arXiv subjects

A. J. Bracken

Publications and source records attributed to A. J. Bracken.

At least 19 recordsLinked to original sources

arXiv2020

Solutions of Painlevé II on real intervals: novel approximating sequences

Novel sequences of approximants to solutions of Painlevé II on finite intervals of the real line, with Neumann boundary conditions, are constructed. Numerical experiments strongly suggest convergence of these sequences in a surprisingly wide range of cases, even ones where ordinary perturbation series fail to converge. These sequences are here labeled extraordinary because of their unusual properties. Each element of such a sequence is defined on its own interval. As the sequence (apparently) converges to a solution of the corresponding boundary value problem for Painlevé II, these intervals themselves (apparently) converge to the defining interval for that problem, and an associated sequence of constants (apparently) converges to the constant term in the Painlevé II equation itself. Each extraordinary sequence is constructed in a nonlinear fashion from a perturbation series approximation to the solution of a supplementary boundary value problem, involving a generalization of Painlevé II that arises in studies of electrodiffusion.

math-ph↗

arXiv2017

Airy series solution of Painlevé II in electrodiffusion: conjectured convergence

A perturbation series solution is constructed in terms of Airy functions for a nonlinear two-point boundary-value problem arising in an established model of steady electrodiffusion in one dimension, for two ionic species carrying equal and opposite charges. The solution includes a formal determination of the associated electric field, which is known to satisfy a form of the Painlevé II differential equation. Comparisons with the numerical solution of the boundary-value problem show excellent agreement following termination of the series after a sufficient number of terms, for a much wider range of values of the parameters in the model than suggested by previously presented analysis, or admitted by previously presented approximation schemes. These surprising results suggest that for a wide variety of cases, a convergent series expansion is obtained in terms of Airy functions for the Painlevé transcendent describing the electric field. A suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases. It is conjectured that the series does converge for this subset.

math.CA↗

arXiv2016

Differential equations of electrodiffusion: constant field solutions, uniqueness, and new formulas of Goldman-Hodgkin-Katz type

The equations governing one-dimensional, steady-state electrodiffusion are considered when there are arbitrarily many mobile ionic species present, in any number of valence classes, possibly also with a uniform distribution of fixed charges. Exact constant field solutions and new formulas of Goldman-Hodgkin-Katz type are found. All of these formulas are exact, unlike the usual approximate ones. Corresponding boundary conditions on the ionic concentrations are identified. The question of uniqueness of constant field solutions with such boundary conditions is considered, and is re-posed in terms of an autonomous ordinary differential equation of order $n+1$ for the electric field, where $n$ is the number of valence classes. When there are no fixed charges, the equation can be integrated once to give the non-autonomous equation of order $n$ considered previously in the literature including, in the case $n=2$, the form of Painlevé's second equation considered first in the context of electrodiffusion by one of us. When $n=1$, the new equation is a form of Liénard's equation. Uniqueness of the constant field solution is established in this case.

q-bio.QM↗

arXiv2016

Remarks on Quantum Probability Backflow

It is known that for a non-relativistic quantum particle traveling freely on the $x$-axis, the positional probability can flow in the opposite direction to the particle's velocity. The maximum possible amount of such backflow that can occur over any time interval has been determined previously as the largest positive eigenvalue of a certain hermitian observable, with the value $0.0384517\dots$, or about $4\%$ of the total probability on the line. The eigenvalue problem is now considered numerically in the more general case of states with momentum restricted to the range $p_0<p<\infty$, for any given value $p_0$. It is found that the maximum possible backflow decreases monotonically, but never reaches $0$, as $p_0$ increases through positive values; and it increases monotonically, but never reaches $1$, as $p_0$ decreases through negative values. Both of these effects are non-classical. The results allow a simple interpretation of the classical limit, as an effective value of Planck's constant goes to zero and probability backflow becomes impossible.

quant-ph↗

arXiv2014

Waiting for the Quantum Bus

Forty-five years after the discovery of the peculiar quantum effect known as `probability backflow', and twenty years after the greatest possible size of the effect was characterized, an experiment has been proposed recently to observe the effect in a Bose-Einstein condensate. Here the history is described in non-technical terms.

quant-ph↗

arXiv2012

Massless Dirac particle in a stochastic magnetic field: A solvable quantum walk approximation

