Search Results

Historical Development

P.J.E. Peebles

in Quantum Mechanics

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

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 represented by an element in an abstract linear space and its observable properties by operators in the space, is fascinating and exceedingly complicated. The much greater change from the classical world picture of Newtonian mechanics and general relativity to the quantum world picture came in many steps taken by many people, often against the better judgment of participants. There are three major elements in the story. The first is the experimental evidence that the energy of an isolated system can only assume special discrete or quantized values. The second is the idea that the energy is proportional to the frequency of a wave function associated with the system. The third is the connection between the de Broglie relation and energy quantization through the mathematical result that a wave equation with fixed boundary conditions allows only discrete quantized values of the frequency of oscillation of the wave function (as in the fundamental and harmonics of the vibration of a violin string).Less

X-ray wave-fields in free space

David M. Paganin

in Coherent X-Ray Optics

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

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 in vacuum. In this respect, the concepts of a spectral decomposition, the complex analytic signal, and the angular spectrum of plane waves are developed. Several diffraction theories are outlined, including the Fraunhofer diffraction formula, the Fresnel diffraction formula, the Kirchhoff integral, and the Rayleigh-Sommerfeld integrals of the first and second kinds. The theory of partially coherent fields is also discussed, including topics such as random processes, the concept of partial coherence, the mutual coherent function, the van Cittert-Zernike theorem, and the Hanbury Brown-Twiss effect.Less

NUMERICAL METHODS

Miguel Alcubierre

in Introduction to 3+1 Numerical Relativity

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

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 ideas behind finite differencing methods, since this is the most commonly used approach in numerical relativity. It focuses on methods for the numerical solution of systems of evolution equations of essentially ‘hyperbolic’ type, and does not deal with the solution of elliptic equations, such as those needed for obtaining initial data.Less

Global Existence Theorems: Asymptotically Euclidean Data

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

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

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 system, and explains how a conformal transformation of a future causal cone in Minkowski spacetime of dimension greater than or equal to six on to another such light cone gives a global existence theorem of solutions of the vacuum Einstein equations with small data which are Schwarzschild outside of a compact set. The chapter indicates some of the arguments of the book Non-Linear Stability of Minkowski Space, and states some further properties proved in another book by Nicolo and Klainerman. Finally, it sketches the main steps of the proof by Lindblad and Rodnianski of the global existence in wave coordinates, for small initial data.Less

Green's functions and retarded potentials

Mauro Fabrizio and Morro Angelo

in Electromagnetism of Continuous Media: Mathematical Modelling and Applications

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

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 oscillator, the wave equation, and the lossy wave equation. The solution to the inhomogeneous wave equation is derived through a retarded potential and by the Kirchhoff procedure. The field generated by a moving charge is determined through the Lienard-Wiechert potential. A Cauchy problem and a boundary-value problem are investigated for the wave equation and the telegraph equation.Less

Other Hyperbolic-Elliptic Well-Posed Systems

Yvonne Choquet-Bruhat

in General Relativity and the Einstein Equations

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

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 formulations may be important in numerical studies or global existence proofs. Topics covered include Leray–Ohya non-hyperbolicity of Rij = 0, wave equation for K, fourth-order non-strict and strict hyperbolic systems, first-order hyperbolic systems, Bianchi–Einstein equations, Bel–Robinson tensor and energy, and Bel–Robinson energy in a strip.Less

Central Equation of Thermoelasticity with Finite Wave Speeds

Józef Ignaczak and Martin Ostoja‐Starzewski

in Thermoelasticity with Finite Wave Speeds

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

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 are: a decomposition theorem for a central equation of Green‐Lindsay theory, the wave‐like equations with a convolution, and the speed and attenuation of thermoelastic disturbances. The chapter ends with an analysis of the convolution coefficient and kernel in the space of constitutive variables.Less

Harmonic morphisms between semi-Riemannian manifolds

Paul Baird and John C. Wood

in Harmonic Morphisms Between Riemannian Manifolds

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

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 conformal and horizontally weakly conformal maps are discussed; care is taken with the definitions as the subspaces of the tangent spaces involved may be degenerate. It is shown that with appropriate definitions, the characterization of harmonic morphisms as horizontally weakly conformal harmonic maps carries over to the semi-Riemannian case. Certain harmonic morphisms are simply null solutions of the wave equation. The chapter concludes with an explicit local description of all harmonic morphisms between Lorentzian 2-manifolds. In the ‘Notes and comments’ section, the connection with the shear-free ray congruences of mathematical physics is described.Less

Review of quantum mechanics

Alan Corney

in Atomic and Laser Spectroscopy

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

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 solved for spherically symmetric potentials. The orbital angular momentum operator is defined and the concept of intrinsic spin is introduced. The extension of these results to atoms with several electrons using the central-field approximation is introduced. Spectroscopic notation for Russell-Saunders coupling is discussed.Less

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

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)

