A Concise Treatise on Quantum Mechanics in Phase Space
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Research Archive
http://dx.doi.org/10.1108/IJBM-05-2016-0075 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Hertfordshire Research Archive Research Archive Citation for published version: Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel, ‘Wigner's representation of quantum mechanics in integral form and its applications’, Physical Review A, Vol. 95, 022127, published 27 February 2017. DOI: https://doi.org/10.1103/PhysRevA.95.022127 Document Version: This is the Accepted Manuscript version. The version in the University of Hertfordshire Research Archive may differ from the final published version. Users should always cite the published version of record. Copyright and Reuse: ©2017 American Physical Society. This manuscript version is made available under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Enquiries If you believe this document infringes copyright, please contact the Research & Scholarly Communications Team at [email protected] Wigner's representation of quantum mechanics in integral form and its applications Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK (Dated: February 28, 2017) We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current J as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products. -
Weyl Quantization and Wigner Distributions on Phase Space
faculteit Wiskunde en Natuurwetenschappen Weyl quantization and Wigner distributions on phase space Bachelor thesis in Physics and Mathematics June 2014 Student: R.S. Wezeman Supervisor in Physics: Prof. dr. D. Boer Supervisor in Mathematics: Prof. dr H. Waalkens Abstract This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate. -
Phase-Space Description of the Coherent State Dynamics in a Small One-Dimensional System
Open Phys. 2016; 14:354–359 Research Article Open Access Urszula Kaczor, Bogusław Klimas, Dominik Szydłowski, Maciej Wołoszyn, and Bartłomiej J. Spisak* Phase-space description of the coherent state dynamics in a small one-dimensional system DOI 10.1515/phys-2016-0036 space variables. In this case, the product of any functions Received Jun 21, 2016; accepted Jul 28, 2016 on the phase space is noncommutative. The classical alge- bra of observables (commutative) is recovered in the limit Abstract: The Wigner-Moyal approach is applied to investi- of the reduced Planck constant approaching zero ( ! 0). gate the dynamics of the Gaussian wave packet moving in a ~ Hence the presented approach is called the phase space double-well potential in the ‘Mexican hat’ form. Quantum quantum mechanics, or sometimes is referred to as the de- trajectories in the phase space are computed for different formation quantization [6–8]. Currently, the deformation kinetic energies of the initial wave packet in the Wigner theory has found applications in many fields of modern form. The results are compared with the classical trajecto- physics, such as quantum gravity, string and M-theory, nu- ries. Some additional information on the dynamics of the clear physics, quantum optics, condensed matter physics, wave packet in the phase space is extracted from the analy- and quantum field theory. sis of the cross-correlation of the Wigner distribution func- In the present contribution, we examine the dynam- tion with itself at different points in time. ics of the initially coherent state in the double-well poten- Keywords: Wigner distribution, wave packet, quantum tial using the phase space formulation of the quantum me- trajectory chanics. -
P. A. M. Dirac and the Maverick Mathematician
Journal & Proceedings of the Royal Society of New South Wales, vol. 150, part 2, 2017, pp. 188–194. ISSN 0035-9173/17/020188-07 P. A. M. Dirac and the Maverick Mathematician Ann Moyal Emeritus Fellow, ANU, Canberra, Australia Email: [email protected] Abstract Historian of science Ann Moyal recounts the story of a singular correspondence between the great British physicist, P. A. M. Dirac, at Cambridge, and J. E. Moyal, then a scientist from outside academia working at the de Havilland Aircraft Company in Britain (later an academic in Australia), on the ques- tion of a statistical basis for quantum mechanics. A David and Goliath saga, it marks a paradigmatic study in the history of quantum physics. A. M. Dirac (1902–1984) is a pre- neering at the Institut d’Electrotechnique in Peminent name in scientific history. In Grenoble, enrolling subsequently at the Ecole 1962 it was my privilege to acquire a set Supérieure d’Electricité in Paris. Trained as of the letters he exchanged with the then a civil engineer, Moyal worked for a period young mathematician, José Enrique Moyal in Tel Aviv but returned to Paris in 1937, (1910–1998), for the Basser Library of the where his exposure to such foundation works Australian Academy of Science, inaugurated as Georges Darmois’s Statistique Mathéma- as a centre for the archives of the history of tique and A. N. Kolmogorov’s Foundations Australian science. This is the only manu- of the Theory of Probability introduced him script correspondence of Dirac (known to to a knowledge of pioneering European stud- colleagues as a very reluctant correspondent) ies of stochastic processes. -
