arXiv subjects

Donald J. Kouri

Publications and source records attributed to Donald J. Kouri.

11 recordsLinked to original sources

arXiv2024

Coupled Supersymmetry and Ladder Structures Beyond the Harmonic Oscillator

The development of supersymmetric (SUSY) quantum mechanics has shown that some of the insights based on the algebraic properties of ladder operators related to the quantum mechanical harmonic oscillator carry over to the study of more general systems. At this level of generality, pairs of eigenfunctions of so-called partner Hamiltonians are transformed into each other, but the entire spectrum of any one of them cannot be deduced from this intertwining relationship in general -- except in special cases. In this paper, we present a more general structure that provides all eigenvalues for a class of Hamiltonians that do not factor into a pair of operators satisfying canonical commutation relations. Instead of a pair of partner Hamiltonians, we consider two pairs that differ by an overall shift in their spectrum. This is called coupled supersymmetry. In that case, we also develop coherent states and present some uncertainty principles which generalize the Heisenberg uncertainty principle. Coupled SUSY is explicitly realized by an infinite family of differential operators which admit orthonormal bases of eigenfunctions of generalized harmonic oscillators.

math-ph↗

arXiv2021

From the Harmonic Oscillator to Time-Frequency Analysis of Chirp Signals

This paper presents a novel approach to understanding the role of harmonic dynamics and gaining a deeper appreciation for its impact within and outside of quantum mechanics. This includes consequences of harmonic dynamics and the uncertainty principle for anomalous diffusion and for the time-frequency analysis of chirp signals. In this approach, we consider a contact transformation to view a system of canonical variables with coordinate $x$ and momentum $p_x$ in the context of a new system of "generalized" coordinates and momentum. This new system is first studied in the context of non-relativistic quantum mechanics. The classical analog is then explored by use of the Poisson bracket equation. From this, new implications are demonstrated in classical phenomena. One is for a new model of Anomalous and Normal Diffusion. In another, we introduce the concept of the "Mixed Fourier Transform" which explores a new Gaussian Fourier Transform kernel in terms of the generalized variables. This has the ultimate objective of "harmonizing" chirp signals or producing a harmonic signal from an otherwise non-harmonic chirp.

quant-ph↗

arXiv2019

Generalized Fourier transform method for nonlinear anomalous diffusion equation

The solution of a nonlinear diffusion equation is numerically investigated using the generalized Fourier transform method. This equation includes fractal dimensions and power-law dependence on the radial variable and on the diffusion function. The generalized Fourier transform approach is the extension of the Fourier transform method used for normal diffusion equation. The feasibility of the approach is validated by comparing the numerical result with the exact solution for point-source. The merit of numerical method is that it provide a way to calculate anomalous diffusion with an arbitrary initial condition.

physics.comp-ph↗

arXiv2018

Point Transformations and the Relationships Among Anomalous Diffusion, Normal Diffusion and the Central Limit Theorem

We present new connections among anomalous diffusion (AD), normal diffusion (ND) and the Central Limit Theorem. This is done by defining a point transformation to a new position variable, which we postulate to be Cartesian, motivated by considerations from super-symmetric quantum mechanics. Canonically quantizing in the new position and momentum variables according to Dirac gives rise to generalized negative semi-definite and self-adjoint Laplacian operators. These lead to new generalized Fourier transformations and associated probability distributions, which are form invariant under the corresponding transform. The new Laplacians also lead us to generalized diffusion equations, which imply a connection to the CLT. We show that the derived diffusion equations capture all of the Fractal and Non-Fractal Diffusion equations of O'Shaughnessy and Procaccia. However, we also obtain new equations that cannot (so far as we are able to tell) be expressed as examples of the O'Shaughnessy and Procaccia equations. These equations also possess asymptotics that are related to a CLT but with bi-modal distributions as limits. The results show, in part, that experimentally measuring the diffusion scaling law can determine the point transformation (for monomial point transformations). We also show that AD in the original, physical position is actually ND when viewed in terms of displacements in an appropriately transformed position variable. Finally, we show that there is a new, anomalous diffusion possible for bi-modal probability distributions that also display attractor behavior which is the consequence of an underlying CLT.

math-ph↗

arXiv2017

Automatic picking of P and S arrivals using a minimal uncertainty wavelet approach

