Position Probability Density Research Articles

The Newton-Wigner and Wightman localization of the photon

A quantum theory of the photon is developed in a natural manner. Newton-Wigner and Wightman demonstrated that the photon could not be strictly localized according to natural criteria. These investigations involved the identification of an elementary system with a uirrep of the Poincare group. We identify a particle with the localized measurement of the states satisfying the uirrep. In the case of zero mass and unit spin, the photon is identified with those components of the state that can be localized. A c-number four-vector potential and Lorentz condition are derived from the relativistic wave equation. The Wightman localization is demonstrated for the three independent space components of the vector potential, and the photon is identified with these components. A position operator and probability density follow immediately from the localization. A consequence of the subjective definition of a photon is that the transformations of the vector potential are unitary, and hence the unitary scalar product can be obtained for the four-vector potential. A Hilbert space is defined for the three space components of the vector potential. A position operator and probability density are derived from the scalar product, which compare directly with those obtained from the localization and the non-relativistic theory. As the longitudinal and scalar polarizations do not contribute to the measured transition probability, they are considered virtual. Lastly, a conserved four-vector current is derived from the scalar product. The possibility of observing a strict localization of the photon in the laboratory is suggested.

Feynman–Kac kernels in Markovian representations of the Schrödinger interpolating dynamics

Probabilistic solutions of the so-called Schrödinger boundary data problem provide for a unique Markovian interpolation between any two strictly positive probability densities designed to form the input–output statistics data for the process taking place in a finite-time interval. The key issue is to select the jointly continuous in all variables positive Feynman–Kac kernel, appropriate for the phenomenological (physical) situation. We extend the existing formulations of the problem to cases when the kernel is not a fundamental solution of a parabolic equation, and prove the existence of a continuous Markovian interpolation in this case. Next, we analyze the compatibility of this stochastic evolution with the original parabolic dynamics, which is assumed to be governed by the temporally adjoint pair of (parabolic) partial differential equations, and prove that the pertinent random motion is a diffusion process. In particular, in conjunction with Born’s statistical interpretation postulate in quantum theory, we consider stochastic processes which are compatible with the Schrödinger picture quantum evolution.

Direct-Sequence Spread-Spectrum Receiver-Code Acquisition by Envelope Correlator

In this paper, the problem of recovering information from a receiver by Code-Division Multiple Access (CDMA) is investigated. To recover information from a spreading code modulation, despreading is performed by applying a replica of the original spreading code signal. If there is no apriori information about the time position of the spreading code modulation at the receiver, the probability density of the true position can be takeen to have a uniform distribution. A code synchronization process is employed to recover the original information.

Ergodic mixing for turbulent drift motion in an inhomogeneous magnetic field

The turbulent E×B drift of a test particle in an inhomogeneous magnetic field is not reducible to a simple diffusion, but rather leads to a biased diffusion producing an inhomogeneous density distribution (pinch effect). The statistical properties of the long-time chaotic two-dimensional drift motion of a charged particle in the magnetic field B(x,y) and the time-dependent electrostatic potential φ(x,y,t) are studied by numerical symplectic integration. For a conditionally periodic potential with two or more incommensurate frequencies, an ergodic behavior is demonstrated in which the probability density of the particle position is proportional to the magnetic field B. The accuracy of this prediction is found to be independent of the number Nω of the incommensurate frequencies for Nω≥2. The relation of this result with the Kolmogorov-Arnold-Moser theory is discussed.

Positive definite probability densities and probability distributions

In the first short survey article [H.-J. Rossberg,Wiss. Z. Karl Marx Univ.,37, 366–374 (1988)] the author indicated that positive definite probability densities deserve attention and that the investigation of this class leads to serious new mathematical problems. Later [R. Riedel and H.-J. Rossberg,Metrika, (submitted for publication)] it turned out that it is worthwhile to consider an even larger class of distribution functions, namely those having characteristic function f≥0; they were first briefly considered in [P. Levy,C. R. Acad. Sci.,265, 249–252 (1967)]. We survey the basic ideas and the present state of the developing new theory; moreover, we formulate open problems. Some new results are proved. For the sake of brevity we do not enter into a discussion of continuation problems and limit theorems; for these subjects see [H.-J. Rossberg,Appendix to Gnedenko B. W., Einfuhrung in die Wahrscheinlichkeitstheorie, 9th ed., Akademie-Verlag, Berlin (1991)], [H.-J. Rossberg,Math. Nachr.,141, 227–232 (1989)], and [H.-J. Rossberg,Lect. Notes Math.,1412, 296–308 (1989)].