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

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 general null frames, in the context of axially symmetric polarized spacetimes. The chapter translates the null structure and null Bianchi identities associated to an S-foliation in the reduced picture. It starts with general, Z-invariant, S-foliation, before considering the special case of geodesic foliations. The chapter then looks at perturbations of Schwarzschild and invariant quantities. It investigates how the main Ricci and curvature quantities change relative to frame transformations, i.e., linear transformations which take null frames into null frames. Finally, the chapter presents wave equations for the invariant quantities.Less

Regge-Wheeler Type Equations

Sergiu Klainerman and Jérémie Szeftel

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

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

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 of Dafermos and Rodnianski for ψ‎. In particular, it obtains as an immediate corollary the proof of Theorem 5.17 in the case s = 0 (i.e., without commutating the equation of ψ‎ with derivatives). It also uses a variation of the method of [5] to derive slightly stronger weighted estimates and prove Theorem 5.18 in the case s = 0. Finally, commuting the equation of ψ‎ with derivatives, the chapter completes the proof of Theorem 5.17 by controlling higher order derivatives of ψ‎.Less

Seismic Waves and Sources

Heiner Igel

in Computational Seismology: A Practical Introduction

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

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 (e.g., P and S waves, surface waves), and boundary and initial conditions, are presented with a view to their impact on numerical solutions. The sources of seismic energy (point and finite sources, moment tensor) are introduced. Several properties of the wave equation (superposition principle, reciprocity) are presented with numerical examples combined with a discussion of how these properties can be used to make simulation tasks more efficient.Less

First steps in classical field theory

Andrew M. Steane

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

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

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 features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.Less

Photons

P. K. Basu

in Theory of Optical Processes in Semiconductors: Bulk and Microstructures

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

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 a quantum mechanical theory of radiation. The quantum theory of light, like classical theory, is also based on Maxwell's equations. However, in quantum theory the electric field F and the magnetic field H are treated as operators. This chapter develops the correspondence between the two and outlines the method for converting field vectors F and H into quantum mechanical operators. Topics discussed include wave equation in a rectangular cavity, quantization of the electric field, time-dependent perturbation theory, interaction of an electron with the magnetic field, and second-order perturbation theory. Exercises are provided at the end of the chapter.Less

Electromagnetic fields and waves in vacuum

J. Pierrus

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

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

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 laws are generalized where appropriate to include the time-dependent charge and current densities ρ‎(r, t) and J(r, t) respectively. The result is a set of four coupled differential equations—known as Maxwell’s equations— which provide the foundation upon which the theory of classical electrodynamics is based. One of the most important aspects which emerges from Maxwell’s theory is the prediction of electromagnetic waves, and an entire spectrum of electromagnetic radiation. Some of the properties of these waves travelling in unbounded vacuum are considered, as well as their polarization states, energy and momentum conservation in the electromagnetic field and also applications to wave guides and transmission lines.Less

Transformations, matrices and operators

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

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

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 eigenvalues. The linear wave equation quadratic in the infinitesimal generators of space and time translations is derived. Rotation operators and matrix groups are introduced.Less

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

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

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 been a widespread belief that global in time solutions of dispersive equations, asymptotically in time, decouple into a sum of finitely many modulated solitons, a free radiation term, and a term that goes to 0 at infinity. Such a result should hold for globally well-posed equations, or in general, with the additional condition that the solution does not blow-up. When blow-up may occur such decompositions are always expected to be unstable.Less

Relativistic quantum mechanics

Franco Strocchi

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

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

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 relativistic particle interactions by contact interactions (such as field-mediated interactions) rather than by interactions at a distance. It is argued that QFT emerges as a possible solution, fully under mathematical control in the free case.Less

Continuous-Valued Cellular Automata in Two Dimensions

Rudy Rucker

in New Constructions in Cellular Automata

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

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 nonlinear wave equations. In addition we cast some of Hans Meinhardt’s activator-inhibitor reaction-diffusion rules into two dimensions. Some illustrative runs of CAPOW, a. CA simulator, are presented. A cellular automaton, or CA, is a computation made up of finite elements called cells. Each cell contains the same type of state. The cells are updated in parallel, using a rule which is homogeneous, and local. In slightly different words, a CA is a computation based upon a grid of cells, with each cell containing an object called a state. The states are updated in discrete steps, with all the cells being effectively updated at the same time. Each cell uses the same algorithm for its update rule. The update algorithm computes a cell’s new state by using information about the states of the cell’s nearby space-time neighbors, that is, using the state of the cell itself, using the states of the cell’s nearby neighbors, and using the recent prior states of the cell and its neighbors. The states do not necessarily need to be single numbers, they can also be data structures built up from numbers. A CA is said to be discrete valued if its states are built from integers, and a CA is continuous valued if its states are built from real numbers. As Norman Margolus and Tommaso Toffoli have pointed out, CAs are well suited for modeling nature [7]. The parallelism of the CA update process mirrors the uniform flow of time. The homogeneity of the CA update rule across all the cells corresponds to the universality of natural law. And the locality of CAs reflect the fact that nature seems to forbid action at a distance. The use of finite space-time elements for CAs are a necessary evil so that we can compute at all. But one might argue that the use of discrete states is an unnecessary evil. Less