Content-Type: multipart/mixed; boundary="-------------0502100318172" This is a multi-part message in MIME format. ---------------0502100318172 Content-Type: text/plain; name="05-61.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-61.keywords" Rigged Hilbert space ---------------0502100318172 Content-Type: text/plain; name="iopams.sty" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="iopams.sty" %% %% This is file `iopams.sty' %% File to include AMS fonts and extra definitions for bold greek %% characters for use with iopart.cls %% \NeedsTeXFormat{LaTeX2e} \ProvidesPackage{iopams}[1997/02/13 v1.0] \RequirePackage{amsgen}[1995/01/01] \RequirePackage{amsfonts}[1995/01/01] \RequirePackage{amssymb}[1995/01/01] \RequirePackage{amsbsy}[1995/01/01] % \iopamstrue % \newif\ifiopams in iopart.cls & iopbk2e.cls % % allows optional text to be in author guidelines % % Bold lower case Greek letters % \newcommand{\balpha}{\boldsymbol{\alpha}} \newcommand{\bbeta}{\boldsymbol{\beta}} \newcommand{\bgamma}{\boldsymbol{\gamma}} \newcommand{\bdelta}{\boldsymbol{\delta}} \newcommand{\bepsilon}{\boldsymbol{\epsilon}} \newcommand{\bzeta}{\boldsymbol{\zeta}} \newcommand{\bfeta}{\boldsymbol{\eta}} \newcommand{\btheta}{\boldsymbol{\theta}} \newcommand{\biota}{\boldsymbol{\iota}} \newcommand{\bkappa}{\boldsymbol{\kappa}} \newcommand{\blambda}{\boldsymbol{\lambda}} \newcommand{\bmu}{\boldsymbol{\mu}} \newcommand{\bnu}{\boldsymbol{\nu}} \newcommand{\bxi}{\boldsymbol{\xi}} \newcommand{\bpi}{\boldsymbol{\pi}} \newcommand{\brho}{\boldsymbol{\rho}} \newcommand{\bsigma}{\boldsymbol{\sigma}} \newcommand{\btau}{\boldsymbol{\tau}} \newcommand{\bupsilon}{\boldsymbol{\upsilon}} \newcommand{\bphi}{\boldsymbol{\phi}} \newcommand{\bchi}{\boldsymbol{\chi}} \newcommand{\bpsi}{\boldsymbol{\psi}} \newcommand{\bomega}{\boldsymbol{\omega}} \newcommand{\bvarepsilon}{\boldsymbol{\varepsilon}} \newcommand{\bvartheta}{\boldsymbol{\vartheta}} \newcommand{\bvaromega}{\boldsymbol{\varomega}} \newcommand{\bvarrho}{\boldsymbol{\varrho}} \newcommand{\bvarzeta}{\boldsymbol{\varsigma}} %NB really sigma \newcommand{\bvarsigma}{\boldsymbol{\varsigma}} \newcommand{\bvarphi}{\boldsymbol{\varphi}} % % Bold upright capital Greek letters % \newcommand{\bGamma}{\boldsymbol{\Gamma}} \newcommand{\bDelta}{\boldsymbol{\Delta}} \newcommand{\bTheta}{\boldsymbol{\Theta}} \newcommand{\bLambda}{\boldsymbol{\Lambda}} \newcommand{\bXi}{\boldsymbol{\Xi}} \newcommand{\bPi}{\boldsymbol{\Pi}} \newcommand{\bSigma}{\boldsymbol{\Sigma}} \newcommand{\bUpsilon}{\boldsymbol{\Upsilon}} \newcommand{\bPhi}{\boldsymbol{\Phi}} \newcommand{\bPsi}{\boldsymbol{\Psi}} \newcommand{\bOmega}{\boldsymbol{\Omega}} % % Bold versions of miscellaneous symbols % \newcommand{\bpartial}{\boldsymbol{\partial}} \newcommand{\bell}{\boldsymbol{\ell}} \newcommand{\bimath}{\boldsymbol{\imath}} \newcommand{\bjmath}{\boldsymbol{\jmath}} \newcommand{\binfty}{\boldsymbol{\infty}} \newcommand{\bnabla}{\boldsymbol{\nabla}} \newcommand{\bdot}{\boldsymbol{\cdot}} % % Symbols for caption % \renewcommand{\opensquare}{\mbox{$\square$}} \renewcommand{\opentriangle}{\mbox{$\vartriangle$}} \renewcommand{\opentriangledown}{\mbox{$\triangledown$}} \renewcommand{\opendiamond}{\mbox{$\lozenge$}} \renewcommand{\fullsquare}{\mbox{$\blacksquare$}} \newcommand{\fulldiamond}{\mbox{$\blacklozenge$}} \newcommand{\fullstar}{\mbox{$\bigstar$}} \newcommand{\fulltriangle}{\mbox{$\blacktriangle$}} \newcommand{\fulltriangledown}{\mbox{$\blacktriangledown$}} \endinput %% %% End of file `iopams.sty'. ---------------0502100318172 Content-Type: application/x-tex; name="ejp.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ejp.tex" \documentclass[12pt]{iopart} %\usepackage{graphicx} %\usepackage{verbatim} \usepackage{iopams} \usepackage{amsthm} \usepackage{epsf} \eqnobysec %%%%%%%%%%%%%%%%%%% DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\Sw}{{\cal S}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swt}{{\cal S}^{\times}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swp}{{\cal S}^{\prime}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\rhsSwt}{\Sw \subset L^2 \subset \Swt} \newcommand{\rhsSwp}{\Sw \subset L^2 \subset \Swp} \newcommand{\Czi}{C_0^{\infty}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhpm}{\widehat{{\cal S}}_{\pm}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmt}{\widehat{{\cal S}}_{\pm}^{\times}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpml}{\widehat{{\cal S}}_{\pm;l}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmr}{\widehat{{\cal S}}_{\pm;r}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmlt}{\widehat{{\cal S}}_{\pm;l}^{\times}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmrt}{\widehat{{\cal S}}_{\pm;r}^{\times}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmp}{\widehat{{\cal S}}_{\pm}^{\prime}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmlp}{\widehat{{\cal S}}_{\pm;l}^{\prime}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swhpmrp}{\widehat{{\cal S}}_{\pm;r}^{\prime}(\mathbb R \frac{\ }{\ }\{ a,b \})} \newcommand{\Swh}{ \widehat{{\cal S}}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swht}{ \widehat{{\cal S}}^{\times}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhp}{\widehat{{\cal S}}^{\prime}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhhpm}{\widehat{\widehat{{\cal S}}\,}\! _{\pm}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhhpmt}{\widehat{\widehat{{\cal S}}\,}\! _{\pm} \, \hskip-.32cm^{\times}(\mathbb R \frac{\ }{\ } \{ a,b \} )} \newcommand{\Swhhpmp}{\widehat{\widehat{{\cal S}}\,}\! _{\pm} \,\hskip-.22cm ^{\prime}(\mathbb R \frac{\ }{\ } \{ a,b \} )} %%%%%%%%%%%%%%%% END OF DEFINITIONS %%%%%%%%%%%%%%%% \begin{document} %\bibliographystyle{revtex} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ARROWS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\llra{\relbar\joinrel\longrightarrow} %THIS IS LONG \def\mapright#1{\smash{\mathop{\llra}\limits_{#1}}} %ARROW ON LINE \def\mapup#1{\smash{\mathop{\llra}\limits^{#1}}} %CAN PUT SOMETHING OVER IT \def\mapupdown#1#2{\smash{\mathop{\llra}\limits^{#1}_{#2}}} %over&under it% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% END ARROWS %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%% These are the AMS constructs for multiline limits %%%%%%%%%%%%% % \catcode`\@=11 \def\BF#1{{\bf {#1}}} \def\NEG#1{{\rlap/#1}} \def\Let@{\relax\iffalse{\fi\let\\=\cr\iffalse}\fi} \def\vspace@{\def\vspace##1{\crcr\noalign{\vskip##1\relax}}} \def\multilimits@{\bgroup\vspace@\Let@ \baselineskip\fontdimen10 \scriptfont\tw@ \advance\baselineskip\fontdimen12 \scriptfont\tw@ \lineskip\thr@@\fontdimen8 \scriptfont\thr@@ \lineskiplimit\lineskip \vbox\bgroup\ialign\bgroup\hfil$\m@th\scriptstyle{##}$\hfil\crcr} \def\Sb{_\multilimits@} \def\endSb{\crcr\egroup\egroup\egroup} \def\Sp{^\multilimits@} \let\endSp\endSb % %%%%%%%%%%%%%%%%%%%%END of explanations for multiline limits %%%%%%%%%%%%%%%%% \title[The RHS in Quantum Mechanics]{The role of the rigged Hilbert space in Quantum Mechanics} \author{Rafael de la Madrid} \address{Departamento de F\'\i sica Te\'orica, Facultad de Ciencias, Universidad del Pa\'\i s Vasco, 48080 Bilbao, Spain \\ E-mail: {\texttt{wtbdemor@lg.ehu.es}}} \date{\small{January 4, 2005}} \begin{abstract} There is compelling evidence that, when continuous spectrum is present, the natural mathematical setting for Quantum Mechanics is the rigged Hilbert space rather than just the Hilbert space. In particular, Dirac's bra-ket formalism is fully implemented by the rigged Hilbert space rather than just by the Hilbert space. In this paper, we provide a pedestrian introduction to the role the rigged Hilbert space plays in Quantum Mechanics, by way of a simple, exactly solvable example. The procedure will be constructive and based on a recent publication. We also provide a thorough discussion on the physical significance of the rigged Hilbert space. \end{abstract} \pacs{03.65.-w, 02.30.Hq} \maketitle \section{Introduction} \setcounter{equation}{0} \label{sec:introduction} It has been known for several decades that Dirac's bra-ket formalism is mathematically justified not by the Hilbert space alone, but by the rigged Hilbert space (RHS). This is the reason why there is an increasing number of Quantum Mechanics textbooks that already include the rigged Hilbert space as part of their contents (see, for example, Refs.~\cite{ATKINSON}-\cite{KUKULIN}). Despite the importance of the RHS, there is still a lack of simple examples for which the corresponding RHS is constructed in a didactical manner. Even worse, there is no pedagogical discussion on the physical significance of the RHS. In this paper, we use the one-dimensional (1D) rectangular barrier potential to introduce the RHS at the graduate student level. As well, we discuss the physical significance of each of the ingredients that form the RHS. The construction of the RHS of such a simple model will unambiguously show that the RHS is needed at the most basic level of Quantum Mechanics. The present paper is complemented by a previous publication, Ref.~\cite{FOCO}, to which we shall refer the reader interested in a detailed mathematical account on the construction of the RHS of the 1D rectangular barrier. For a general background on the Hilbert and the rigged Hilbert space methods, the reader may consult Ref.~\cite{DIS} and references therein. Dirac's bra-ket formalism was introduced by Dirac in his classic monograph~\cite{DIRAC}. Since its inception, Dirac's abstract algebraic model of {\it bras} and {\it kets} (from the bracket notation for the inner product) proved to be of great calculational value, although there were serious difficulties in finding a mathematical justification for the actual calculations within the Hilbert space, as Dirac~\cite{DIRAC} and von Neumann~\cite{VON} themselves state in their books~\cite{QUOTEVONDIRAC}. As part of his bra-ket formalism, Dirac introduced the so-called Dirac delta function, a formal entity without a counterpart in the classical theory of functions. It was L.~Schwartz who gave a precise meaning to the Dirac delta function as a functional over a space of test functions~\cite{SCHWARTZ}. This led to the development of a new branch of functional analysis, the theory of distributions. By combining von Neumann's Hilbert space with the theory of distributions, I.~Gelfand and collaborators introduced the RHS~\cite{GELFAND,MAURIN}. It was already clear to the creators of the RHS that their formulation was the mathematical support of Dirac's bra-ket formalism~\cite{CITEMAURIN}. The RHS made its first appearance in the Physics literature in the 1960s~\cite{ROBERTS,ANTOINE,B60}, when some physicists also realized that the RHS provides a rigorous mathematical rephrasing of all of the aspects of Dirac's bra-ket formalism. Nowadays, there is a growing consensus that the RHS, rather than the Hilbert space alone, is the natural mathematical setting of Quantum Mechanics~\cite{QUOTEBALLENTINE}. A note on semantics. The word ``rigged'' in rigged Hilbert space has a nautical connotation, such as the phrase ``fully rigged ship;'' it has nothing to do with any unsavory practice such as ``fixing'' or predetermining a result. The phrase ``rigged Hilbert space'' is a direct translation of the phrase ``osnashchyonnoe Hilbertovo prostranstvo'' from the original Russian. A more faithful translation would be ``equipped Hilbert space.'' Indeed, the rigged Hilbert space is just the Hilbert space equipped with distribution theory---in Quantum Mechanics, to rig a Hilbert space means simply to equip that Hilbert space with distribution theory. Thus, the RHS is not a replacement but an enlargement of the Hilbert space. The RHS is {\it neither} an extension {\it nor} an interpretation of the physical principles of Quantum Mechanics, but rather the most natural, concise and logic language to formulate Quantum Mechanics. The RHS is simply a mathematical tool to extract and process the information contained in observables that have continuous spectrum. Observables with discrete spectrum and a finite number of eigenvectors (e.g., spin) do not need the RHS. For such observables, the Hilbert space is sufficient. Actually, as we shall explain, in general only unbounded observables with continuous spectrum need the RHS. The usefulness of the RHS is not simply restricted to accounting for Dirac's bra-ket formalism. The RHS has also proved to be a very useful research tool in the quantum theory of scattering and decay (see Ref.