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Historical Development

P.J.E. Peebles

in Quantum Mechanics

Published in print:
2019
Published Online:
May 2021
ISBN:
9780691209821
eISBN:
9780691206738
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691209821.003.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter presents the origins of quantum mechanics. The story of how people hit on the highly non-intuitive world picture of quantum mechanics, in which the physical state of a system is ... More


X-ray wave-fields in free space

David M. Paganin

in Coherent X-Ray Optics

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198567288
eISBN:
9780191717963
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567288.003.0001
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter considers the theory of classical X-ray wave-fields in free space, taking the Maxwell equations as a starting point. Vacuum wave equations (d’Alembert equations) are developed for X-rays ... More


NUMERICAL METHODS

Miguel Alcubierre

in Introduction to 3+1 Numerical Relativity

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199205677
eISBN:
9780191709371
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199205677.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

There are many ways to solve partial differential equations numerically. The most popular methods are: finite differencing, finite elements, and spectral methods. This chapter describes the main ... More


Global Existence Theorems: Asymptotically Euclidean Data

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

Published in print:
2008
Published Online:
May 2009
ISBN:
9780199230723
eISBN:
9780191710872
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230723.003.0015
Subject:
Mathematics, Applied Mathematics

This chapter shows how the Penrose transform can be used to prove global existence of solutions of various semilinear field equations. It outlines the foundation points of Friedrich's conformal ... More


Green's functions and retarded potentials

Mauro Fabrizio and Morro Angelo

in Electromagnetism of Continuous Media: Mathematical Modelling and Applications

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198527008
eISBN:
9780191713316
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198527008.003.0002
Subject:
Mathematics, Applied Mathematics

This chapter gives a concise and self-contained presentation of the use of Green’s functions to solve linear problems with a given point source. Specific functions are determined for the harmonic ... More


Other Hyperbolic-Elliptic Well-Posed Systems

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

Published in print:
2008
Published Online:
May 2009
ISBN:
9780199230723
eISBN:
9780191710872
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199230723.003.0008
Subject:
Mathematics, Applied Mathematics

This chapter presents well-posed hyperbolic or hyperbolic-elliptic systems that lead to the same local existence andgeometric uniqueness theorems as the wave gauge choice. However, these different ... More


Central Equation of Thermoelasticity with Finite Wave Speeds

Józef Ignaczak and Martin Ostoja‐Starzewski

in Thermoelasticity with Finite Wave Speeds

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199541645
eISBN:
9780191716164
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199541645.003.0006
Subject:
Mathematics, Applied Mathematics, Mathematical Physics

This chapter focuses on a central equation of thermoelasticity with finite wave speeds, i.e. a partial differential equation which, most remarkably, has a similar form for both theories. Given next ... More


Harmonic morphisms between semi-Riemannian manifolds

Paul Baird and John C. Wood

in Harmonic Morphisms Between Riemannian Manifolds

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198503620
eISBN:
9780191708435
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198503620.003.0014
Subject:
Mathematics, Pure Mathematics

This final chapter discusses how the main definitions and results need to be modified in the semi-Riemannian case. In this context, harmonic maps include the strings of mathematical physics. Weakly ... More


Review of quantum mechanics

Alan Corney

in Atomic and Laser Spectroscopy

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211456
eISBN:
9780191705915
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211456.003.0003
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter derives the quantum mechanical wave functions which describe the energy levels of simple atoms. Schrödinger's equation is introduced as well and the angular part of the equation is ... More


Relativistic wave equations

Victor Galitski, Boris Karnakov, Vladimir Kogan, and Victor Galitski, Jr.

in Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199232710
eISBN:
9780191774973
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199232710.003.0015
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter deals with problems related to the Klein-Gordon equation and the Dirac equation.


Preliminaries

Sergiu Klainerman and Jérémie Szeftel

in Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations: (AMS-210)

Published in print:
2020
Published Online:
May 2021
ISBN:
9780691212425
eISBN:
9780691218526
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691212425.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter discusses the main quantities, equations, and basic tools needed in the following chapters. It is the main reference kit providing all main null structure and null Bianchi equations, in ... More


Regge-Wheeler Type Equations

Sergiu Klainerman and Jérémie Szeftel

in Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations: (AMS-210)

Published in print:
2020
Published Online:
May 2021
ISBN:
9780691212425
eISBN:
9780691218526
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691212425.003.0010
Subject:
Mathematics, Geometry / Topology

This chapter explores estimates for Regge-Wheeler type wave equations used in Theorem M1. It first proves basic Morawetz estimates for ψ‎. The chapter then proves rp-weighted estimates in the spirit ... More


Seismic Waves and Sources

Heiner Igel

in Computational Seismology: A Practical Introduction

Published in print:
2016
Published Online:
January 2017
ISBN:
9780198717409
eISBN:
9780191835070
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198717409.003.0002
Subject:
Physics, Geophysics, Atmospheric and Environmental Physics

The specific partial differential equation with which the numerical methods are introduced in the remainder of the volume is presented in detail. Fundamental consequences of the elastic wave equation ... More


First steps in classical field theory

Andrew M. Steane

in Relativity Made Relatively Easy Volume 2: General Relativity and Cosmology

Published in print:
2021
Published Online:
December 2021
ISBN:
9780192895646
eISBN:
9780191943911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780192895646.003.0027
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some ... More


Photons

P. K. Basu

in Theory of Optical Processes in Semiconductors: Bulk and Microstructures

Published in print:
2003
Published Online:
January 2010
ISBN:
9780198526209
eISBN:
9780191706790
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526209.003.0003
Subject:
Physics, Atomic, Laser, and Optical Physics

The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires ... More


Electromagnetic fields and waves in vacuum

J. Pierrus

in Solved Problems in Classical Electromagnetism: Analytical and Numerical Solutions with Comments

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198821915
eISBN:
9780191861055
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821915.003.0007
Subject:
Physics, Condensed Matter Physics / Materials

In previous chapters four experimental laws of electromagnetism were encountered: Gauss’s law in electrostatics, Gauss’s law in magnetism, Faraday’s law and Ampere’s law. Now, in this chapter, these ... More


Transformations, matrices and operators

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199641390
eISBN:
9780191747786
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199641390.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Linear algebra of matrices is discussed in the context of rotations in space. Determinants are used to find solutions of simultaneous linear equations, to invert matrices and to find matrix ... More


The Energy Critical Wave Equation in 3D

Carlos Kenig

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691159416
eISBN:
9781400848935
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691159416.003.0009
Subject:
Mathematics, Numerical Analysis

This chapter discusses the energy critical nonlinear wave equation in 3 space dimensions. It mainly focuses on soliton resolution for radial solutions of nonlinear waves. For a long time there has ... More


Relativistic quantum mechanics

Franco Strocchi

in An Introduction to Non-Perturbative Foundations of Quantum Field Theory

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199671571
eISBN:
9780191751073
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199671571.003.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

The chapter focuses on the problems at the origin of QFT: i) the conflict between locality and energy positivity, which affects relativistic wave equations in general, and ii) the need of describing ... More


Continuous-Valued Cellular Automata in Two Dimensions

Rudy Rucker

in New Constructions in Cellular Automata

Published in print:
2003
Published Online:
November 2020
ISBN:
9780195137170
eISBN:
9780197561652
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/9780195137170.003.0016
Subject:
Computer Science, Systems Analysis and Design

We explore a variety of two-dimensional continuous-valued cellular automata (CAs). We discuss how to derive CA schemes from differential equations and look at CAs based on several kinds of ... More


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