The so-called "minimum uncertainty" or mu-wavelet approach has been shown to be very effective. Prior work has shown that using special algorithms based on a mu-wavelet decomposition of a seismic signal allows for arrival detection of various waves on a single-component seismogram with good precision. However, the question of interpretation of the type of wave arriving has not been addressed. The problem lies in the fact that the algorithm detects more than two arrivals even on seismograms with little noise, the peaks corresponding to P and S wave arrivals on the indicator function are not always the strongest, and the peak corresponding to the P wave arrival is not always the first peak detected. We propose to use a signal/noise relationship function in a running window as a weight coefficient of the indicator function obtained using the mu-wavelet transformation, which allows to isolate specifically those arrivals which correspond to significant changes in the signal amplitude. To interpret wave arrivals on linear registration systems (well observation systems for monitoring hydrolic fracturing, ultrasound investigations on models and samples of geological material), an algorithm was also developed to isolate arrivals that best correspond to arrivals on all registration points that have observation systems. As a result, we propose an algorithm that allows to automatically isolate arrivals of S and P waves on "frac" event recordings of a well observation system, and provides agreement with manual detection that doesn't exceed four discretes.

physics.geo-ph↗

arXiv2017

Canonical Quantization, Quasi-Hermiticity, Observables and the Construction of Complete Basis Sets

We consider the problem of designing a variety of "system guided" basis sets for quantum mechanical anharmonic oscillators. Using ideas based on supersymmetric quantum mechanics, we design canonical transformations of the usual position and momentum to generate generalized "Cartesian-like positions, W and momenta, p_W" with unit Poisson brackets. These are quantized following Dirac, leading to an infinite family of potential "operator observables". The fundamental issue is that all but one of the operators are not Hermitian (formally self-adjoint) in the original position representation. We show that the resulting operators are "quasi-Hermitian" relative to the x-representation and that all are Hermitian in the W-representation. Depending on how one treats the Jacobian of the canonical transformation in the expression for the classical momentum, p_W, quantization yields a) continuous mutually unbiased bases (MUB) b) orthogonal bases (with Dirac delta normalization) c) biorthogonal bases (with Dirac delta normalization) d) new W-harmonic oscillators yielding standard orthonormal bases (as functions of W) and associated coherent states. The MUB include W-generalized Fourier transform kernels whose eigenvectors are the W-harmonic oscillator eigenstates, with the spectrum (p/m i, p/m i). The W, p_W satisfy the uncertainty product relation: Delta-W * Delta-p_W gte 1/2.

quant-ph↗

arXiv2016

Fourier and Beyond: Invariance Properties of a Family of Integral Transforms

The Fourier transform is typically seen as closely related to the additive group of real numbers, its characters and its Haar measure. In this paper, we propose an alternative viewpoint; the Fourier transform can be uniquely characterized by an intertwining relation with dilations and by having a Gaussian as an eigenfunction. This broadens the perspective to an entire family of Fourier-like transforms that are uniquely identified by the same dilation property and having Gaussian-like functions as eigenfunctions. We show that these transforms share many properties with the Fourier transform, particularly unitarity, periodicity and eigenvalues. We also establish short-time analogues of these transforms and show a reconstruction property and an orthogonality relation for the short-time transforms.

math.FA↗

arXiv2015

Harmonic Oscillators, Heisenberg's Uncertainty Principle and Simultaneous Measurement Precision for Position and Momentum

At the recent QSCP XIX, the author claimed a procedure of using a scaled Fourier transform (the scaling being determined by the detailed interaction and particle mass for a harmonic oscillator) to achieve simultaneous resolution of position and momentum greater than the standard Heisenberg value of 1/2. The procedure is, in fact, invalid for quantum mechanics. The purpose of this paper is simply to give the correct analysis of the uncertainty product, thereby clarifying the error made.

quant-ph↗

arXiv2010

Quantum dynamics and super-symmetric quantum mechanics

In my talk I will present an overview of our recent work involving the use of supersymmetric quantum mechanics (SUSY-QM). I begin by discussing the mathematical underpinnings of SUSY-QM and then discuss how we have used this for developing novel theoretical and numerical approaches suitable for studying molecular systems. I will conclude by discussing our attempt to extend SUSY-QM to multiple dimensions.

quant-ph↗

arXiv2009

A Supersymmetric Approach to Excited States via Quantum Monte Carlo

We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schrödinger equation. Since the ground-state of each partner potential is node-less, we avoid any ``node''-problem typically associated with the Monte Carlo technique. While we provide an example of using this approach to determine the tunneling states in a double-well potential, the method is applicable to any 1D potential problem. We conclude by discussing the extension to higher dimensions.

hep-th↗

arXiv2000

Characterization of the Emergence of Order in an Oscillated Granular Layer

The formation of textured patterns has been predicted to occur in two stages. The first is an early time, domain-forming stage with dynamics characterized by a disorder function $\barδ(β) \sim t^{-σ_{E}}$, with $σ_{E} = {1/2}β$; this decay is universal. Coarsening of domains occurs in the second stage, in which $\barδ(β) \sim t^{-σ_{L}}$, where $σ_{L}$ is a nonlinear function of $β$ whose form is system and model dependent. Our experiments on a vertically oscillated granular layer are in accord with theory, yielding $σ_{E}\approx 0.5β$, and $σ_{L}$ a nonlinear function of $β$.

nlin.PS↗