~\cite{DIS} and references therein), and in the construction of generalized spectral decompositions of chaotic maps~\cite{AT93,SUCHANECKI}. In fact, it seems that the RHS is the natural language to deal with problems that involve continuous and resonance spectra. Loosely speaking, a rigged Hilbert space (also called a Gelfand triplet) is a triad of spaces \begin{equation} {\mathbf \Phi} \subset {\cal H} \subset {\mathbf \Phi}^{\times} \label{RHStIntro} \end{equation} such that $\cal H$ is a Hilbert space, $\mathbf \Phi$ is a dense subspace of $\cal H$~\cite{DENSE}, and $\mathbf \Phi ^{\times}$ is the space of antilinear functionals over $\mathbf \Phi$~\cite{FUNCTIONAL}. Mathematically, $\mathbf \Phi$ is the space of test functions, and $\mathbf \Phi ^{\times}$ is the space of distributions. The space $\mathbf \Phi ^{\times}$ is called the antidual space of $\mathbf \Phi$. Associated with the RHS~(\ref{RHStIntro}), there is always another RHS, \begin{equation} {\mathbf \Phi} \subset {\cal H} \subset {\mathbf \Phi}^{\prime} \, , \label{RHSpIntro} \end{equation} where ${\mathbf \Phi}^{\prime}$ is called the dual space of ${\mathbf \Phi}$ and contains the linear functionals over $\mathbf \Phi$~\cite{FUNCTIONAL}. The basic reason why we need the spaces ${\mathbf \Phi}^{\prime}$ and ${\mathbf \Phi}^{\times}$ is that the bras and kets associated with the elements in the continuous spectrum of an observable belong, respectively, to ${\mathbf \Phi}^{\prime}$ and ${\mathbf \Phi}^{\times}$ rather than to ${\cal H}$. The basic reason reason why we need the space $\mathbf \Phi$ is that unbounded operators are not defined on the whole of ${\cal H}$ but only on dense subdomains of ${\cal H}$ that are not invariant under the action of the observables. Such non-invariance makes expectation values, uncertainties and commutation relations not well defined on the whole of $\cal H$. The space $\mathbf \Phi$ is the largest subspace of the Hilbert space on which such expectation values, uncertainties and commutation relations are well defined. The original formulation of the RHS~\cite{GELFAND,MAURIN} does not provide a systematic procedure to construct the RHS generated by the Hamiltonian of the Schr\"odinger equation, since the space $\mathbf \Phi$ is assumed to be given beforehand. Such systematic procedure is important because, after all, claiming that the RHS is the natural setting for Quantum Mechanics is about the same as claiming that, when the Hamiltonian has continuous spectrum, the natural setting for the solutions of the Schr\"odinger equation is the RHS rather than just the Hilbert space. The task of developing a systematic procedure to construct the RHS generated by the Schr\"odinger equation was undertaken in Ref.~\cite{DIS}. The method proposed in Ref.~\cite{DIS}, which was partly based on Refs.~\cite{ROBERTS,ANTOINE,B60}, has been applied to two simple three-dimensional potentials, see Refs.~\cite{JPA02,FP02}, to the three-dimensional free Hamiltonian, see Ref.~\cite{IJTP03}, and to the 1D rectangular barrier potential, see Ref.~\cite{FOCO}. In this paper, we present the method of Ref.~\cite{DIS} in a didactical manner. The organization of the paper is as follows. In Sec.~\ref{sec:why}, we outline the major reasons why the RHS provides the mathematical setting for Quantum Mechanics. In Sec.~\ref{sec:e1dsbp}, we recall the basics of the 1D rectangular potential model. Section~\ref{sec:crhs} provides the RHS of this model. In Sec.~\ref{sec:phymean}, we discuss the physical meaning of each of the ingredients that form the RHS. In Sec.~\ref{sec:gener}, we discuss the relation of the Hilbert space spectral measures with the bras and kets, as well as the limitations of our method to construct RHSs. Finally, Sec.~\ref{sec:conclusions} contains the conclusions to the paper. \section{Motivating the rigged Hilbert space} %\setcounter{equation}{0} \label{sec:why} The {\it linear superposition principle} and the {\it probabilistic interpretation} of Quantum Mechanics are two major guiding principles in our understanding of the microscopic world. These two principles suggest that the space of states be a linear space (which accounts for the superposition principle) endowed with a scalar product (which is used to calculate probability amplitudes). A linear space endowed with a scalar product is called a Hilbert space and is usually denoted by $\cal H$~\cite{HSDEF}. In Quantum Mechanics, observable quantities are represented by linear, self-adjoint operators acting on $\cal H$. The eigenvalues of an operator represent the possible values of the measurement of the corresponding observable. These eigenvalues, which mathematically correspond to the spectrum of the operator, can be discrete (as the energies of a particle in a box), continuous (as the energies of a free, unconstrained particle), or a combination of discrete and continuous (as the energies of the Hydrogen atom). When the spectrum of an observable $A$ is discrete and $A$ is bounded~\cite{UNB}, then $A$ is defined on the whole of $\cal H$ and the eigenvectors of $A$ belong to $\cal H$. In this case, $A$ can be essentially seen as a matrix. This means that, as far as discrete spectrum is concerned, there is no need to extend $\cal H$. However, quantum mechanical observables are in general unbounded~\cite{UNB} and their spectrum has in general a continuous part. In order to deal with continuous spectrum, textbooks usually follow Dirac's bra-ket formalism, which is a heuristic generalization of the linear algebra of Hermitian matrices used for discrete spectrum. As we shall see, the mathematical methods of the Hilbert space are not sufficient to make sense of the prescriptions of Dirac's formalism, the reason for which we shall extend the Hilbert space to the rigged Hilbert space. For pedagogical reasons, we recall the essentials of the linear algebra of Hermitian matrices before proceeding with Dirac's formalism. \subsection{Hermitian matrices} If the measurement of an observable $A$ (e.g., spin) yields a discrete, finite number $N$ of results $a_n$, $n=1, 2, \ldots , N$, then $A$ is realized by a Hermitian matrix on a Hilbert space $\cal H$ of dimension $N$. Since $\cal H$ is an $N$-dimensional linear space, there are $N$ linearly independent vectors $\{ e_n \} _{n=1}^N$ that form an orthonormal basis system for $\cal H$. We denote these basis vectors $e_n$ also by $|e_n\rangle$. The scalar products of the elements of the basis system are written in one of the following ways: \begin{equation} e_n \cdot e_m \equiv (e_n,e_m)\equiv \langle e_n|e_m\rangle = \delta_{nm} \, , \qquad n,m=1,2,\ldots , N \, , \label{dcnII} \end{equation} where $\delta_{nm}$ is the Kronecker delta. As the basis system for the space $\cal H$, it is always possible to choose the eigenvectors of $A$. Therefore, one can choose basis vectors $e_n\in {\cal H}$ which also fulfill \begin{equation} Ae_n=a_n e_n \, . \end{equation} Since $A$ is Hermitian, the eigenvalues $a_n$ are real. The eigenvectors $e_n$ are often labeled by their eigenvalues $a_n$ and denoted by \begin{equation} e_n\equiv |a_n\rangle \, , \end{equation} and