a particle’s changing thermal environment. The exact quantum mechanical trajectory distributions and the velocity field of the Gaussian wave packet, as well as Born’s rule, are thus all derived solely from classical physics.     

On some integrable systems in the extended lobachevsky space

Some classical and quantum-mechanical problems previously studied in Lobachevsky space are generalized to the extended Lobachevsky space (unification of the real, imaginary Lobachevsky spaces and absolute). Solutions of the Schrödinger equation with Coulomb potential in two coordinate systems of the imaginary Lobachevsky space are considered. The problem of motion of a charged particle in the homogeneous magnetic field in the imaginary Lobachevsky space is treated both classically and quantum mechanically. In the classical case, Hamilton-Jacoby equation is solved by separation of variables, and constraints for integrals of motion are derived. In the quantum case, solutions of Klein-Fock-Gordon equation are found.     

Quantum dynamics of hydrogen atom in complex space

We show in this paper that the electron’s quantum dynamics in hydrogen atom can be modeled exactly by quantum Hamilton-Jacobi formalism. It is found that the quantizations of energy, angular momentum, and the action variable ∫p dq are all originated from the electron’s complex motion, and that the shell structure observed in hydrogen atom is indeed originated from the structure of the complex quantum potential, from which the quantum forces acting upon the electron can be uniquely determined, the stability of atomic configuration can be justified, and the electron’s complex trajectories can be derived accordingly. Based on the derived electron’s trajectory, we can explain why the electron appears at some positions with large probability, while at some other positions with small probability. The positions with maximum probability predicted by standard quantum mechanics are found to be just the stable equilibrium points of the electron’s non-linear complex dynamics. The electron’s trajectories in hydrogen atom are discovered to be very diverse and strongly state-dependent; some of them are open and non-periodic, while some are closed and periodic. Over such a great diversity of orbits, commensurability condition ensuring the existence of closed orbit will be derived and the de Broglie’s standing wave pattern will be identified. Along the investigation of the electron’s orbits in hydrogen atom, we will also clarify why old quantum mechanics using the concept of classical orbit can correctly predict the energy quantization of hydrogen atom and meanwhile why it is not applicable to general quantum system. Finally, the internal mechanism of how the precessing, non-conical eigen-trajectories can evolve continuously to the classical, non-precessing, conical orbits as n → ∞ is explained in detail.     

Quantum particles from classical statistics

Quantum particles and classical particles are described in a common setting of classical statistical physics. The property of a particle being “classical” or “quantum” ceases to be a basic conceptual difference. The dynamics differs, however, between quantum and classical particles. We describe position, motion and correlations of a quantum particle in terms of observables in a classical statistical ensemble. On the other side, we also construct explicitly the quantum formalism with wave function and Hamiltonian for classical particles. For a suitable time evolution of the classical probabilities and a suitable choice of observables all features of a quantum particle in a potential can be derived from classical statistics, including interference and tunneling. Besides conceptual advances, the treatment of classical and quantum particles in a common formalism could lead to interesting cross‐fertilization between classical statistics and quantum physics.     

Space-time trajectories of quantum particles

The concept of trajectory is extended theoretically from classical mechanics through nonrelativistic and relativistic quantum mechanics. Forced motion of the particle as might be caused by an electromagnetic field is included in the equations. A new interpretation of the electromagnetic potential and the gauge transformation is presented. Using this formal structure, the problem of collecting particles into packets using trajectories is studied for both quantum mechanics and classical mechanics. Quantum mechanical trajectories are found to be significantly more restricted than those allowed by classical physics. The uncertainty principle comes from the second-order nature of the field equation without recourse to statistical arguments. The trajectories of particles in a quantum state can be calculated explicitly from the wave function (also see article in Volume 20, Number 6).     

Classical Mechanics Without Determinism

Classical statistical particle mechanics in the configuration space can be represented by a nonlinear Schrödinger equation. Even without assuming the existence of deterministic particle trajectories, the resulting quantum-like statistical interpretation is sufficient to predict all measurable results of classical mechanics. In the classical case, the wave function that satisfies a linear equation is positive, which is the main source of the fundamental difference between classical and quantum mechanics.     

On the Electromagnetic Origin of Inertia and Inertial Mass

We address the problem of inertial property of matter through analysis of the motion of an extended charged particle. Our approach is based on the continuity equation for momentum (Newton’s second law) taking due account of the vector potential and its convective derivative. We obtain a development in terms of retarded potentials allowing an intuitive physical interpretation of its main terms. The inertial property of matter is then discussed in terms of a kind of induction law related to the extended charged particle’s own vector potential. Moreover, it is obtained a force term that represents a drag force acting on the charged particle when in motion relatively to its own vector potential field lines. The time rate of variation of the particle’s vector potential leads to the acceleration inertia reaction force, equivalent to the Schott term responsible for the source of the radiation field. We also show that the velocity dependent term of the particle’s vector potential is connected with the relativistic increase of mass with velocity and generates a longitudinal stress force that is the source of electric field lines deformation. In the framework of classical electrodynamics, we have shown that the electron mass has possibly a complete electromagnetic origin and the obtained covariant equation solves the “4/3 mass paradox” for a spherical charge distribution.     

