Schr dinger Diffusion ProcessesSchr dinger Diffusion Processes



The book presumes a graduate level of knowledge in mathematics or physics and represents a relevant and demanding application of today's advanced probability theory.

Author: Robert Aebi

Publisher: Birkhäuser

ISBN: 3034898746

Category:

Page: 186

View: 315

In 1931 Erwin Schrödinger considered the following problem: A huge cloud of independent and identical particles with known dynamics is supposed to be observed at finite initial and final times. What is the "most probable" state of the cloud at intermediate times? The present book provides a general yet comprehensive discourse on Schrödinger's question. Key roles in this investigation are played by conditional diffusion processes, pairs of non-linear integral equations and interacting particles systems. The introductory first chapter gives some historical background, presents the main ideas in a rather simple discrete setting and reveals the meaning of intermediate prediction to quantum mechanics. In order to answer Schrödinger's question, the book takes three distinct approaches, dealt with in separate chapters: transformation by means of a multiplicative functional, projection by means of relative entropy, and variation of a functional associated to pairs of non-linear integral equations. The book presumes a graduate level of knowledge in mathematics or physics and represents a relevant and demanding application of today's advanced probability theory.

Schr dinger Equations and Diffusion TheorySchr dinger Equations and Diffusion Theory



--- This book is a self-contained, very well-organized monograph recommended to researchers and graduate students in the field of probability theory, functional analysis and quantum dynamics. (...) what is written in this book may be ...

Author: Masao Nagasawa

Publisher: Springer Science & Business Media

ISBN: 9783034805605

Category:

Page: 319

View: 837

Schrödinger Equations and Diffusion Theory addresses the question “What is the Schrödinger equation?” in terms of diffusion processes, and shows that the Schrödinger equation and diffusion equations in duality are equivalent. In turn, Schrödinger’s conjecture of 1931 is solved. The theory of diffusion processes for the Schrödinger equation tells us that we must go further into the theory of systems of (infinitely) many interacting quantum (diffusion) particles. The method of relative entropy and the theory of transformations enable us to construct severely singular diffusion processes which appear to be equivalent to Schrödinger equations. The theory of large deviations and the propagation of chaos of interacting diffusion particles reveal the statistical mechanical nature of the Schrödinger equation, namely, quantum mechanics. The text is practically self-contained and requires only an elementary knowledge of probability theory at the graduate level. --- This book is a self-contained, very well-organized monograph recommended to researchers and graduate students in the field of probability theory, functional analysis and quantum dynamics. (...) what is written in this book may be regarded as an introduction to the theory of diffusion processes and applications written with the physicists in mind. Interesting topics present themselves as the chapters proceed. (...) this book is an excellent addition to the literature of mathematical sciences with a flavour different from an ordinary textbook in probability theory because of the author’s great contributions in this direction. Readers will certainly enjoy the topics and appreciate the profound mathematical properties of diffusion processes. (Mathematical Reviews)​

Schr dinger Diffusion ProcessesSchr dinger Diffusion Processes



Let us agree on the terminology that diffusion processes in SchrödingerNagasawa representation are called Schrödinger processes from now on. Nevertheless we are always aware that they are general diffusion processes represented in a ...

Author: Robert Aebi

Publisher: Birkhäuser

ISBN: 9783034890274

Category:

Page: 186

View: 552

In 1931 Erwin Schrödinger considered the following problem: A huge cloud of independent and identical particles with known dynamics is supposed to be observed at finite initial and final times. What is the "most probable" state of the cloud at intermediate times? The present book provides a general yet comprehensive discourse on Schrödinger's question. Key roles in this investigation are played by conditional diffusion processes, pairs of non-linear integral equations and interacting particles systems. The introductory first chapter gives some historical background, presents the main ideas in a rather simple discrete setting and reveals the meaning of intermediate prediction to quantum mechanics. In order to answer Schrödinger's question, the book takes three distinct approaches, dealt with in separate chapters: transformation by means of a multiplicative functional, projection by means of relative entropy, and variation of a functional associated to pairs of non-linear integral equations. The book presumes a graduate level of knowledge in mathematics or physics and represents a relevant and demanding application of today's advanced probability theory.

