From Classical Hamiltonian Flow to Quantum Theory: Derivation of the Schrödinger Equation

It is shown how the essentials of quantum theory, i.e., the Schrödinger equation and the Heisenberg uncertainty relations, can be derived from classical physics. Next to the empirically grounded quantisation of energy and momentum, the only input is given by the assumption of fluctuations in energy and momentum to be added to the classical motion. Extending into the relativistic regime for spinless particles, this procedure leads also to a derivation of the Klein-Gordon equation. Comparing classical Hamiltonian flow with quantum theory, then, the essential difference is given by a vanishing divergence of the velocity of the probability current in the former, whereas the latter results from a much less stringent requirement, i.e., that only the average over fluctuations and positions on the average divergence be identical to zero.     

Influence of quantum fluctuations on the stochastic dynamics of the channeling effect of relativistic electrons and positrons

A system of equations describing multiple scattering by crystal nuclei and electrons, and also quantum fluctuations of the coordinate and momentum operators of fast charged particles has been obtained. Quantum corrections to the classical equations of motion have been determined in quasiclassical approximations. A computer simulation of 855-MeV electron and positron motion in the (110) planar channel of a Si crystal has been carried out. The inclusion of quantum fluctuations in the equation of motion affects the dynamics of the electron channeling effect in planar crystal channels considerably; in particular, intense dechanneling (heating) occurs. Intense rechanneling (cooling) occurs in the case of positrons.     

On the quantum motion of a generalized time-dependent forced harmonic oscillator

In this work, we use linear invariants and the dynamical invariant method to obtain exact solutions of the Schrödinger equation for the generalized time-dependent forced harmonic oscillator in terms of solutions of a second order ordinary differential equation that describes the amplitude of the classical unforced damped oscillator. In addition, we construct Gaussian wave packet solutions and calculate the fluctuations in coordinate and momentum as well as the quantum correlations between coordinate and momentum. It is shown that the width of the Gaussian packet, fluctuations and correlations do not depend on the external force. As a particular case, we consider the forced Caldirola-Kanai oscillator.     

Classical and quantum self amplified spontaneous emission in a high gain free electron laser

We describe the quantum theory and the photon statistics of self amplified spontaneous emission (SASE) in a high gain free electron laser (FEL) using Glauber's quantum theory of coherence. We generalize a previous theory by taking into account many-mode effects and the initiation process resulting from classical shot noise, quantum noise, an injected coherent field and coherent bunching. In particular, we define the concept of quantum SASE which is appropriate when the initial quantum fluctuations dominate over the classical shot noise. We also discuss the conditions for the observation. Quantum SASE is a new quantum phenomenon in which the single electron uncertainty fluctuations of the conjugate variables position and momentum produce exponential amplification of the vacuum field.     

On the motion of classical three-body system with consideration of quantum fluctuations

We obtained the systemof stochastic differential equations which describes the classicalmotion of the three-body system under influence of quantum fluctuations. Using SDEs, for the joint probability distribution of the total momentum of bodies system were obtained the partial differential equation of the second order. It is shown, that the equation for the probability distribution is solved jointly by classical equations, which in turn are responsible for the topological peculiarities of tubes of quantum currents, transitions between asymptotic channels and, respectively for arising of quantum chaos.     

Quantum effects in a homogeneous dust cloud collapse

The status of a classical space-time singularity, when quantum effects are taken into account, has remained a matter of intense interest ever since the epochmaking paper of DeWitt [1] on quantum gravity. We examine here the evolution of quantum fluctuations in the vicinity of the singularity arising out of the classical collapse of a homogeneous dust cloud. As opposed to the pathintegral method used to quantize the conformal degree of freedom (see, e.g., [3] or [4]), we use here the traditional operator approach to the quantum theory which is much more direct and appealing while achieving an additional generalization that the wave function of the system is assumed to have a completely general form. It is shown that the quantum uncertainty diverges in the limit of approach to the classically singular epoch and that nonsingular, nonclassical states can occur with finite probability.     

Quantum structures due to fluctuations of the measurement situation

We analyze the meaning of the nonclassical aspects of quantum structures. We proceed by introducing a simple mechanistic macroscopic experimental situation that gives rise to quantum-like structures. We use this situation as a guiding example for our attempts to explain the origin of the nonclassical aspects of quantum structures. We see that the quantum probabilities can be introduced as a consequence of the presence of fluctuations on the experimental apparatuses, and show that the full quantum structure can be obtained in this way. We define the classical limit as the physical situation that arises when the fluctuations on the experiment apparatuses disappear. In the limit case we come to a classical structure, but in between we find structures that are neither quantum nor classical. In this sense, our approach not only gives an explanation for the nonclassical structure of quantum theory, but also makes it possible to define and study the structure describing the intermediate new situations. By investigating how the nonlocal quantum behavior disappears during the limiting process, we can explain theapparentlocality of the classical macroscopic world. We come to the conclusion that quantum structures are the ordinary structures of reality, and that our difficulties of becoming aware of this fact are due to prescientific prejudices, some of which we point out.     