A massless Dirac particle is considered, moving along the x-axis while Pauli-coupled by its anomalous magnetic moment to a piecewise constant magnetic field along the same axis, with stochastically varying sign. The motion is approximated as a quantum walk with unitary noise, for which the evolution can be found exactly. Initially ballistic, the motion approaches a classical diffusion on a time-scale determined by the speed of light, the size of the magnetic moment, the strength of the field and the time interval between changes in its direction. It is suggested that a process of this type could occur in the Sun's corona, significantly affecting the solar fluxes of one or more neutrino types.

quant-ph↗

arXiv2012

Discrete Randomness in Discrete Time Quantum Walk: Study via Stochastic Averaging

The role of classical noise in quantum walks (QW) on integers is investigated in the form of discrete dichotomic random variable affecting its reshuffling matrix parametrized as a SU2)/U(1) coset element. Analysis in terms of quantum statistical moments and generating functions, derived by the completely positive trace preserving (CPTP) map governing evolution, reveals a pronounced eventual transition in walk's diffusion mode, from a quantum ballistic regime with rate O(t) to a classical diffusive regime with rate O(\surdt), when condition (strength of noise parameter)^{2}\times(number of steps)=1, is satisfied. The role of classical randomness is studied showing that the randomized QW, when treated on the stochastic average level by means of an appropriate CPTP averaging map, turns out to be equivalent to a novel quantized classical walk without randomness. This result emphasizes the dual role of quantization/randomization in the context of classical random walk.

quant-ph↗

arXiv2012

Bäcklund flux-quantization in a model of electrodiffusion based on Painlevé II

A previously-established model of steady one-dimensional two-ion electrodiffusion across a liquid junction is reconsidered. It involves three coupled first-order nonlinear ordinary differential equations, and has the second-order Painlevé II equation at its core. Solutions are now grouped by Bäcklund transformations into infinite sequences, partially labelled by two Bäcklund invariants. Each sequence is characterized by evenly-spaced quantized fluxes of the two ionic species, and hence evenly-spaced quantization of the electric current-density. Finite subsequences of exact solutions are identified, with positive ionic concentrations and quantized fluxes, starting from a solution with zero electric field found by Planck, and suggesting an interpretation as a ground state plus excited states of the system. Positivity of ionic concentrations is established whenever Planck's charge-neutral boundary-conditions apply. Exact solutions are obtained for the electric field and ionic concentrations in well-stirred reservoirs outside each face of the junction, enabling the formulation of more realistic boundary-conditions. In an approximate form, these lead to radiation boundary conditions for Painlevé II. Illustrative numerical solutions are presented, and the problem of establishing compatibility of boundary conditions with the structure of flux-quantizing sequences is discussed.

math-ph↗

arXiv2012

Planck's Other Quanta: Corpuscular Electrodiffusion

When Bäcklund transformations are applied repeatedly to any solution of the nonlinear equations describing electrodiffusion through a liquid junction separating two infinite well-stirred layers, they give rise to quantized ionic fluxes across the junction. A particular exact solution is shown to imply that these fluxes consist of quanta of electric charge.

math-ph↗

arXiv2010

The quantum state vector in phase space and Gabor's windowed Fourier transform

Representations of quantum state vectors by complex phase space amplitudes, complementing the description of the density operator by the Wigner function, have been defined by applying the Weyl-Wigner transform to dyadic operators, linear in the state vector and anti-linear in a fixed `window state vector'. Here aspects of this construction are explored, with emphasis on the connection with Gabor's `windowed Fourier transform'. The amplitudes that arise for simple quantum states from various choices of window are presented as illustrations. Generalized Bargmann representations of the state vector appear as special cases, associated with Gaussian windows. For every choice of window, amplitudes lie in a corresponding linear subspace of square-integrable functions on phase space. A generalized Born interpretation of amplitudes is described, with both the Wigner function and a generalized Husimi function appearing as quantities linear in an amplitude and anti-linear in its complex conjugate. Schrödinger's time-dependent and time-independent equations are represented on phase space amplitudes, and their solutions described in simple cases.

quant-ph↗

arXiv2006

Free Dirac evolution as a quantum random walk

Any positive-energy state of a free Dirac particle that is initially highly-localized, evolves in time by spreading at speeds close to the speed of light. This general phenomenon is explained by the fact that the Dirac evolution can be approximated arbitrarily closely by a quantum random walk, where the roles of coin and walker systems are naturally attributed to the spin and position degrees of freedom of the particle. Initially entangled and spatially localized spin-position states evolve with asymptotic two-horned distributions of the position probability, familiar from earlier studies of quantum walks. For the Dirac particle, the two horns travel apart at close to the speed of light.

quant-ph↗

arXiv2006

What is semiquantum mechanics?