SchrodingerSchrodinger



In his second paper, Schrödinger discussed the quantum dynamics' of the Dirac electron in a coulombic field, as in the hydrogen atom. ... i.e., when wi(x) is exactly the value that would be reached by the forward diffusion process.

Author: Walter Moore

Publisher: Cambridge University Press

ISBN: 9781107569911

Category:

Page: 528

View: 535

This is a biography of the great scientist, Erwin Schrödinger (author of What is Life?), which draws upon recollections of his family and friends, as well as on contemporary records, diaries and letters. It aims to reveal the fundamental motives that drove him.

Seminar on Stochastic Analysis Random Fields and ApplicationsSeminar on Stochastic Analysis Random Fields and Applications



[2] R. Aebi, Schroedinger Diffusion Processes, Probability and Its Applications, Birkhäuser, 1996. [3] S. Albeverio, Wiener and Feynman path integrals and their applications, in: V. Mandrekar and P. R. Masani, Eds., Proceedings of the ...

Author: Robert Dalang

Publisher: Birkhäuser

ISBN: 9783034886819

Category:

Page: 300

View: 172

A collection of 20 refereed research or review papers presented at a six-day seminar in Switzerland. The contributions focus on stochastic analysis, its applications to the engineering sciences, and stochastic methods in financial models, which was the subject of a minisymposium.

Nonlinear Klein gordon And Schrodinger Systems Theory And ApplicationsNonlinear Klein gordon And Schrodinger Systems Theory And Applications



(1) holds thus for u = XA, hence for general u, as seen by approximation through linear combinations of indicator functions of Borel sets. X is the prototype of a diffusion process, in particular is Markov (also with respect to stopping ...

Author: Vazquez Luis

Publisher: World Scientific

ISBN: 9789814548090

Category:

Page: 380

View: 349

Modern theory of elliptic operators, or simply elliptic theory, has been shaped by the Atiyah-Singer Index Theorem created 40 years ago. Reviewing elliptic theory over a broad range, 32 leading scientists from 14 different countries present recent developments in topology; heat kernel techniques; spectral invariants and cutting and pasting; noncommutative geometry; and theoretical particle, string and membrane physics, and Hamiltonian dynamics.The first of its kind, this volume is ideally suited to graduate students and researchers interested in careful expositions of newly-evolved achievements and perspectives in elliptic theory. The contributions are based on lectures presented at a workshop acknowledging Krzysztof P Wojciechowski's work in the theory of elliptic operators.

Solving the Schr dinger EquationSolving the Schr dinger Equation



Thus, the same types of diffusion processes can be set up to solve for the 1-DM, at least in principle. The 1-DM description is a highly reduced one, however, and the 1-DM possesses a relatively simple structure.

Author: Paul Popelier

Publisher: World Scientific

ISBN: 9781908977762

Category:

