Rigorous Time Slicing Approach to Feynman Path Integrals (Mathematical Physics Studies) - Image 1

£85.74

Springer Rigorous Time Slicing Approach to Feynman Path Integrals (Mathematical Physics Studies)

Name: Rigorous Time Slicing Approach to Feynman Path Integrals (Mathematical Physics Studies)
Brand: Springer
Price: 89.23 GBP
Availability: InStock

Differential Equations

Price History & Forecast

Last 87 days • 87 data points (No recent data available)

Historical

Generating forecast...

Price Distribution

Price distribution over 87 days • 4 price levels

Days at Price

Current Price

Price Analysis

Most common price: £91 (36 days, 41.4%)

Price range: £86 - £92

Price levels: 4 different prices over 87 days

Description

This book proves that Feynman's original definition of the path integral actually converges to the fundamental solution of the Schrödinger equation at least in the short term if the potential is differentiable sufficiently many times and its derivatives of order equal to or higher than two are bounded. The semi-classical asymptotic formula up to the second term of the fundamental solution is also proved by a method different from that of Birkhoff. A bound of the remainder term is also proved.The Feynman path integral is a method of quantization using the Lagrangian function, whereas Schrödinger's quantization uses the Hamiltonian function. These two methods are believed to be equivalent. But equivalence is not fully proved mathematically, because, compared with Schrödinger's method, there is still much to be done concerning rigorous mathematical treatment of Feynman's method. Feynman himself defined a path integral as the limit of a sequence of integrals over finite-dimensional spaces which is obtained by dividing the time interval into small pieces. This method is called the time slicing approximation method or the time slicing method.This book consists of two parts. Part I is the main part. The time slicing method is performed step by step in detail in Part I. The time interval is divided into small pieces. Corresponding to each division a finite-dimensional integral is constructed following Feynman's famous paper. This finite-dimensional integral is not absolutely convergent. Owing to the assumption of the potential, it is an oscillatory integral. The oscillatory integral techniques developed in the theory of partial differential equations are applied to it. It turns out that the finite-dimensional integral gives a finite definite value. The stationary phase method is applied to it. Basic properties of oscillatory integrals and the stationary phase method are explained in the book in detail.Those finite-dimensional integralsform a sequence of approximation of the Feynman path integral when the division goes finer and finer. A careful discussion is required to prove the convergence of the approximate sequence as the length of each of the small subintervals tends to 0. For that purpose the book uses the stationary phase method of oscillatory integrals over a space of large dimension, of which the detailed proof is given in Part II of the book. By virtue of this method, the approximate sequence converges to the limit. This proves that the Feynman path integral converges. It turns out that the convergence occurs in a very strong topology. The fact that the limit is the fundamental solution of the Schrödinger equation is proved also by the stationary phase method. The semi-classical asymptotic formula naturally follows from the above discussion.A prerequisite for readers of this book is standard knowledge of functional analysis. Mathematical techniques required here are explained and proved from scratch in Part II, which occupies a large part of the book, because they are considerably different from techniques usually used in treating the Schrödinger equation.

Product Specifications

Brand: Springer
Format: hardcover
ASIN: 4431565515
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Mathematical Analysis > Differential Equations
Domain: Amazon UK
Release Date: 06 July 2017
Listed Since: 23 March 2017

Barcode

No barcode data available

Springer Rigorous Time Slicing Approach to Feynman Path Integrals (Mathematical Physics Studies)

Price History & Forecast

Price Distribution

Price Analysis

Description

Product Specifications

Barcode

Similar Products You Might Like

Schrödinger Equations and Diffusion Theory: 86 (Monographs in Mathematics, 86)

Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities: 200 (Cambridge Studies in Advanced Mathematics, Series Number 200)

Semilinear Schrodinger Equations (Courant Lecture Notes)

Direct and Inverse Scattering for the Matrix Schrödinger Equation: 203 (Applied Mathematical Sciences, 203)

Multiple Scattering in Solids (Graduate Texts in Contemporary Physics)

Quantum Mechanics: Detailed Historical, Mathematical and Computational Approaches

Back-of-the-envelope Quantum Mechanics: With Extensions To Many-body Systems, Integrable Pdes, And Rare And Exotic Methods (second Edition)

Spectral and Scattering Theory for Second Order Partial Differential Operators (Chapman & Hall/CRC Monographs and Research Notes in Mathematics)

Self-adjoint Extensions in Quantum Mechanics: General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials: 62 (Progress in Mathematical Physics, 62)

Coherent States, Wavelets, and Their Generalizations (Theoretical and Mathematical Physics)

One-Dimensional Ergodic Schrodinger Operators: I. General Theory: 1 (Graduate Studies in Mathematics)

Linear Operators and their Spectra: 106 (Cambridge Studies in Advanced Mathematics, Series Number 106)

Correlated Random Systems: Five Different Methods: CIRM Jean-MorletChair, Spring 2013: 2143 (Lecture Notes in Mathematics, 2143)

Symmetries in Science X (Language of Science)

Quantum Mechanics for Scientists and Engineers: 0

Lectures On Quantum Mechanics

Relativistic Quantum Mechanics: 0 (Theoretical and Mathematical Physics)

Lectures Of Sidney Coleman On Quantum Field Theory: Foreword By David Kaiser

Tunneling Dynamics in Open Ultracold Bosonic Systems: Numerically Exact Dynamics – Analytical Models – Control Schemes (Springer Theses)

Quantum‐Enhanced Sensing Based on Time Reversal of Entangling Interactions (Springer Theses)

XVIITH INTERNATIONAL CONGRESS ON MATHEMATICAL PHYSICS

The challenging concept of time in quantum mechanics: Time in Quantum mechanics

Almost Periodic and Almost Automorphic Functions in Abstract Spaces

Inverse Problems in Scattering: An Introduction: 23 (Solid Mechanics and Its Applications, 23)