they are represented by column vectors. For each column eigenvector $e_n\equiv |a_n\rangle$, there also exists a row eigenvector $\tilde{e}_n\equiv \langle a_n|$ that is a left eigenvector of $A$, \begin{equation} \tilde{e}_n A = a_n \tilde{e}_n \, . \end{equation} Thus, when $A$ is a Hermitian matrix acting on an $N$-dimensional Hilbert space $\mathcal H$, for each eigenvalue $a_n$ of $A$ there exist a right (i.e., column) eigenvector of $A$ \begin{equation} A|a_n\rangle =a_n|a_n \rangle \, , \quad n=1,2,\ldots,N \, , \label{fddeII} \end{equation} and also a left (i.e., row) eigenvector of $A$ \begin{equation} \langle a_n|A =a_n \langle a_n| \, , \quad n=1,2,\ldots,N \, , \label{fddeIIl} \end{equation} such that these row and column eigenvectors are orthonormal, \begin{equation} \langle a_n|a_m \rangle =\delta _{nm} \, , \quad n,m=1,2,\ldots ,N \, , \label{orthonomr} \end{equation} and such that every vector $\varphi \in {\cal H}$ can be written as \begin{equation} \varphi=\sum^N_{n=1}|a_n\rangle \langle a_n|\varphi \rangle \, . \label{fddsdII} \end{equation} Equation~(\ref{fddsdII}) is called the eigenvector expansion of $\varphi$ with respect to the eigenvectors of $A$. The complex numbers $\langle a_n|\varphi \rangle$ are the components of the vector $\varphi$ with respect to the basis of eigenvectors of $A$. Physically, $\langle a_n|\varphi \rangle$ represents the probability amplitude of obtaining the value $a_n$ in the measurement of the observable $A$ on the state $\varphi$. By acting on both sides of Eq.~(\ref{fddsdII}) with $A$, and recalling Eq.~(\ref{fddeII}), we obtain that \begin{equation} A \varphi=\sum^N_{n=1}a_n |a_n\rangle \langle a_n|\varphi \rangle \, . \label{fddsdIIA} \end{equation} \subsection{Dirac's bra-ket formalism} Dirac's formalism is an elegant, heuristic generalization of the algebra of finite dimensional matrices to the continuous-spectrum, infinite-dimensional case. Four of the most important features of Dirac's formalism are: \begin{enumerate} \item To each element of the spectrum of an observable $A$, there correspond a left and a right eigenvector (for the moment, we assume that the spectrum is non-degenerate). If discrete eigenvalues are denoted by $a_n$ and continuous eigenvalues by $a$, then the corresponding right eigenvectors, which are denoted by the kets $|a_n\rangle$ and $|a \rangle$, satisfy \numparts \begin{equation} A|a_n \rangle =a_n|a_n\rangle \, , \label{dketqueintro} \end{equation} \begin{equation} A|a \rangle =a |a \rangle \, , \label{cketequeintro} \end{equation} \endnumparts and the corresponding left eigenvectors, which are denoted by the bras $\langle a_n|$ and $\langle a|$, satisfy \numparts \begin{equation} \langle a_n|A=a_n \langle a_n| \, , \end{equation} \begin{equation} \langle a|A=a \langle a| \, . \label{braeqeneintro} \end{equation} \endnumparts The bras $\langle a|$ generalize the notion of row eigenvectors, whereas the kets $|a \rangle$ generalize the notion of column eigenvectors. \item In analogy to Eq.~(\ref{fddsdII}), the eigenbras and eigenkets of an observable form a complete basis, that is, any wave function $\varphi$ can be expanded in the so-called Dirac basis expansion: \begin{equation} \varphi = \sum_n |a_n\rangle \langle a_n|\varphi \rangle + \int \rmd a \, |a \rangle \langle a |\varphi \rangle \, . \label{introDirbaexp} \end{equation} In addition, the bras and kets furnish a resolution of the identity, \begin{equation} I = \sum_n |a_n\rangle \langle a_n| + \int \rmd a \, |a \rangle \langle a | \, , \label{introresiden} \end{equation} and, in a generalization of Eq.~(\ref{fddsdIIA}), the action of $A$ can be written as \begin{equation} A = \sum_n a_n |a_n\rangle \langle a_n| + \int \rmd a \, a |a \rangle \langle a | \, . \label{introactionA} \end{equation} \item The bras and kets are normalized according to the following rule: \numparts \begin{equation} \langle a_n|a_m\rangle =\delta _{nm} \, , \end{equation} \begin{equation} \langle a |a ^{\prime} \rangle = \delta (a -a ^{\prime}) \, , \label{deltanorintro} \end{equation} \endnumparts where $\delta _{nm}$ is the Kronecker delta and $\delta (a-a ^{\prime})$ is the Dirac delta. The Dirac delta normalization generalizes the orthonormality~(\ref{orthonomr}) of the eigenvectors of a Hermitian matrix. \item Like in the case of two finite-dimensional matrices, all algebraic operations such as the commutator of two observables $A$ and $B$, \begin{equation} [A,B]=AB-BA \, , \label{comuts} \end{equation} are always well defined. \end{enumerate} \subsection{The need of the rigged Hilbert space} In Quantum Mechanics, observables are usually given by differential operators. In the Hilbert space framework, the formal prescription of an observable leads to the definition of a linear operator as follows: One has to find first the Hilbert space $\cal H$, then one sees on what elements of $\cal H$ the action of the observable makes sense, and finally one checks whether the action of the observable remains in $\cal H$. For example, the position observable $Q$ of a 1D particle is given by \begin{equation} Qf(x)=xf(x) \, . \label{fdopx} \end{equation} The Hilbert space of a 1D particle is given by the collection of square integrable functions, \begin{equation} L^2 =\{ f(x) \, | \ \int_{-\infty}^{\infty}\rmd x \, |f(x)|^2 < \infty \} \, , \label{l2space} \end{equation} and the action of $Q$, although in principle well defined on every element of $L^2$, remains in $L^2$ only for the elements of the following subspace: \begin{equation} {\cal D}(Q)= \{ f(x) \in L^2 \, | \ \int_{-\infty}^{\infty}\rmd x \, |xf(x)|^2 < \infty \} \, . \label{domainQ} \end{equation} The space ${\cal D}(Q)$ is the domain of the position operator. Domain (\ref{domainQ}) is not the whole of $L^2$, since the function $g(x)=1/(x+\rmi)$ belongs to $L^2$ but not to ${\cal D}(Q)$; as well, $Q$ is an unbounded operator, because $\| Qg \| = \infty$; as well, $Q{\cal D}(Q)$ is not included in ${\cal D}(Q)$, since $h(x)=1/(x^2+1)$ belongs to ${\cal D}(Q)$ but $Qh$ does not belong to ${\cal D}(Q)$. The denseness and the non-invariance of the domains of unbounded operators create much trouble in the Hilbert space framework, because one has always to be careful whether formal operations are valid. For example, $Q^2=QQ$ is not defined on the whole of $L^2$, not even on the whole of ${\cal D}(Q)$, but only on those square integrable functions such that $x^2f \in L^2$. Also, the expectation value of the measurement of $Q$ in the state $\varphi$, \begin{equation} (\varphi ,Q\varphi ) \, , \label{exintrodispQ} \end{equation} is not finite for every $\varphi \in L^2$, but only when $\varphi \in {\cal D}(Q)$. Similarly, the uncertainty of the measurement of $Q$ in $\varphi$, \begin{equation} \Delta _{\varphi}Q= \sqrt{ (\varphi ,Q^2\varphi )-(\varphi ,Q\varphi )^2} \, , \label{introdispQ} \end{equation} is not defined on the whole of $L^2$. On the other hand, if we denote the momentum observable by \begin{equation} Pf(x)=-\rmi \hbar \frac{\rmd}{\rmd x}f(x) \, , \label{fdopp} \end{equation} then the product of $P$ and $Q$, $PQ$, is not defined everywhere in the Hilbert space, but only on those square integrable functions for which the quantity \begin{equation} PQf(x) = -\rmi \hbar \frac{\rmd}{\rmd x}xf(x)= -\rmi \hbar \left( f(x)+ xf'(x) \right) \label{pqf} \end{equation} makes sense and is square integrable. Obviously, $PQf$ makes sense only when $f$ is differentiable, and $PQf$ remains in $L^2$ only when $f$, $f'$ and $xf'$ are also in $L^2$; thus, $PQ$ is not defined everywhere in $L^2$ but only on those square integrable functions that satisfy the aforementioned conditions. Similar domain concerns arise in calculating the commutator of $P$ with $Q$. As in the case of the position operator, the domain ${\cal D}(A)$ of an unbounded operator $A$ does not coincide with the whole of $\cal H$~\cite{RS84}, but is just a dense subspace of $\cal H$~\cite{DENSE}; also, in general ${\cal D}(A)$ does not remain invariant under the action of $A$, that is, $A{\cal D}(A)$ is not included in ${\cal D}(A)$. Such non-invariance makes expectation values, \begin{equation} (\varphi ,A\varphi ) \, , \label{exintrodispP} \end{equation} uncertainties, \begin{equation} \Delta _{\varphi}A= \sqrt{ (\varphi ,A^2\varphi )-(\varphi ,A\varphi )^2} \, , \label{introdispA} \end{equation} and algebraic operations such as commutation relations not well defined on the whole of the Hilbert space $\cal H$~\cite{INFENER}. Thus, when the position, momentum and energy operators $Q$, $P$, $H$ are unbounded, it is natural to seek a subspace $\mathbf \Phi$ of $\cal H$ on which all of these physical quantities can be calculated and yield meaningful, finite values. Because the reason why these quantities may not be well defined is that the domains of $Q$, $P$ and $H$ are not invariant under the action of these operators, the subspace $\mathbf \Phi$ must be such that it remains invariant under the actions of $Q$, $P$ and $H$. This is why we take as $\mathbf \Phi$ the intersection of the domains of all the powers of $Q$, $P$ and $H$~\cite{ROBERTS}: \begin{equation} {\mathbf \Phi} =\bigcap _{\Sb n,m=0 \\ A,B=Q,P,H \endSb}^{\infty} {\cal D}(A^nB^m) \, . \label{maximalinvas} \end{equation} This space is known as the maximal invariant subspace of the algebra generated by $Q$, $P$ and $H$, because it is the largest subdomain of the Hilbert space that remains invariant under the action of any power of $Q$, $P$ or $H$, \begin{equation} A {\mathbf \Phi} \subset {\mathbf \Phi} \, , \qquad A=Q,P,H \, . \end{equation} On $\mathbf \Phi$, all physical quantities such as expectation values and uncertainties can be associated well-defined, finite values, and algebraic operations such as the commutation relation (\ref{comuts}) are well defined. In addition, the elements of $\mathbf \Phi$ are represented by smooth, continuous functions that have a definitive value at each point, in contrast to the elements of $\cal H$, which are represented by classes of functions which can vary arbitrarily on sets of zero Lebesgue measure. Not only there are compelling reasons to shrink the Hilbert space $\cal H$ to $\mathbf \Phi$, but, as we are going to explain now, there are also reasons to enlarge $\cal H$ to the spaces ${\mathbf \Phi}^{\times}$ and ${\mathbf \Phi}^{\prime}$ of Eqs.~(\ref{RHStIntro}) and (\ref{RHSpIntro}). When the spectrum of $A$ has a continuous part, prescriptions~(\ref{braeqeneintro}) and (\ref{cketequeintro}) associate a bra $\langle a|$ and a ket $|a\rangle$ to each element $a$ of the continuous spectrum of $A$. Obviously, the bras $\langle a|$ and kets $|a\rangle$ are not in the Hilbert space~\cite{SNHS}, and therefore we need two linear spaces larger than the Hilbert space to accommodate them. It turns out that the bras and kets acquire mathematical meaning as distributions. More specifically, the bras $\langle a|$ are {\it linear} functionals over the space $\mathbf \Phi$, and the kets $|a\rangle$ are {\it antilinear} functionals over the space $\mathbf \Phi$. That is, $\langle a| \in {\mathbf \Phi}^{\prime}$ and $|a \rangle \in {\mathbf \Phi}^{\times}$. In this way, the Gelfand triplets of Eqs.~(\ref{RHStIntro}) and (\ref{RHSpIntro}) arise in a natural way. The Hilbert space $\mathcal H$ arises from the requirement that the wave functions be square normalizable. Aside from providing mathematical concepts such as self-adjointness or unitarity, the Hilbert space plays a very important physical role, namely $\cal H$ selects the scalar product that is used to calculate probability amplitudes. The subspace $\mathbf \Phi$ contains those square integrable functions that should be considered as physical, because any expectation value, any uncertainty and any algebraic operation can be calculated for its elements, whereas this is not possible for the rest of the elements of the Hilbert space. The dual space ${\mathbf \Phi}^{\prime}$ and the antidual space ${\mathbf \Phi}^{\times}$ contain respectively the bras and the kets associated with the continuous spectrum of the observables. These bras and kets can be used to expand any $\varphi \in {\mathbf \Phi}$ as in Eq.