Ballistic Electron Transport in Graphene Nanodevices and Billiards
Ballistic Electron Transport in Graphene Nanodevices and Billiards Dissertation (Cumulative Thesis) for the award of the degree Doctor rerum naturalium of the Georg-August-Universität Göttingen within the doctoral program Physics of Biological and Complex Systems of the Georg-August University School of Science submitted by George Datseris from Athens, Greece Göttingen 2019 Thesis advisory committee Prof. Dr. Theo Geisel Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Examination board Prof. Dr. Theo Geisel (Reviewer) Department of Nonlinear Dynamics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stephan Herminghaus (Second Reviewer) Department of Dynamics of Complex Fluids Max Planck Institute for Dynamics and Self-Organization Dr. Michael Wilczek Turbulence, Complex Flows & Active Matter Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Ulrich Parlitz Biomedical Physics Max Planck Institute for Dynamics and Self-Organization Prof. Dr. Stefan Kehrein Condensed Matter Theory, Physics Department Georg-August-Universität Göttingen Prof. Dr. Jörg Enderlein Biophysics / Complex Systems, Physics Department Georg-August-Universität Göttingen Date of the oral examination: September 13th, 2019 Contents Abstract 3 1 Introduction 5 1.1 Thesis synopsis and outline.............................. 10 2 Fundamental Concepts 13 2.1 Nonlinear dynamics of antidot superlattices..................... 13 2.2 Lyapunov exponents in billiards............................ 20 2.3 Fundamental properties of graphene......................... 22 2.4 Why quantum?..................................... 31 2.5 Transport simulations and scattering wavefunctions................ -
Weierstrass Transform on Howell's Space-A New Theory
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 6 Ser. IV (Nov. – Dec. 2020), PP 57-62 www.iosrjournals.org Weierstrass Transform on Howell’s Space-A New Theory 1 2* 3 V. N. Mahalle , S.S. Mathurkar , R. D. Taywade 1(Associate Professor, Dept. of Mathematics, Bar. R.D.I.K.N.K.D. College, Badnera Railway (M.S), India.) 2*(Assistant Professor, Dept. of Mathematics, Govt. College of Engineering, Amravati, (M.S), India.) 3(Associate Professor, Dept. of Mathematics, Prof. Ram Meghe Institute of Technology & Research, Badnera, Amravati, (M.S), India.) Abstract: In this paper we have explained a new theory of Weierstrass transform defined on Howell’s space. The definition of the Weierstrass transform on this space of test functions is discussed. Some Fundamental theorems are proved and some results are developed. Weierstrass transform defined on Howell’s space is linear, continuous and one-one mapping from G to G c is also discussed. Key Word: Weierstrass transform, Howell’s Space, Fundamental theorem. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 10-12-2020 Date of Acceptance: 25-12-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction A new theory of Fourier analysis developed by Howell, presented in an elegant series of papers [2-6] n supporting the Fourier transformation of the functions on R . This theory is similar to the theory for tempered distributions and to some extent, can be viewed as an extension of the Fourier theory. Bhosale [1] had extended Fractional Fourier transform on Howell’s space. Mahalle et. al. [7, 8] described a new theory of Mellin and Laplace transform defined on Howell’s space. -
Quantum Mechanics Propagator
Quantum Mechanics_propagator This article is about Quantum field theory. For plant propagation, see Plant propagation. In Quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. In Feynman diagrams, which calculate the rate of collisions in quantum field theory, virtual particles contribute their propagator to the rate of the scattering event described by the diagram. They also can be viewed as the inverse of the wave operator appropriate to the particle, and are therefore often called Green's functions. Non-relativistic propagators In non-relativistic quantum mechanics the propagator gives the probability amplitude for a particle to travel from one spatial point at one time to another spatial point at a later time. It is the Green's function (fundamental solution) for the Schrödinger equation. This means that, if a system has Hamiltonian H, then the appropriate propagator is a function satisfying where Hx denotes the Hamiltonian written in terms of the x coordinates, δ(x)denotes the Dirac delta-function, Θ(x) is the Heaviside step function and K(x,t;x',t')is the kernel of the differential operator in question, often referred to as the propagator instead of G in this context, and henceforth in this article. This propagator can also be written as where Û(t,t' ) is the unitary time-evolution operator for the system taking states at time t to states at time t'. The quantum mechanical propagator may also be found by using a path integral, where the boundary conditions of the path integral include q(t)=x, q(t')=x' . -
