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A quantum solution of the classical electrodynamics equations has been found. It is shown that all information on the multiparticle process of creation of scalar pairs of particles by a nonstationary self-acting electric field is contained in solutions of the d’Alembert single-particle equation. The existence of a quantum solution of the d’Alembert equation is determined by the Ehrenfest theorem. In this case, the corresponding solution is independent of the Planck constant. The process of conversion of thermal and acceleration energies into radiation has been investigated. It is demonstrated that a self-acting electric field exhibits elastic behavior. The connection of classical electrodynamics with geometrodynamics is established. The geometrodynamic justification of the appearance of the Hubble magnitude in classical electrodynamics is provided. The probabilities of creation of one-dimensional space and charge during tunneling in time of the effective Planck particle are determined. It has been shown that the fine structure constant α can be interpreted as the probability $$ \alpha ={e}^{-\frac{\pi^2\gamma }{2}} $$ of charge creation without charge and real mass. This implies that the so-called fine structure of mathematical constants can contain information on the interactions of matter, which can be used to solve the problem of information loss in black holes.

Electrons moving at slow speeds much lower that the speed of light are described by a wave function which is a solution of Pauli's equation. This is a low velocity limit of the relativistic Dirac equation. Here we compare two approaches, one which is the more conservative Copenhagen's interpretation denying a trajectory of the electron but allowing a trajectory to the electron expectation value through Ehrenfest theorem. The said expectation value is of course calculated using a solution of Pauli's equation. A less orthodox approach is championed by Bohm, and attributes a velocity field to the electron also derived from the Pauli wave function. It is thus interesting to compare the trajectory followed by the electron according to Bohm and its expectation value according to Ehrenfest. Both similarities and differences will be considered., Comment: 12 pages

We introduce the Poisson bracket operator which is an alternative quantum counterpart of the Poisson bracket. This operator is defined using the operator derivative formulated in quantum analysis and is equivalent to the Poisson bracket in the classical limit. Using this, we derive the quantum canonical equation which describes the time evolution of operators. In the standard applications of quantum mechanics, the quantum canonical equation is equivalent to the Heisenberg equation. At the same time, this equation is applicable to c-number canonical variables and then coincides with the canonical equation in classical mechanics. Therefore the Poisson bracket operator enables us to describe classical and quantum behaviors in a unified way. Moreover, the quantum canonical equation is applicable to non-standard system where the Heisenberg equation is not defined. As an example, we consider the application to the system where a c-number and a q-number particles coexist. The derived dynamics satisfies the Ehrenfest theorem and the energy and momentum conservations., Comment: 25 pages, no figure, the discussions of the Wigner function and Lindblad equation and references were added. Accepted for publication in Phys. Rev. A

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.

The transport of energy inside protein $\alpha$-helices is studied by deriving a system of quantum equations of motion from the Davydov Hamiltonian with the use of the Schr\"odinger equation and the generalized Ehrenfest theorem. Numerically solving the system of quantum equations of motion for different initial distributions of the amide I energy over the peptide groups confirmed the generation of both moving or stationary Davydov solitons. In this simulation the soliton generation, propagation, and stability were found to be dependent on the symmetry of the exciton-phonon interaction Hamiltonian and the initial site of application of the exciton energy., Comment: 18 pages, 9 figures

Elaboration of some fundamental relations in 3-dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator’s time derivative and the time derivative of the mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial-like theorems. As a particular case,the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers’ sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated.

Quantum dynamics of a pulsating mercury drop is investigated. It is demonstrated that the energy of the pulsating mercury drop can be quantized; therefore, it can generate electromagnetic radiation with a discrete spectrum. This allows a quantum generator and a radio-wave amplifier to be developed. The existence of a quantum solution of the classical Newton equation is caused by the nonstationarity of the potential of the corresponding type and by the Ehrenfest theorem. In this case, the corresponding solution is independent of the Planck constant. The quantum solutions of the equations of classical physics possess all attributes of quantum mechanics: wave-particle duality, uncertainty principle, superposition principle, causality principle, quantum interference, radiation with a discrete spectrum, tunneling, and spin effects.

We analyze the phenomenon of semiquantum chaos in two GaAs quantum dots coupled linearly and quadratically by two harmonic potentials. We show how semiquantum dynamics should be derived via the Ehrenfest theorem. The extended Ehrenfest theorem in two dimensions is used to study the system. The numerical simulations reveal that, by varying the interdot distance and coupling parameters, the system can exhibit either periodic or quasi-periodic behavior and chaotic behavior.