~(\ref{introDirbaexp}). Thus, the rigged Hilbert space, rather than the Hilbert space alone, can accommodate prescriptions (\ref{dketqueintro})-(\ref{comuts}) of Dirac's formalism. It should be clear that the rigged Hilbert space is just a combination of the Hilbert space with distribution theory. This combination enables us to deal with singular objects such as bras, kets, or Dirac's delta function, something that is impossible if we only use the Hilbert space. Even though it is apparent that the rigged Hilbert space should be an essential part of the mathematical methods for Quantum Mechanics, one may still wonder if the rigged Hilbert space is a helpful tool in teaching Quantum Mechanics, or rather is a technical nuance. Because basic quantum mechanical operators such as $P$ and $Q$ are in general unbounded operators with continuous spectrum~\cite{RS274}, and because this kind of operators necessitates the rigged Hilbert space, it seems pertinent to introduce the rigged Hilbert space in graduate courses on Quantum Mechanics. From a pedagogical standpoint, however, this section's introduction to the rigged Hilbert space is not sufficient. In the classroom, new concepts are better introduced by way of a simple, exactly solvable example. This is why we shall construct the RHS of the 1D rectangular barrier system. We note that this system does not have bound states, and therefore in what follows we shall not deal with discrete spectrum. \subsection{Representations} In working out specific examples, the prescriptions of Dirac's formalism have to be written in a particular representation. Thus, before constructing the RHS of the 1D rectangular barrier, it is convenient to recall some of the basics of representations. In Quantum Mechanics, the most common of all representations is the position representation, sometimes called the $x$-representation. In the $x$-representation, the position operator $Q$ acts as multiplication by $x$. Since the spectrum of $Q$ is $(-\infty ,\infty )$, the $x$-representation of the Hilbert space $\cal H$ is given by the space $L^2$. In this paper, we shall mainly work in the position representation. In general, given an observable $B$, the $b$-representation is that in which the operator $B$ acts as multiplication by $b$, where the $b$'s denote the eigenvalues of $B$. If we denote the spectrum of $B$ by ${\rm Sp}(B)$, then the $b$-representation of the Hilbert space $\cal H$ is given by the space $L^2( {\rm Sp}(B),\rmd b)$, which is the space of square integrable functions $f(b)$ with $b$ running over ${\rm Sp}(B)$. In the $b$-representation, the restrictions to purely continuous spectrum of prescriptions~(\ref{dketqueintro})-(\ref{introresiden}) become \numparts \begin{equation} \langle b|A|a \rangle =a \langle b|a \rangle \, , \label{bqueintro} \end{equation} \begin{equation} \langle a|A|b\rangle =a \langle a|b\rangle \, , \label{bbraeqeneintro} \end{equation} \begin{equation} \langle b| \varphi \rangle = \int \rmd a \, \langle b|a \rangle \langle a |\varphi \rangle \, , \label{bintroDirbaexp} \end{equation} \begin{equation} \delta (b-b')=\langle b|b'\rangle = \int \rmd a \, \langle b|a \rangle \langle a |b'\rangle \, . \label{bintroresiden} \end{equation} \endnumparts The ``scalar product'' $\langle b|a\rangle$ is obtained from Eq.~(\ref{bqueintro}) as the solution of a differential eigenequation in the $b$-representation. The $\langle b|a\rangle$ can also be seen as transition elements from the $a$- to the $b$-representation. Mathematically, the $\langle b|a\rangle$ are to be treated as distributions, and therefore they often appear as kernels of integrals. In this paper, we shall encounter a few of these ``scalar products'' such as $\langle x| p\rangle$, $\langle x| x'\rangle$ and $\langle x| E^{\pm} \rangle _{\rm l,r}$. \section{Example: The one-dimensional rectangular barrier potential} %\setcounter{equation}{0} \label{sec:e1dsbp} The example we consider in this paper is supposed to represent a spinless particle moving in one dimension and impinging on a rectangular barrier. The observables relevant to this system are the position $Q$, the momentum $P$, and the Hamiltonian $H$. In the position representation, $Q$ and $P$ are respectively realized by the differential operators (\ref{fdopx}) and (\ref{fdopp}), whereas $H$ is realized by \begin{equation} Hf(x)= \left( -\frac{\hbar ^2}{2m}\frac{\rmd ^2}{\rmd x^2}+V(x) \right) f(x) \, , \label{fdoph} \end{equation} where \begin{equation} V(x)=\left\{ \begin{array}{ll} 0 &-\infty 0$, we can find a $\varphi$ in $S$ such that $\| f-\varphi \| < \epsilon$. In physical terms, this inequality means that we can replace $f$ by $\varphi$ within an accuracy $\epsilon$. \bibitem{FUNCTIONAL} A function $F: {\mathbf \Phi} \to {\mathbb C}$ is called a linear [respectively antilinear] functional over $\mathbf \Phi$ if for any complex numbers $\alpha , \beta$ and for any $\varphi , \psi \in \mathbf \Phi$, it holds that $F(\alpha \varphi +\beta \psi)=\alpha F(\varphi) +\beta F(\psi )$ [respectively $F(\alpha \varphi +\beta \psi)=\alpha ^* F(\varphi) +\beta ^*F(\psi )$]. \bibitem{JPA02} R.~de la Madrid, J.~Phys.~A: Math.~Gen.~{\bf 35}, 319--342 (2002); {\sf quant-ph/0110165}. \bibitem{FP02} R.~de la Madrid, A.~Bohm, and M.~Gadella, Fortsch.~Phys.~{\bf 50}, 185--216 (2002); {\sf quant-ph/0109154}. \bibitem{IJTP03} R.~de la Madrid, Int.~J.~Theor.~Phys.~{\bf 42}, 2441--2460 (2003); {\sf quant-ph/0210167}. \bibitem{HSDEF} Strictly speaking, a Hilbert space possesses additional properties (e.g., it must be complete with respect to the topology induced by the scalar product). For a more technical definition of the Hilbert space, see for example Ref.~\cite{DIS}. \bibitem{UNB} An operator $A$ is bounded if there is some finite $K$ such that $\| Af \| Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} % \NeedsTeXFormat{LaTeX2e} \ProvidesClass{iopart}[1996/06/10 v0.0 IOP Journals LaTeX article class] \newcommand\@ptsize{0} \newif\if@restonecol \newif\if@titlepage \newif\ifiopams \@titlepagefalse \DeclareOption{a4paper} {\setlength\paperheight {297mm}% \setlength\paperwidth {210mm}} \DeclareOption{letterpaper} {\setlength\paperheight {11in}% \setlength\paperwidth {8.5in}} \DeclareOption{landscape} {\setlength\@tempdima {\paperheight}% \setlength\paperheight {\paperwidth}% \setlength\paperwidth {\@tempdima}} \DeclareOption{10pt}{\renewcommand\@ptsize{0}} \DeclareOption{11pt}{\renewcommand\@ptsize{2}} % No 11pt version \DeclareOption{12pt}{\renewcommand\@ptsize{2}} \DeclareOption{draft}{\setlength\overfullrule{5pt}} \DeclareOption{final}{\setlength\overfullrule{0pt}} \DeclareOption{titlepage}{\@titlepagetrue} \DeclareOption{notitlepage}{\@titlepagefalse} \ExecuteOptions{letterpaper,final} \ProcessOptions \DeclareMathAlphabet{\bi}{OML}{cmm}{b}{it} \DeclareMathAlphabet{\bcal}{OMS}{cmsy}{b}{n} \input{iopart1\@ptsize.clo} \setlength\lineskip{1\p@} \setlength\normallineskip{1\p@} \renewcommand\baselinestretch{} \setlength\parskip{0\p@ \@plus \p@} \@lowpenalty 51 \@medpenalty 151 \@highpenalty 301 \setlength\parindent{2em} \setcounter{topnumber}{8} \renewcommand\topfraction{1} \setcounter{bottomnumber}{3} \renewcommand\bottomfraction{.99} \setcounter{totalnumber}{8} \renewcommand\textfraction{0.01} \renewcommand\floatpagefraction{.8} \setcounter{dbltopnumber}{6} \renewcommand\dbltopfraction{1} \renewcommand\dblfloatpagefraction{.8} % \pretolerance=5000 \tolerance=8000 % % Headings for all pages apart from first % \def\ps@headings{\let\@oddfoot\@empty \let\@evenfoot\@empty \def\@evenhead{\thepage\hfil\itshape\rightmark}% \def\@oddhead{{\itshape\leftmark}\hfil\thepage}% \let\@mkboth\markboth \let\sectionmark\@gobble \let\subsectionmark\@gobble} % % Headings for first page % \def\ps@myheadings{\let\@oddfoot\@empty\let\@evenfoot\@empty \let\@oddhead\@empty\let\@evenhead\@empty \let\@mkboth\@gobbletwo \let\sectionmark\@gobble \let\subsectionmark\@gobble} % % \maketitle just ends page % \newcommand\maketitle{\newpage} % % Article titles % % Usage: \title[Short title]{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \renewcommand{\title}{\@ifnextchar[{\@stitle}{\@ftitle}} \def\@stitle[#1]#2{\markboth{#1}{#1}% \thispagestyle{myheadings}% \vspace*{3pc}{\exhyphenpenalty=10000\hyphenpenalty=10000 \Large\raggedright\noindent \bf#2\par}} \def\@ftitle#1{\markboth{#1}{#1}% \thispagestyle{myheadings}% \vspace*{3pc}{\exhyphenpenalty=10000\hyphenpenalty=10000 \Large\raggedright\noindent \bf#1\par}} % % Can use \paper instead of \title % \let\paper=\title % % Generic title command for articles other than papers % % Usage: \article[Short title]{ARTICLE TYPE}{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \newcommand{\article}{\@ifnextchar[{\@sarticle}{\@farticle}} \def\@sarticle[#1]#2#3{\markboth{#1}{#1}% \thispagestyle{myheadings}% \vspace*{.5pc}% {\parindent=\mathindent \bf #2\par}% \vspace*{1.5pc}% {\exhyphenpenalty=10000\hyphenpenalty=10000 \Large\raggedright\noindent \bf#3\par}}% \def\@farticle#1#2{\markboth{#2}{#2}% \thispagestyle{myheadings}% \vspace*{.5pc}% {\parindent=\mathindent \bf #1\par}% \vspace*{1.5pc}% {\exhyphenpenalty=10000\hyphenpenalty=10000 \Large\raggedright\noindent \bf#2\par}}% % % Letters to the Editor % % Usage \letter{Full title} % No short title is required for Letters % \def\letter#1{\article[Letter to the Editor]{LETTER TO THE EDITOR}{#1}} % % Review articles % % Usage: \review[Short title]{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \def\review{\@ifnextchar[{\@sreview}{\@freview}} \def\@sreview[#1]#2{\@sarticle[#1]{REVIEW ARTICLE}{#2}} \def\@freview#1{\@farticle{REVIEW ARTICLE}{#1}} % % Topical Review % % Usage: \topical[Short title]{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \def\topical{\@ifnextchar[{\@stopical}{\@ftopical}} \def\@stopical[#1]#2{\@sarticle[#1]{TOPICAL REVIEW}{#2}} \def\@ftopical#1{\@farticle{TOPICAL REVIEW}{#1}} % % Comments % % Usage: \comment[Short title]{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \def\comment{\@ifnextchar[{\@scomment}{\@fcomment}} \def\@scomment[#1]#2{\@sarticle[#1]{COMMENT}{#2}} \def\@fcomment#1{\@farticle{COMMENT}{#1}} % % Rapid Communications % % Usage: \rapid[Short title]{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \def\rapid{\@ifnextchar[{\@srapid}{\@frapid}} \def\@srapid[#1]#2{\@sarticle[#1]{RAPID COMMUNICATION}{#2}} \def\@frapid#1{\@farticle{RAPID COMMUNICATION}{#1}} % % Notes % % Usage: \note[Short title]{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \def\note{\@ifnextchar[{\@snote}{\@fnote}} \def\@snote[#1]#2{\@sarticle[#1]{NOTE}{#2}} \def\@fnote#1{\@farticle{NOTE}{#1}} % % Preliminary Communications % % Usage: \prelim[Short title]{Full title} % [Short title] is optional; use where title is too long % or contains footnotes, 50 characters maximum % \def\prelim{\@ifnextchar[{\@sprelim}{\@fprelim}} \def\@sprelim[#1]#2{\@sarticle[#1]{PRELIMINARY COMMUNICATION}{#2}} \def\@fprelim#1{\@farticle{PRELIMINARY COMMUNICATION}{#1}} % % List of authors % % Usage \author[Short form]{List of all authors} % The short form excludes footnote symbols linking authors to addresses % and is used for running heads in printed version (but not on preprints) % \renewcommand{\author}{\@ifnextchar[{\@sauthor}{\@fauthor}} \def\@sauthor[#1]#2{\markright{#1} % for production only \vspace*{1.5pc}% \begin{indented}% \item[]\normalsize\bf\raggedright#2 \end{indented}% \smallskip} \def\@fauthor#1{%\markright{#1} for production only \vspace*{1.5pc}% \begin{indented}% \item[]\normalsize\bf\raggedright#1 \end{indented}% \smallskip} % % Affiliation (authors address) % % Usage: \address{Address of first author} % \address{Address of second author} % Use once for each address, use symbols \dag \ddag \S \P $\|$ % to connect authors with addresses % \newcommand{\address}[1]{\begin{indented} \item[]\rm\raggedright #1 \end{indented}} % % American Mathematical Society Classification Numbers % Usage: \ams{57.XX, 58.XX} % \def\ams#1{\vspace{10pt} \begin{indented} \item[]\rm AMS classification scheme numbers: #1\par \end{indented}} % % A single Physics & Astronomy Classification Number % Usage \pacno{31.10} % \def\pacno#1{\vspace{10pt} \begin{indented} \item[]\rm PACS number: #1\par \end{indented}} % % Physics & Astronomy Classification Numbers (more than one) % Usage \pacs{31.10, 31.20T} % \def\pacs#1{\vspace{10pt} \begin{indented} \item[]\rm PACS numbers: #1\par \end{indented}} % % Submission details. If \jl command used journals name printed % otherwise Institute of Physics Publishing % \def\submitted{\vspace{28pt plus 10pt minus 18pt} \noindent{\small\rm Submitted to: {\it \journal}\par}} % \def\submitto#1{\vspace{28pt plus 10pt minus 18pt} \noindent{\small\rm Submitted to: {\it #1}\par}} % % For articles (other than Letters) not divided into sections % Usage \nosections Start of text % \def\nosections{\vspace{30\p@ plus12\p@ minus12\p@} \noindent\ignorespaces} % % Acknowledgments (no heading if letter) % Usage \ack for Acknowledgments, \ackn for Acknowledgement % \def\ack{\ifletter\bigskip\noindent\ignorespaces\else \section*{Acknowledgments}\fi} \def\ackn{\ifletter\bigskip\noindent\ignorespaces\else \section*{Acknowledgment}\fi} % % Footnotes: symbols selected in order \dag (1), \ddag (2), \S (3), % $\|$ (4), $\P$ (5), $^+$ (6), $^*$ (7), \sharp (8), \dagger\dagger (9) % unless optional argument of [] use to specify required symbol, % 1=\dag, 2=\ddag, etc % Usage: \footnote{Text of footnote} % \footnote[3]{Text of footnote} % \def\footnoterule{}% \setcounter{footnote}{1} \long\def\@makefntext#1{\parindent 1em\noindent \makebox[1em][l]{\footnotesize\rm$\m@th{\fnsymbol{footnote}}$}% \footnotesize\rm #1} \def\@makefnmark{\hbox{${\fnsymbol{footnote}}\m@th$}} \def\@thefnmark{\fnsymbol{footnote}} \def\footnote{\@ifnextchar[{\@xfootnote}{\stepcounter{\@mpfn}% \begingroup\let\protect\noexpand \xdef\@thefnmark{\thempfn}\endgroup \@footnotemark\@footnotetext}} \def\@xfootnote[#1]{\setcounter{footnote}{#1}% \addtocounter{footnote}{-1}\footnote} \def\@fnsymbol#1{\ifcase#1\or \dagger\or \ddagger\or \S\or \|\or \P\or ^{+}\or ^{\tsty *}\or \sharp \or \dagger\dagger \else\@ctrerr\fi\relax} % % IOP Journals % \newcounter{jnl} \newcommand{\jl}[1]{\setcounter{jnl}{#1}} \def\journal{\ifnum\thejnl=0 Institute of Physics Publishing\fi \ifnum\thejnl=1 J. Phys.