Uncertainty features of microscopic particles described by nonlinear Schrüdinger equation

The uncertainty relationship between position and momentum of the microscopic particles is calculated by nonlinear quantum theory in which the states of the particles are described by a nonlinear Schrüdinger equation. The results show that the uncertainty relation differs from that in the quantum mechanics and has a minimum value in this case. This means that the position and momentum of the particles could be determined simultaneously to a certain degree, which could be caused by the wave–corpuscle duality of the microscopic particles described by the nonlinear Schrüdinger equation.     

Fractals and quantum mechanics

A new application of a fractal concept to quantum physics has been developed. The fractional path integrals over the paths of the Levy flights are defined. It is shown that if fractality of the Brownian trajectories leads to standard quantum mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics. The fractional quantum mechanics has been developed via the new fractional path integrals approach. A fractional generalization of the Schrodinger equation has been discovered. The new relationship between the energy and the momentum of the nonrelativistic fractional quantum-mechanical particle has been established, and the Levy wave packet has been introduced into quantum mechanics. The equation for the fractional plane wave function has been found. We have derived a free particle quantum-mechanical kernel using Fox's H-function. A fractional generalization of the Heisenberg uncertainty relation has been found. As physical applications of the fractional quantum mechanics we have studied a free particle in a square infinite potential well, the fractional "Bohr atom" and have developed a new fractional approach to the QCD problem of quarkonium. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum mechanics. (c) 2000 American Institute of Physics.     

Admissible states in quantum phase space

We address the question of which phase space functionals might represent a quantum state. We derive necessary and sufficient conditions for both pure and mixed phase space quantum states. From the pure state quantum condition we obtain a formula for the momentum correlations of arbitrary order and derive explicit expressions for the wave functions in terms of time-dependent and independent Wigner functions. We show that the pure state quantum condition is preserved by the Moyal (but not by the classical Liouville) time evolution and is consistent with a generic stargenvalue equation. As a by-product Baker's converse construction is generalized both to an arbitrary stargenvalue equation, associated to a generic phase space symbol, as well as to the time-dependent case. These results are properly extended to the mixed state quantum condition, which is proved to imply the Heisenberg uncertainty relations. Globally, this formalism yields the complete characterization of the kinematical structure of Wigner quantum mechanics. The previous results are then succinctly generalized for various quasi-distributions. Finally, the formalism is illustrated through the simple examples of the harmonic oscillator and the free Gaussian wave packet. As a by-product, we obtain in the former example an integral representation of the Hermite polynomials.     

Gibbs ensembles in relativistic classical and quantum mechanics

Classical and quantum Gibbs ensembles are constructed for equilibrium statistical mechanics in the framework of an extension to many-body theory of a relativistic mechanics proposed by Stueckelberg. In addition to the usual chemical potential in the grand canonical ensemble, there is a new potential corresponding to the mass degree of freedom of relativistic systems. It is shown that in the nonrelativistic limit the relativistic ensembles we have obtained reduce to the usual ones, and mass fluctuations for the free-particle gas approach the fluctuations in N. The ultrarelativistic limit of the canonical ensemble for the free-particle gas differs from the corresponding limit of the ensemble proposed by Jüttner and Pauli. Due to the mass degree of freedom, the quantum counting of states is different from that of the nonrelativistic theory. If the mass distribution is sufficiently sharp, the thermodynamical effects of this multiplicity will not be large. There may, however, be detectable effects such as a shift in the Fermi level and the critical temperature for Bose-Einstein condensation, and some change in specific heats.     

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

The fundamental equations of equilibrium quantum statistical mechanics are derived in the context of a measure-theoretic approach to the quantum mechanical ergodic problem. The method employed is an extension, to quantum mechanical systems, of the techniques developed by R. M. Lewis for establishing the foundations of classical statistical mechanics. The existence of a complete set of commuting observables is assumed, but no reference is made a priori to probability or statistical ensembles. Expressions for infinite-time averages in the microcanonical, canonical, and grand canonical ensembles are developed which reduce to conventional quantum statistical mechanics for systems in equilibrium when the total energy is the only conserved quantity. No attempt is made to extend the formalism at this time to deal with the difficult problem of the approach to equilibrium.     