Inspired by the recent work of Filho et al., a Hermitian momentum operator is introduced in a general curved space with diagonal metric. The modified Hamiltonian associated with this new momentum is calculated and discussed. Furthermore, granting the validity of the Heisenberg equation in a curved space, the Ehrenfest theorem is generalized and interpreted with the new position-dependent differential operator in a curved space. The modified Hamiltonian leads to a modified time-independent Schr\"odinger equation, which is solved explicitly for a free particle in the Poincar\'e upper half-plane geometry. It is shown that a "free particle" does not behave as it is totally free due to curved background geometry., Comment: 8 pages, 2 figures

We generalise the celebrated semiclassical wavepacket approach from the adiabatic to the non-adiabatic regime. A unified description covering both of these regimes is particularly desired for systems with spatially varying band structures where band gaps of various sizes are simultaneously present, e.g. in moir\'{e} patterns. For a single wavepacket, alternative to the previous derivation by Lagrangian variational approach, we show that the same semiclassical equations of motion can be obtained by introducing a spatial-texture-induced force operator similar to the Ehrenfest theorem. For semiclassically computing the current, the ensemble of wavepackets based on adiabatic dynamics is shown to well correspond to a phase-space fluid for which the fluid's mass and velocity are two distinguishable properties. This distinction is not inherited to the ensemble of wavepackets with the non-adiabatic dynamics. We extend the adiabatic kinetic theory to the non-adiabatic regime by taking into account decoherence, whose joint action with electric field favours certain form of inter-band coherence. The steady-state density matrix as a function of the phase-space variables is then phenomenologically obtained for calculating the transport current. The result, applicable with a finite electric field, expectedly reproduces the known adiabatic limit by taking the electric field to be infinitesimal, and therefore attains a unified description from the adiabatic to the non-adiabatic situations., Comment: 16 pages, 1 figure

A frequently encountered source of systematic error in quantum computations is imperfections in the control pulses which are the classical fields that control qubit gate operations. From an analysis of the quantum-mechanical time-evolution operator of the spin wavefunction, it has been demonstrated that composite pulses can mitigate certain systematic errors and an appealing geometric interpretation was developed for the design of error-suppressing composite pulses. Here we show that these same pulse sequences can be obtained within a quasi-classical framework. This raises the question of whether error-correction procedures exist that exploit entanglement in a manner that can not be reproduced in the quasi-classical formulation.

Based on the Ehrenfest theorem, the time-dependent expectation value of a momentum operator can be evaluated equivalently in two ways. The integrals appearing in the expressions are taken over two different functions. In one case, the integrand is the quantum mechanical flux density j̲ , and in the other, a different quantity j̃ ̲ appears, which also has the units of a flux density. The quantum flux density j̲ is related to the probability density ρ via the continuity equation, and j̃ ̲ may as well be used to define a density ρ̃ that fulfills a continuity equation. Employing a model for the coupled dynamics of an electron and a proton, we document the properties of the densities and flux densities. It is shown that although the mean momentum derived from the two quantities is identical, the various functions exhibit a very different coordinate and time-dependence. In particular, it is found that the flux density j̃ ̲ directly monitors temporal changes in the probability density, and the density ρ̃ carries information about wave packet dispersion occurring in different spatial directions.

One of the most accurate methods for solving the time-dependent Schr\"{o}dinger equation uses a combination of the dynamic Fourier method with the split-operator algorithm on a tensor-product grid. To reduce the number of required grid points, we let the grid move together with the wavepacket, but find that the na\"ive algorithm based on an alternate evolution of the wavefunction and grid destroys the time reversibility of the exact evolution. Yet, we show that the time reversibility is recovered if the wavefunction and grid are evolved simultaneously during each kinetic or potential step; this is achieved by using the Ehrenfest theorem together with the splitting method. The proposed algorithm is conditionally stable, symmetric, time-reversible, and conserves the norm of the wavefunction. The preservation of these geometric properties is shown analytically and demonstrated numerically on a three-dimensional harmonic model and collinear model of He-H$_{2}$ scattering. We also show that the proposed algorithm can be symmetrically composed to obtain time-reversible integrators of an arbitrary even order. We observed $10000$-fold speedup by using the tenth- instead of the second- order method to obtain a solution with a time discretization error below $10^{-9}$. Moreover, using the adaptive grid instead of the fixed grid resulted in a 64-fold reduction in the required number of grid points in the harmonic system and made it possible to simulate the He-H$_{2}$ scattering for six times longer, while maintaining reasonable accuracy. Applicability of the algorithm to high-dimensional quantum dynamics is demonstrated using the strongly anharmonic eight-dimensional H\'{e}non--Heiles model., Comment: Added new subsection III C and Fig. 9 with results on the eight-dimensional H\'{e}non--Heiles model, added new Appendix A and Fig. 10 to discuss the exponential convergence of the wavefunction with the increasing number of grid points