Brownian motion of a classical particle in quantum environment

The Klein–Kramers equation, governing the Brownian motion of a classical particle in a quantum environment under the action of an arbitrary external potential, is derived. Quantum temperature and friction operators are introduced and at large friction the corresponding Smoluchowski equation is obtained. Introducing the Bohm quantum potential, this Smoluchowski equation is extended to describe the Brownian motion of a quantum particle in quantum environment.     

Classical particle dynamics in quantum space

It is suggested that if space-time is quantized at small distances, then even at the classical level particle motion in space is complicated and described by a nonlinear equation. In the quantum space the Lagrangian function or energy of the particle consists of two parts: the usual kinetic terms, and a rotation term determined by the square of the inner angular momentum-a torsion torque caused by the quantum nature of space. Rotational energy and rotational motion of the particle disappear in the limitl0, wherel the value of the fundamental length. In the free particle case, in addition to the rectilinear motion, the particle undergoes a rotation given by the inner angular momentum. Different possible types of particle motion are discussed. Thus, the scheme may shed light on the appearance of rotating or twisting, stochastic, and turbulent types of motion in classical physics and, perhaps, on the notion of spin in quantum physics within the framework of the quantum character of space-time at small distances.     

Quantum and classical dissipation of charged particles

A Hamiltonian approach is presented to study the two dimensional motion of damped electric charges in time dependent electromagnetic fields. The classical and the corresponding quantum mechanical problems are solved for particular cases using canonical transformations applied to Hamiltonians for a particle with variable mass. Green’s function is constructed and, from it, the motion of a Gaussian wave packet is studied in detail.     

Wave Function in Classical Statistical Mechanics

The possibility to formulate classical statistical mechanics in terms of the complex wave function and density matrix obeying the evolution equation is discussed. It is shown that the modulus squared of the introduced wave function of the classical particle has the same physical meaning as the modulus squared of the wave function of the quantum particle. The tomographic probabilities are studied for both classical and quantum states. Integrals of motion and their relation to the propagators are discussed.     

Conduction bands in classical periodic potentials

The energy of a quantum particle cannot be determined exactly unless there is an infinite amount of time to perform the measurement. This paper considers the possibility that ΔE, the uncertainty in the energy, may be complex. To understand the effect of a particle having a complex energy, the behaviour of a classical particle in a one-dimensional periodic potential V(x) = −cos(x) is studied. On the basis of detailed numerical simulations it is shown that if the energy of such a particle is allowed to be complex, the classical motion of the particle can exhibit two qualitatively different behaviours: (i) The particle may hop from classically allowed site to nearest-neighbour classically allowed site in the potential, behaving as if it were a quantum particle in an energy gap and undergoing repeated tunnelling processes or (ii) the particle may behave as a quantum particle in a conduction band and drift at a constant average velocity through the potential as if it were undergoing resonant tunnelling. The classical conduction bands for this potential are determined numerically with high precision.     

Motion of Charged Particle in Electric and Magnetic Fields in 3D Noncommutative Spaces and Related Problems

In three-dimensional noncommutative phase space, the energy spectrum and wave functions for the motion of a charged particle in a magnetic field are derived. Due to the momentum–momentum noncommutativity, the particle feels an effective magnetic field in a new direction. When an external electric field perpendicular to this effective magnetic field is applied, the Hall conductivity can be calculated. To get the Hall conductivity, one should define the electric currents from the probability currents in quantum mechanics rather than extending the classical electric currents to quantum mechanics directly. When the electric field is not perpendicular to the effective magnetic field, it is difficult to define the Hall conductivity.     

Superoperator nonequilibrium Green’s function theory of many-body systems; applications to charge transfer and transport in open junctions

Nonequilibrium Green’s functions provide a powerful tool for computing the dynamical response and particle exchange statistics of coupled quantum systems. We formulate the theory in terms of the density matrix in Liouville space and introduce superoperator algebra that greatly simplifies the derivation and the physical interpretation of all quantities. Expressions for various observables are derived directly in real time in terms of superoperator nonequilibrium Green’s functions (SNGF), rather than the artificial time-loop required in Schwinger’s Hilbert-space formulation. Applications for computing interaction energies, charge densities, average currents, current induced fluorescence, electroluminescence and current fluctuation (electron counting) statistics are discussed.