Semiclassical approximations to quantum dynamics are almost as old as quantum mechanics itself. In the approach pioneered by Wigner, the evolution of his quasiprobability density function on phase space is expressed as an asymptotic series in increasing powers of Planck's constant, with the classical Liouvillean evolution as leading term. Successive semiclassical approximations to quantum dynamics are defined by successive terms in the series. We consider a complementary approach, which explores the quantum-clssical interface from the other direction. Classical dynamics is formulated in Hilbert space, with the Groenewold quasidensity operator as the image of the Liouville density on phase space. The evolution of the Groenewold operator is then expressed as an asymptotic series in increasing powers of Planck's constant. Successive semiquantum approximations to classical dynamics are defined by successive terms in this series, with the familiar quantum evolution as leading term.

quant-ph↗

arXiv2005

Semiquantum versus semiclassical mechanics for simple nonlinear systems

Quantum mechanics has been formulated in phase space, with the Wigner function as the representative of the quantum density operator, and classical mechanics has been formulated in Hilbert space, with the Groenewold operator as the representative of the classical Liouville density function. Semiclassical approximations to the quantum evolution of the Wigner function have been defined, enabling the quantum evolution to be approached from a classical starting point. Now analogous semiquantum approximations to the classical evolution of the Groenewold operator are defined, enabling the classical evolution to be approached from a quantum starting point. Simple nonlinear systems with one degree of freedom are considered, whose Hamiltonians are polynomials in the Hamiltonian of the simple harmonic oscillator. The behaviour of expectation values of simple observables and of eigenvalues of the Groenewold operator, are calculated numerically and compared for the various semiclassical and semiquantum approximations.

quant-ph↗

arXiv2005

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

quant-ph↗

arXiv2005

Massive neutrinos, massless neutrinos, and so(4,2)invariance

Dirac's equation for a massless particle is conformal invariant, and accordingly has an so(4,2)invariance algebra. It is known that although Dirac's equation for a massive spin 1/2 particle is not conformal invariant, it too has an so(4,2) invariance algebra. It is shown here that the algebra of operators associated with a 4-component massless particle, or two flavors of 2-component massless particles, can be deformed into the algebra of operators associated with a spin 1/2 particle with positive rest mass. It is speculated that this may be exploited to describe massless neutrino mixing.

hep-th↗

arXiv2004

Bounds on integrals of the Wigner function: the hyperbolic case

Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue problem and that the bounds differ from those on classical Liouville distributions. In particular, we show that the total ``quasiprobability'' on such a region can be greater than 1 or less than zero.

quant-ph↗

arXiv2004

Non-positivity of Groenewold operators

A central feature in the Hilbert space formulation of classical mechanics is the quantisation of classical Liouville densities, leading to what may be termed term Groenewold operators. We investigate the spectra of the Groenewold operators that correspond to Gaussian and to certain uniform Liouville densities. We show that when the classical coordinate-momentum uncertainty product falls below Heisenberg's limit, the Groenewold operators in the Gaussian case develop negative eigenvalues and eigenvalues larger than 1. However, in the uniform case, negative eigenvalues are shown to persist for arbitrarily large values of the classical uncertainty product.

quant-ph↗

arXiv2003

One dimensional quantum walk with unitary noise

The effect of unitary noise on the discrete one-dimensional quantum walk is studied using computer simulations. For the noiseless quantum walk, starting at the origin (n=0) at time t=0, the position distribution Pt(n) at time t is very different from the Gaussian distribution obtained for the classical random walk. Furthermore, its standard deviation, sigma(t) scales as sigma(t) ~ t, unlike the classical random walk for which sigma(t) ~ sqrt{t}. It is shown that when the quantum walk is exposed to unitary noise, it exhibits a crossover from quantum behavior for short times to classical-like behavior for long times. The crossover time is found to be T ~ alpha^(-2) where alpha is the standard deviation of the noise.

quant-ph↗