Non-existence of positive phase-space distribution functions in quantum mechanics

We show that in space dimension ≥2, there cannot exist a positive definite phase-space distribution function whose marginals coincide with the quantum position probability density, momentum probability density and joint probability densities of commuting components of position and momentum.

CAUSAL QUANTUM MECHANICS TREATING POSITION AND MOMENTUM SYMMETRICALLY

De Broglie and Bohm formulated a causal quantum mechanics with a phase space density whose integral over momentum reproduces the position probability density of the usual statistical quantum theory. We propose a causal quantum theory with a joint probability distribution such that the separate probability distributions for position and momentum agree with the usual quantum theory. Unlike the Wigner distribution the suggested distribution is positive-definite and obeys the Liouville condition.

Position and contextuality in Bohm's causal completion of quantum mechanics

Abstract We argue that, in Bohm's formulation, the assumption stipulating the position probability density to be ψ2 is both necessary and sufficient for satisfying the requirement of “noncontextuality” (absence of any dependence on the specifics of the measurement procedure) of the observable probabilities at the ensemble level. The reasons as to why position needs to have a special status in a causal completion of quantum mechanics are also discussed.

Analysis of the angular distribution of electrons injected through a quantum point contact by use of a Green’s function with a weak-magnetic-field approximation

Based on work by Molenkamp et al., we measured transfer efficiency through two quantum point contacts (QPC's) as a function of magnetic field applied perpendicularly to two-dimensional electron gas. We have reported that the results depend on the number of modes in the QPC. Previously, we had analyzed our results assuming classical mechanical trajectories of an electron. This time, to analyze our results, we studied the formulation of a Green's function using weak-magnetic-field approximation and constructed a mirror-image method in a weak magnetic field. Based on this, we calculated the position probability densities of an electron near the detector. The calculated probability densities agreed qualitatively with our results. Using quantum mechanics, we showed that our measured results equal the electron-wave diffraction pattern in a two-dimensional electron gas. We also showed that our experiments correspond to measurements of the angular distribution of electrons injected through a QPC.

Quantum beam collapse in a self-focusing medium

A qualitative theory of quantum beam collapse is given for an input coherent state whose mean power equals the critical power for self-focusing. The probability for beam collapse is shown to be 0.5, and the probability density for the axial position of the self-focusing singularity is derived.

Axial transport of granular solids in horizontal rotating cylinders. Part 1: Theory

A model for the axial transport of solids in a horizontal rotating cylinder is obtained. The physical basis of the model is a modified version of the single particle trajectory model of Saeman, which includes random deviations from the most probable trajectory. According to the model, each particle executes a one-dimensional random walk, and the stochastic evolution equation for the probability density of the axial position of the particle with time is derived. The average axial velocity (ν) and the axial dispersion coefficient ( D a) are obtained in the course of the derivation in terms of the system parameters, and an unknown parameter, δ, which represents the maximum deviation of the trajectory from the model of Saeman. The model predictions are compared with the experimental data of Hehl et al. Reasonable agreement is obtained between theory and experiments for ν, and the parameter δ is back-calculated from experimental values of D a. The results give a better understanding of the mechanisms of axial transport in rotary cylinders.

A posteriori probability source localization in an uncertain sound speed, deep ocean environment

The a posteriori source position probability density function for a narrow-band source in an uncertain acoustic environment is derived. The implementation of this probability density function (pdf) is termed the optimum uncertain field processor (OUFP). It is shown that the OUFP is a generalization of matched-field processing to situations in which there is uncertainty about the environment. The robustness of the OUFP is illustrated through performance comparisons to a matched-field algorithm.

Quantum Dynamics and Non-Inertial Frames of References. II

Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.

Electronic distribution, position probability density and 'clouds of points'

A method for visualising electronic densities is derived directly from the definition of position probability density. It is based on the generation of a random cloud of points so that the density of points at a given position is equal to the electronic density at that position. There are three main advantages to this approach: (a) it allows one to view the inner parts of the electronic distribution, (b) the rotation of the graph is easily accomplished using this method and (c) the overall visual impression is the closest that can be obtained to reality. A set of FORTRAN 77 subroutines is given, which has been applied for drawing hydrogen-like and hybrid orbitals.