14 pags., 5 figs., 1 tab., Recently, two generalized nonlinear Schrödinger equations have been proposed by Chavanis (Eur. Phys. J. Plus 132, 286 (2017)) by applying Nottale's theory of scale relativity relying on a fractal space-time to describe dissipation in quantum systems. Several existing nonlinear equations are then derived and discussed in this context leading to a continuity equation with an extra source/sink term which violates Ehrenfest theorem. An extension to describe stochastic dynamics is also carried out by including thermal fluctuations or noise of the environment. These two generalized nonlinear equations are analyzed within the Bohmian mechanics framework to describe the corresponding dissipative and stochastic dynamics in terms of quantum trajectories. Several applications of this second generalized equation which can be considered as a generalized Kostin equation have been carried out. The first application consists of the study of the position-momentum uncertainty principle in a dissiaptive dynamics. After, the so-called Brownian-Bohmian motion is investigated by calculating classical and quantum diffusion coefficients. And as a third example, transmission through a transient (time-dependent) parabolic repeller is studied where the interesting phenomenon of early arrival is observed even in the stochastic dynamics although the magnitude of early arrival is reduced by friction., SVM acknowledges support from the University of Qom and SMA support from the Ministerio de Ciencia, Innovacion y Universidades (Spain) under the Project FIS2017-83473-C2-1-P.

Newtonian and Schrodinger dynamics can be formulated in a physically meaningful way within the same Hilbert space framework. This fact was recently used to discover an unexpected relation between classical and quantum motions that goes beyond the results provided by the Ehrenfest theorem. The Newtonian dynamics was shown to be the Schrodinger dynamics of states constrained to a submanifold of the space of states, identified with the classical phase space of the system. Quantum observables are identified with vector fields on the space of states. The commutators of observables are expressed through the curvature of the space. The resulting embedding of the Newtonian and Schrodinger dynamics into a unified geometric framework is rigid in the sense that the Schrodinger dynamics is a unique extension of the Newtonian one. Under the embedding, the normal distribution of measurement results associated with a classical measurement implies the Born rule for the probability of transition of quantum states. The mathematics of the discovered relationship between the classical and the quantum is reviewed and investigated here in detail, and applied to the process of measurement of spin and position observables., Comment: published version. arXiv admin note: text overlap with arXiv:1805.07510

Elaboration of some fundamental relations in three dimensional quantum mechanics is considered taking into account the restricted character of areas in radial distance. In such cases the boundary behavior of the radial wave function and singularity of operators at the origin of coordinates contribute to these relations. We derive the relation between the average value of the operator time derivative and the time derivative of mean value of this operator, which is usually considered to be the same by definition. The deviation from the known result is deduced and manifested by extra term, which depends on the boundary behavior mentioned above. The general form for this extra term takes place in the hypervirial like theorems. As a particular case, the virial theorem for Coulomb and oscillator potentials is considered and correction to the Kramers sum rule is derived. Moreover the corrected Ehrenfest theorem is deduced and its consistency with real physical picture is demonstrated., 31 pages