Memristor-Based Approximation of Gaussian Filter
Memristor-based Approximation of Gaussian Filter Alex Pappachen James, Aidyn Zhambyl, Anju Nandakumar Electrical and Computer Engineering department Nazarbayev University School of Engineering Astana, Kazakhstan Email: [email protected], [email protected], [email protected] Abstract—Gaussian filter is a filter with impulse response of fourth section some mathematical analysis will be provided to Gaussian function. These filters are useful in image processing obtained results and finally the paper will be concluded by of 2D signals, as it removes unnecessary noise. Also, they could stating the result of the done work. be helpful for data transmission (e.g. GMSK modulation). In practice, the Gaussian filters could be approximately designed II. GAUSSIAN FILTER AND MEMRISTOR by several methods. One of these methods are to construct Gaussian-like filter with the help of memristors and RLC circuits. A. Gaussian Filter Therefore, the objective of this project is to find and design Gaussian filter is such filter which convolves the input signal appropriate model of Gaussian-like filter, by using mentioned devices. Finally, one possible model of Gaussian-like filter based with the impulse response of Gaussian function. In some on memristor designed and analysed in this paper. sources this process is also known as the Weierstrass transform [2]. The Gaussian function is given as in equation 1, where µ Index Terms—Gaussian filters, analog filter, memristor, func- is the time shift and σ is the scale. For input signal of x(t) tion approximation the output is the convolution of x(t) with Gaussian, as shown in equation 2. -
The Moyal Equation in Open Quantum Systems
The Moyal Equation in Open Quantum Systems Karl-Peter Marzlin and Stephen Deering CAP Congress Ottawa, 13 June 2016 Outline l Quantum and classical dynamics l Phase space descriptions of quantum systems l Open quantum systems l Moyal equation for open quantum systems l Conclusion Quantum vs classical Differences between classical mechanics and quantum mechanics have fascinated researchers for a long time Two aspects: • Measurement process: deterministic vs contextual • Dynamics: will be the topic of this talk Quantum vs classical Differences in standard formulations: Quantum Mechanics Classical Mechanics Class. statistical mech. complex wavefunction deterministic position probability distribution observables = operators deterministic observables observables = random variables dynamics: Schrödinger Newton 2 Boltzmann equation for equation for state probability distribution However, some differences “disappear” when different formulations are used. Quantum vs classical First way to make QM look more like CM: Heisenberg picture Heisenberg equation of motion dAˆH 1 = [AˆH , Hˆ ] dt i~ is then similar to Hamiltonian mechanics dA(q, p, t) @A @H @A @H = A, H , A, H = dt { } { } @q @p − @p @q Still have operators as observables Second way to compare QM and CM: phase space representation of QM Most popular phase space representation: the Wigner function W(q,p) 1 W (q, p)= S[ˆ⇢](q, p) 2⇡~ iq0p/ Phase space S[ˆ⇢](q, p)= dq0 q + 1 q0 ⇢ˆ q 1 q0 e− ~ h 2 | | − 2 i Z The Weyl symbol S [ˆ ⇢ ]( q, p ) maps state ⇢ ˆ to a function of position and momentum (= phase space) Phase space The Wigner function is similar to the distribution function of classical statistical mechanics Dynamical equation = Liouville equation. -
Eigenfunctions and Eigenoperators of Cyclic Integral Transforms with Application to Gaussian Beam Propagation