A new method for the computation of particulate dispersion in turbulent two-phase flows

This article describes a new and powerful method for the calculation of the dispersion of a particulate phase in a turbulent flow. It is expected to benefit significantly the mathematical modeling of oil- and coal-fired combustors and of the spread of particles in atmospheric flows. Existing predictive codes rely almost exclusively on a stochastic treatment of the particle-turbulence interaction in which the trajectories of particles of representative sizes emerging from representative injection locations are computed. The problem is that more than 1000 particles from each size group and injection location must be tracked in order to determine meaningful position and velocity probability density functions (pdfs). When this is attempted a computer economy disaster is inevitable. In the new approach an equation, which correctly embodies particle momentum conservation, is derived for the evolution of the probability of particle velocity and by implication, its position. The pdf is discretized within the range of physically probable velocities. The equation is used to evolve the probability for each discretization from a known initial condition. In this manner the pdfs of the particle velocities and positions may be determined for all cells within the computational domain for a fraction of the cost of that of a conventional stochastic calculation. In this article the derivation of the pdf evolution equation is summarized. The validity and usefulness of the formulation are then demonstrated through its application to three test cases.

Position and quantum theory

It is customarily maintained that the concept of unmeasured (i.e., unobserved) position is without meaning in quantum theory. However, the vast literature on the subject, as shown here through the typical example of the double-slit experiment, really does not provide an iron-clad proof that this is indeed so. The question of the possible meaning of unobserved position is thus open. The problem is then resolved by showing quite rigorously that, after all, unobserved position is meaningless in quantum theory. The criterion for unobserved position to be meaningful in any theory (probabilistic or not) is taken to be that the probability density for position must satisfy the Einstein-Chapman-Kolmogorov equation with a positive-semidefinite kernel which is also properly normalized.

The Parseval Equation as a Tool for the Continuation of Characteristic Functions and Positive Definite Probability Densities

The Parseval Equation as a Tool for the Continuation of Characteristic Functions and Positive Definite Probability Densities

Critique and correction of the textbook comparison between classical and quantum harmonic oscillator probability densities

This comparison, although well meant, is unsuitable for demonstrating in introductory quantum mechanics the convergence of classical and quantum predictions for large quantum numbers, for it really tries to match ‘‘classical apples’’ with ‘‘quantum pears’’! First, it compares a classical oscillatory state with a quantum stationary state, instead of comparing it to the corresponding quantum nonstationary state. Second, it compares probabilities of different quality, namely, a classical probability of the classical oscillator (which could be circumvented by specifying the initial conditions precisely) with a quantum probability of the quantum oscillator (which cannot be avoided by any means). However, it is shown that both defects can be nicely corrected, illuminating the fact that—if properly formulated— classical and quantum oscillators have more in common than the textbooks are trying to make the beginner believe. The associated quantum position probability density, because it is now truly analogous to the classical situation, agrees much better with the classical curve than the textbook one does, and its analytical form also explains why the textbook quantum curve, derived from an unwarranted comparison, nevertheless resembles the classical curve in some sense.

The effect of a one-dimensional potential of finite range on the statistical parameters of an incident ensemble of particles. II. Localized states

If an ensemble of particles moving parallel to the x axis in the positive direction impinges on a piecewise continuous potential confined to the interval [−a,a] (a>0) it will divide into a transmitted ensemble and a reflected ensemble. It is shown that the classical results for the means, variances, and covariance of position and velocity of the transmitted and reflected ensembles hold in quantum mechanics if the position and momentum probability densities of the incident ensemble are assumed to be sufficiently localized, both in momentum and position. It is also assumed that the support of the probability density of momentum is bounded and positive, and on it the moduli and arguments of the transmission and reflection coefficients are sufficiently well behaved.

Anisotropy and anharmonicity of atomic fluctuations in proteins: analysis of a molecular dynamics simulation.

Positional probability density functions (pdf) for the atomic fluctuations are determined from a molecular dynamics simulation for hen egg-white lysozyme. Most atoms are found to have motions that are highly anisotropic but only slightly anharmonic. The largest deviations from harmonic motion are in the direction of the largest rms fluctuations in the local principal axis frame. Backbone atoms tend to be more nearly harmonic than sidechain atoms. The atoms with the largest anharmonicities tend to have pdfs with multiple peaks, each of which is close to harmonic. Several model pdfs are evaluated on the basis of how well they fit probability densities from the dynamics simulations when parameterized in terms of the moments of the distribution. Gram-Charlier and Edgeworth perturbation expansions, which have been successful in describing the motions of small molecules in crystals, are shown to be inadequate for the distributions found in the dynamics of proteins. Multipeaked distribution functions are found to be more appropriate.