Page: 376

View: 874

The Schrödinger equation is the master equation of quantum chemistry. The founders of quantum mechanics realised how this equation underpins essentially the whole of chemistry. However, they recognised that its exact application was much too complicated to be solvable at the time. More than two generations of researchers were left to work out how to achieve this ambitious goal for molecular systems of ever-increasing size. This book focuses on non-mainstream methods to solve the molecular electronic Schrödinger equation. Each method is based on a set of core ideas and this volume aims to explain these ideas clearly so that they become more accessible. By bringing together these non-standard methods, the book intends to inspire graduate students, postdoctoral researchers and academics to think of novel approaches. Is there a method out there that we have not thought of yet? Can we design a new method that combines the best of all worlds? Contents:Intracule Functional Theory (Deborah L Crittenden and Peter M W Gill)Explicitly Correlated Electronic Structure Theory (Frederick R Manby)Solving Problems with Strong Correlation Using the Density Matrix Renormalization Group (DMRG) (Garnet Kin-Lic Chan and Sandeep Sharma)Reduced-Density-Matrix Theory for Many-electron Correlation (David A Mazziotti)Finite Size Scaling for Criticality of the Schrödinger Equation (Sabre Kais)The Generalized Sturmian Method (James Avery and John Avery)Slater-Type Orbital Basis Sets: Reliable and Rapid Solution of the Schrödinger Equation for Accurate Molecular Properties (Philip E Hoggan)Modern Ab Initio Valence Bond Methods (Philippe C Hiberty & Sason Shaik)Quantum Monte Carlo Approaches for Tackling Electronic Correlation (Massimo Mella and Gabriele Morosi)Solving the Schrödinger Equation on Real-Space Grids and with Random Walks (Thomas L Beck and Joel H Dedrick)Changes in Dense Linear Algebra Kernels: Decades-Long Perspective (Piotr Luszczek, Jakub Kurzak, and Jack Dongarra) Readership: Graduate students, postdoctoral researchers and academics in the fields of computational chemistry, theoretical chemistry, super computing, molecular physics and solid state physics. Keywords:Quantum Mechanics;Quantum Chemistry;Molecular Electronic Schrödinger EquationKey Features:Unusual combination of methods/techniquesA thought-provoking and didactic exposé, not a review, nor a textbookLooks at the future

Gravity and Quantum Non LinearityGravity and Quantum Non Linearity



The Diffusion Equation . 1 Relating the Schroedinger equation to random walk and diffusion . Dispensing with particles and replacing them with probability envelopes generated by a signalling process . The first step in quantising the ...

Author: Timothy Reeves

Publisher: Authors On Line Ltd

ISBN: 0755201442

Category:

Page: 156

View: 397

For many years Maxwell's equations were mute testimony to the Lorentz coordinate transformations of special relativity, but the significance of this transformation was not fully understood until relativity itself had been discovered. The author of this book claims that an oversight has once again occurred in fundamental physics. A direct link between Quantum Mechanics and Gravity - perhaps the biggest scientific enigma of all time - has been overlooked. In an entirely novel approach to Quantum-Gravity, this book suggests that Gravity is the manifestation of a very weak non-linearity in Quantum Mechanics; a non-linearity only apparent in the presence of large quantities of matter or very condensed wave functions. The reader will discover how this non-linearity is a natural consequence of a particular interpretation of the Quantum Equations. This book however, does not share the same genre of totalising theories that emphatically and triumphantly declare a new dawn. Instead its thesis can only present itself as a fascinating hypothesis submitted for consideration. Should the theory presented here prove its mettle, then it would provide the key to opening up new vistas of theoretical and experimental research, thus helping to clear the conceptual log jam in which contemporary fundamental physics finds itself.

Remote Viewing observations of Atoms Quarks Remote Viewing observations of Atoms Quarks



Schroedinger's Equation dψ/dt = (hi/4πm) [d2ψ/dx2] is functionally exactly like Fick's 2nd Law of Diffusion dC/dt = (const ... Their movement through such a “space” (even in a vacuum) must then be by a diffusion process and hence Fick's ...

Author: M.G. Hocking

Publisher: 4-D Books Ltd

ISBN: 9780952109969

Category:

Page: 78

View: 839

The fine internal structure of the atomic nucleus and elementary particles may never be discovered by instrumental methods. This paper describes detailed observations of their structures by “remote viewing”. This is a mind-based faculty for magnifying objects, requiring specialized meditation training. This paper reconciles these observations with modern physics. Remote viewing has been verified by experiments published in Nature by Targ & Puthoff in 1974, with a possibility of only 1 in a million that it could have occurred by chance.