Wave function of the quantum black hole

We show that the Wald Noether-charge entropy is canonically conjugate to the opening angle at the horizon. Using this canonical relation, we extend the Wheeler–DeWitt equation to a Schrödinger equation in the opening angle, following Carlip and Teitelboim. We solve the equation in the semiclassical approximation by using the correspondence principle and find that the solutions are minimal uncertainty wavefunctions with a continuous spectrum for the entropy and therefore also of the area of the black hole horizon. The fact that the opening angle fluctuates away from its classical value of 2π indicates that the quantum black hole is a superposition of horizonless states. The classical geometry with a horizon serves only to evaluate quantum expectation values in the strict classical limit.     

Effective classical stochastic theory for quantum tunneling

The ensemble mean equations for a classical particle moving stochastically obtain the form of fluid equations. When applying the Madelung transformation to write the Schrödinger equation in a fluid-like form we find that the equations are equivalent to the classical ensemble mean equations if an additional force is added to the equations. The latter can be expressed as a pressure gradient force of a fluctuating pressure with zero mean. Here we analyze the mechanism of quantum tunneling through a rectangular potential barrier from this perspective. We find that despite of the vanishing of the mean of the pressure fluctuations their local non zero gradients enable the tunneling by balancing the counter external potential gradients at the two sides of the potential barrier. Consequently, for stationary solutions, the ensemble mean kinetic energy remains unchanged across the boundaries of the barrier.     

The motion of wavelets—An interpretation of the Schrödinger equation

There are stable wavelets which satisfy the Schrödinger equation. The motion of a wavelet is determined by a set of ordinary differential equations. In a certain limit, a wavelet turns out to be the known representation of a classical material point. A de Broglie wave is constructed by superposing similar free wavelets. Conventional energy eigensolutions of the Schrödinger equation can be interpreted as ensembles of wavelets. If the dynamics of wavelets form the quantum mechanical counterpart of Newton's dynamics of particles, then conventional quantum mechanics is the counterpart of Gibbs's mechanics of ensembles. In this way, conventional quantum mechanics is reinterpreted on a deterministic basis. A difficulty of quantum field theory is predictable from this point of view.     

Generalized coherent states for solvable quantum systems with degenerate discrete spectra and their nonclassical properties

In this paper, the generalized coherent state for quantum systems with degenerate spectra is introduced. Then, the nonclassicality features and number-phase entropic uncertainty relation of two particular degenerate quantum systems are studied. Finally, using the Gazeau-Klauder coherent states approach, the time evolution of some of the nonclassical properties of the coherent states corresponding to the considered physical systems are discussed.     

Observation of quantum particles on a large space-time scale

A quantum particle observed on a sufficiently large space-time scale can be described by means of classical particle trajectories. The joint distribution for large-scale multiple-time position and momentum measurements on a nonrelativistic quantum particle moving freely inR ^v is given by straight-line trajectories with probabilities determined by the initial momentum-space wavefunction. For large-scale toroidal and rectangular regions the trajectories are geodesics. In a uniform gravitational field the trajectories are parabolas. A quantum counting process on free particles is also considered and shown to converge in the large-space-time limit to a classical counting process for particles with straight-line trajectories. If the quantum particle interacts weakly with its environment, the classical particle trajectories may undergo random jumps. In the random potential model considered here, the quantum particle evolves according to a reversible unitary one-parameter group describing elastic scattering off static randomly distributed impurities (a quantum Lorentz gas). In the large-space-time weak-coupling limit a classical stochastic process is obtained with probability one and describes a classical particle moving with constant speed in straight lines between random jumps in direction. The process depends only on the ensemble value of the covariance of the random field and not on the sample field. The probability density in phase space associated with the classical stochastic process satisfies the linear Boltzmann equation for the classical Lorentz gas, which, in the limith0, goes over to the linear Landau equation. Our study of the quantum Lorentz gas is based on a perturbative expansion and, as in other studies of this system, the series can be controlled only for small values of the rescaled time and for Gaussian random fields. The discussion of classical particle trajectories for nonrelativistic particles on a macroscopic spacetime scale applies also to relativistic particles. The problem of the spatial localization of a relativistic particle is avoided by observing the particle on a sufficiently large space-time scale.     

Electron Rydberg wave packets in one-dimensional atoms

An expression for the transition probability or form factor in one-dimensional Rydberg atom irradiated by short half-cycle pulse was constructed. In applicative contexts, our expression was found to be more useful than the corresponding result given by Landau and Lifshitz. Using the new expression for the form factor, the motion of a localized quantum wave packet was studied with particular emphasis on its revival and super-revival properties. Closed form analytical expressions were derived for expectation values of the position and momentum operators that characterized the widths of the position and momentum distributions. Transient phase-space localization of the wave packet produced by the application of a single impulsive kick was explicitly demonstrated. The undulation of the uncertainty product as a function of time was studied in order to visualize how the motion of the wave packet in its classical trajectory spreads throughout the orbit and the system becomes nonclassical. The process, however, repeats itself such that the atom undergoes a free evolution from a classical, to a nonclassical, and back to a classical state.