The radiative response of the classical electron is commonly described by the Lorentz-Abraham-Dirac (LAD) equation. Dirac's derivation of this equation is based on energy and momentum conservation laws and on regularization of the field singularities and infinite energies of the point charge by subtraction of certain quantities: "We ... shall try to get over difficulties associated with the infinite energy of the process by a process of direct omission or subtraction of unwanted terms". To substantiate Dirac's approach and clarify the mass renormalization, we introduce the point charge as a limit of extended charges contracting to a point; the fulfillment of conservation laws follows from the relativistic covariant Lagrangian formulation of the problem. We derive the relativistic point charge dynamics described by the LAD equation from the extended charge dynamics in a localization limit by a method which can be viewed as a refinement of Dirac's approach in the spirit of Ehrenfest theorem. The model exhibits the mass renormalization as the cancellation of Coulomb energy with the Poincar\'e cohesive energy. The value of the renormalized mass is not postulated as an arbitrary constant, but is explicitly calculated. The analysis demonstrates that the local energy-momentum conservation laws yield dynamics of a point charge which involves three constants: mass, charge and radiative response coefficient {\theta}. The value of {\theta} depends on the composition of the adjacent potential which generates Poincar\'e forces. The classical value of the radiative response coefficient is singled out by the global requirement that the adjacent potential does not affect the radiated energy balance and affects only the local energy balance involved in the renormalization., Comment: More details are added in the introduction concerning similarities and differences between the original Dirac's approach and our treatment. Several typos are corrected

We investigate the ballistic zitterbewegung dynamics and the Landau-Zener tunneling between edge and bulk states of wave packets in two-dimensional topological insulators. In bulk, we use the Ehrenfest theorem to show that an external in-plane electric field not only drifts the packet longitudinally, but also induces a transverse finite side-jump for both trivial and topological regimes. For finite ribbons of width $W$, we show that the Landau-Zener tunneling between bulk and edge states vanishes for large $W$ as their electric field-induced coupling decays with $W^{-3/2}$. This is demonstrated by expanding the time-dependent Schr\"odinger equation in terms of Houston states. Hence we cannot picture the quantum spin Hall states as arising from the zitterbewegung bulk trajectories `leaking' into the edge states, as proposed in Phys. Rev. B 87, 161115 (2013)., Comment: 7 pages, 5 figures

In this paper we derive various identities involving the {\it action} functional which enters the path-integral formulation of quantum mechanics. They provide some kind of generalisations of the Ehrenfest theorem giving correlations between powers of the action and its functional derivatives., 4 pages, few parts deleted

In the case of an adiabatic motion in molecules, electrons adjust to the smoothly changing geometry of the nuclei. Although then the Born-Oppenheimer (BO) approximation is valid, it fails in predicting the time-dependence of electron momenta because, within its product ansatz for the wave function, the respective expectation values are zero. It is shown that this failure can be circumvented using the Ehrenfest theorem. Here we extend our former work [T. Schaupp et al., Eur. Phys. J. B 91, 97 (2018)] and regard models in higher dimensions and for more particles. We solve the time-dependent Schrödinger equation for the combined nuclear-electronic motion and compare the results to those derived from BO wave functions. For all situations, it is found that the time-dependent BO electronic momenta are in excellent agreement with the numerically exact results.

In this Letter, the nature of phase transition at the critical point of P-V criticality in the extended phase space of RN-AdS black holes has been investigated. By treating the cosmological constant and its conjugate quantity as thermodynamic pressure and volume respectively, we introduce the original expressions of Ehrenfest equations directly into the black hole research instead of utilizing the analogy of Ehrenfest equations. We carry out an analytical check of Ehrenfest equations and prove that both Ehrenfest equations are satisfied. So the black hole undergoes a second order phase transition at the critical point. This result is consistent with the nature of liquid–gas phase transition at the critical point, hence deepening the understanding of the analogy of charged AdS black holes and liquid–gas systems.

The time derivative of the position of a relativistic wave packet is evaluated. It is found that it is equal to the mean value of the momentum of the wave packet divided by the mass of the particle. The equation derived represents a relativistic version of the second Ehrenfest theorem.

The states of radiation of a single frequency and polarization propagating in free space are very conveniently represented by those of a quantum mechanical simple harmonic oscillator. This fact is exploited, together with the Ehrenfest theorem, to map the time evolution of the radiation to the dynamics of the oscillator. This enables a graphical comparison of the behaviour of classical and nonclassical states of radiation.

Ginzburg–Landau theory for studying phase transitions of higher order has been derived using coarse graining and lattice formulation within Ehrenfest thermodynamics. Our developed Hamiltonian leads directly to the functional of the system. We studied the evolution of the order parameter using our developed model equations for third and fourth order phase transitions. The periodic nature of the system can be likened to spatially varying periodic soliton/antisoliton lattice of holes in condensate. This is different from what one observes for any conventional solitary wave in the second order phase regime.