NORTHWESTERN UNIVERSITY Eigenfunctions and Eigenoperators of Cyclic Integral Transforms with Application to Gaussian Beam Propagation A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS for the degree DOCTOR OF PHILOSOPHY Field of Applied Mathematics By Matthew S. McCallum EVANSTON, ILLINOIS December 2008 2 c Copyright by Matthew S. McCallum 2008 All Rights Reserved 3 ABSTRACT Eigenfunctions and Eigenoperators of Cyclic Integral Transforms with Application to Gaussian Beam Propagation Matthew S. McCallum An integral transform which reproduces a transformable input function after a finite number N of successive applications is known as a cyclic transform. Of course, such a transform will reproduce an arbitrary transformable input after N applications, but it also admits eigenfunction inputs which will be reproduced after a single application of the transform. These transforms and their eigenfunctions appear in various applications, and the systematic determination of eigenfunctions of cyclic integral transforms has been a problem of interest to mathematicians since at least the early twentieth century. In this work we review the various approaches to this problem, providing generalizations of published expressions from previous approaches. We then develop a new formalism, differential eigenoperators, that reduces the eigenfunction problem for a cyclic transform to an eigenfunction problem for a corresponding ordinary differential equation. In this way we are able to relate eigenfunctions of integral equations to boundary-value problems, 4 which are typically easier to analyze. We give extensive examples and discussion via the specific case of the Fourier transform. We also relate this approach to two formalisms that have been of interest to the math- ematical physics community – hyperdifferential operators and linear canonical transforms. -
A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms
The Generalized Fourier Transform: A Unified Framework for the Fourier, Laplace, Mellin and Z Transforms Pushpendra Singha,∗, Anubha Guptab, Shiv Dutt Joshic aDepartment of ECE, National Institute of Technology Hamirpur, Hamirpur, India. bSBILab, Department of ECE, Indraprastha Institute of Information Technology Delhi, Delhi, India cDepartment of EE, Indian Institute of Technology Delhi, Delhi, India Abstract This paper introduces Generalized Fourier transform (GFT) that is an extension or the generalization of the Fourier transform (FT). The Unilateral Laplace transform (LT) is observed to be the special case of GFT. GFT, as proposed in this work, contributes significantly to the scholarly literature. There are many salient contribution of this work. Firstly, GFT is applicable to a much larger class of signals, some of which cannot be analyzed with FT and LT. For example, we have shown the applicability of GFT on the polynomially decaying functions and super exponentials. Secondly, we demonstrate the efficacy of GFT in solving the initial value problems (IVPs). Thirdly, the generalization presented for FT is extended for other integral transforms with examples shown for wavelet transform and cosine transform. Likewise, generalized Gamma function is also presented. One interesting application of GFT is the computation of generalized moments, for the otherwise non-finite moments, of any random variable such as the Cauchy random variable. Fourthly, we introduce Fourier scale transform (FST) that utilizes GFT with the topological isomorphism of an exponential map. Lastly, we propose Generalized Discrete-Time Fourier transform (GDTFT). The DTFT and unilateral z-transform are shown to be the special cases of the proposed GDTFT. The properties of GFT and GDTFT have also been discussed. -
About the Concept of Quantum Chaos
entropy Concept Paper About the Concept of Quantum Chaos Ignacio S. Gomez 1,*, Marcelo Losada 2 and Olimpia Lombardi 3 1 National Scientific and Technical Research Council (CONICET), Facultad de Ciencias Exactas, Instituto de Física La Plata (IFLP), Universidad Nacional de La Plata (UNLP), Calle 115 y 49, 1900 La Plata, Argentina 2 National Scientific and Technical Research Council (CONICET), University of Buenos Aires, 1420 Buenos Aires, Argentina; [email protected] 3 National Scientific and Technical Research Council (CONICET), University of Buenos Aires, Larralde 3440, 1430 Ciudad Autónoma de Buenos Aires, Argentina; olimpiafi[email protected] * Correspondence: nachosky@fisica.unlp.edu.ar; Tel.: +54-11-3966-8769 Academic Editors: Mariela Portesi, Alejandro Hnilo and Federico Holik Received: 5 February 2017; Accepted: 23 April 2017; Published: 3 May 2017 Abstract: The research on quantum chaos finds its roots in the study of the spectrum of complex nuclei in the 1950s and the pioneering experiments in microwave billiards in the 1970s. Since then, a large number of new results was produced. Nevertheless, the work on the subject is, even at present, a superposition of several approaches expressed in different mathematical formalisms and weakly linked to each other. The purpose of this paper is to supply a unified framework for describing quantum chaos using the quantum ergodic hierarchy. Using the factorization property of this framework, we characterize the dynamical aspects of quantum chaos by obtaining the Ehrenfest time. We also outline a generalization of the quantum mixing level of the kicked rotator in the context of the impulsive differential equations. Keywords: quantum chaos; ergodic hierarchy; quantum ergodic hierarchy; classical limit 1.