\ A: Math.\ Gen.\ \fi \ifnum\thejnl=2 J. Phys.\ B: At.\ Mol.\ Opt.\ Phys.\ \fi \ifnum\thejnl=3 J. Phys.:\ Condens. Matter\ \fi \ifnum\thejnl=4 J. Phys.\ G: Nucl.\ Part.\ Phys.\ \fi \ifnum\thejnl=5 Inverse Problems\ \fi \ifnum\thejnl=6 Class. Quantum Grav.\ \fi \ifnum\thejnl=7 Network: Comput.\ Neural Syst.\ \fi \ifnum\thejnl=8 Nonlinearity\ \fi \ifnum\thejnl=9 J. Opt. B: Quantum Semiclass. Opt.\ \fi \ifnum\thejnl=10 Waves Random Media\ \fi \ifnum\thejnl=11 J. Opt. A: Pure Appl. Opt.\ \fi \ifnum\thejnl=12 Phys. Med. Biol.\ \fi \ifnum\thejnl=13 Modelling Simul.\ Mater.\ Sci.\ Eng.\ \fi \ifnum\thejnl=14 Plasma Phys. Control. Fusion\ \fi \ifnum\thejnl=15 Physiol. Meas.\ \fi \ifnum\thejnl=16 Combust. Theory Modelling\ \fi \ifnum\thejnl=17 High Perform.\ Polym.\ \fi \ifnum\thejnl=18 Public Understand. Sci.\ \fi \ifnum\thejnl=19 Rep.\ Prog.\ Phys.\ \fi \ifnum\thejnl=20 J.\ Phys.\ D: Appl.\ Phys.\ \fi \ifnum\thejnl=21 Supercond.\ Sci.\ Technol.\ \fi \ifnum\thejnl=22 Semicond.\ Sci.\ Technol.\ \fi \ifnum\thejnl=23 Nanotechnology\ \fi \ifnum\thejnl=24 Measur.\ Sci.\ Technol.\ \fi \ifnum\thejnl=25 Plasma.\ Sources\ Sci.\ Technol.\ \fi \ifnum\thejnl=26 Smart\ Mater.\ Struct.\ \fi \ifnum\thejnl=27 J.\ Micromech.\ Microeng.\ \fi \ifnum\thejnl=28 Distrib.\ Syst.\ Engng\ \fi \ifnum\thejnl=29 Bioimaging\ \fi \ifnum\thejnl=30 J.\ Radiol. Prot.\ \fi \ifnum\thejnl=31 Europ. J. Phys.\ \fi \ifnum\thejnl=32 J. Opt. A: Pure Appl. Opt.\ \fi \ifnum\thejnl=33 New. J. Phys.\ \fi} % % E-mail addresses (to provide links from headers) % \def\eads#1{\vspace*{5pt}\address{E-mail: #1}} \def\ead#1{\vspace*{5pt}\address{E-mail: \mailto{#1}}} \def\mailto#1{{\tt #1}} % % Switches % \newif\ifletter % \setcounter{secnumdepth}{3} \newcounter {section} \newcounter {subsection}[section] \newcounter {subsubsection}[subsection] \newcounter {paragraph}[subsubsection] \newcounter {subparagraph}[paragraph] \renewcommand\thesection {\arabic{section}} \renewcommand\thesubsection {\thesection.\arabic{subsection}} \renewcommand\thesubsubsection {\thesubsection.\arabic{subsubsection}} \renewcommand\theparagraph {\thesubsubsection.\arabic{paragraph}} \renewcommand\thesubparagraph {\theparagraph.\arabic{subparagraph}} \def\@chapapp{Section} \newcommand\section{\@startsection {section}{1}{\z@}% {-3.5ex \@plus -1ex \@minus -.2ex}% {2.3ex \@plus.2ex}% {\reset@font\normalsize\bfseries\raggedright}} \newcommand\subsection{\@startsection{subsection}{2}{\z@}% {-3.25ex\@plus -1ex \@minus -.2ex}% {1.5ex \@plus .2ex}% {\reset@font\normalsize\itshape\raggedright}} \newcommand\subsubsection{\@startsection{subsubsection}{3}{\z@}% {-3.25ex\@plus -1ex \@minus -.2ex}% {-1em \@plus .2em}% {\reset@font\normalsize\itshape}} \newcommand\paragraph{\@startsection{paragraph}{4}{\z@}% {3.25ex \@plus1ex \@minus.2ex}% {-1em}% {\reset@font\normalsize\itshape}} \newcommand\subparagraph{\@startsection{subparagraph}{5}{\parindent}% {3.25ex \@plus1ex \@minus .2ex}% {-1em}% {\reset@font\normalsize\itshape}} \def\@sect#1#2#3#4#5#6[#7]#8{\ifnum #2>\c@secnumdepth \let\@svsec\@empty\else \refstepcounter{#1}\edef\@svsec{\csname the#1\endcsname. }\fi \@tempskipa #5\relax \ifdim \@tempskipa>\z@ \begingroup #6\relax \noindent{\hskip #3\relax\@svsec}{\interlinepenalty \@M #8\par}% \endgroup \csname #1mark\endcsname{#7}\addcontentsline {toc}{#1}{\ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}\fi #7}\else \def\@svsechd{#6\hskip #3\relax %% \relax added 2 May 90 \@svsec #8\csname #1mark\endcsname {#7}\addcontentsline {toc}{#1}{\ifnum #2>\c@secnumdepth \else \protect\numberline{\csname the#1\endcsname}\fi #7}}\fi \@xsect{#5}} % \def\@ssect#1#2#3#4#5{\@tempskipa #3\relax \ifdim \@tempskipa>\z@ \begingroup #4\noindent{\hskip #1}{\interlinepenalty \@M #5\par}\endgroup \else \def\@svsechd{#4\hskip #1\relax #5}\fi \@xsect{#3}} \setlength\leftmargini{2em} \setlength\leftmarginii{2em} \setlength\leftmarginiii{1.8em} \setlength\leftmarginiv{1.6em} \setlength\leftmarginv{1em} \setlength\leftmarginvi{1em} \setlength\leftmargin{\leftmargini} \setlength\labelsep{0.5em} \setlength\labelwidth{\leftmargini} \addtolength\labelwidth{-\labelsep} \@beginparpenalty -\@lowpenalty \@endparpenalty -\@lowpenalty \@itempenalty -\@lowpenalty \renewcommand\theenumi{\roman{enumi}} \renewcommand\theenumii{\alph{enumii}} \renewcommand\theenumiii{\arabic{enumiii}} \renewcommand\theenumiv{\Alph{enumiv}} \newcommand\labelenumi{(\theenumi)} \newcommand\labelenumii{(\theenumii)} \newcommand\labelenumiii{\theenumiii.} \newcommand\labelenumiv{(\theenumiv)} \renewcommand\p@enumii{(\theenumi)} \renewcommand\p@enumiii{(\theenumi.\theenumii)} \renewcommand\p@enumiv{(\theenumi.\theenumii.\theenumiii)} \newcommand\labelitemi{$\m@th\bullet$} \newcommand\labelitemii{\normalfont\bfseries --} \newcommand\labelitemiii{$\m@th\ast$} \newcommand\labelitemiv{$\m@th\cdot$} \newenvironment{description} {\list{}{\labelwidth\z@ \itemindent-\leftmargin \let\makelabel\descriptionlabel}} {\endlist} \newcommand\descriptionlabel[1]{\hspace\labelsep \normalfont\bfseries #1} \newenvironment{abstract}{% \vspace{16pt plus3pt minus3pt} \begin{indented} \item[]{\bfseries \abstractname.}\quad\rm\ignorespaces} {\end{indented}\if@titlepage\newpage\else\vspace{18\p@ plus18\p@}\fi} \newenvironment{verse} {\let\\=\@centercr \list{}{\itemsep \z@ \itemindent -1.5em% \listparindent\itemindent \rightmargin \leftmargin \advance\leftmargin 1.5em}% \item[]} {\endlist} \newenvironment{quotation} {\list{}{\listparindent 1.5em% \itemindent \listparindent \rightmargin \leftmargin \parsep \z@ \@plus\p@}% \item[]} {\endlist} \newenvironment{quote} {\list{}{\rightmargin\leftmargin}% \item[]} {\endlist} \newenvironment{titlepage} {% \@restonecolfalse\newpage \thispagestyle{empty}% \if@compatibility \setcounter{page}{0} \else \setcounter{page}{1}% \fi}% {\newpage\setcounter{page}{1}} \def\appendix{\@ifnextchar*{\@appendixstar}{\@appendix}} \def\@appendix{\eqnobysec\@appendixstar} \def\@appendixstar{\@@par \ifnumbysec % Added 30/4/94 to get Table A1, \@addtoreset{table}{section} % Table B1 etc if numbering by \@addtoreset{figure}{section}\fi % section \setcounter{section}{0} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{equation}{0} \setcounter{figure}{0} \setcounter{table}{0} \def\thesection{Appendix \Alph{section}} \def\theequation{\ifnumbysec \Alph{section}.\arabic{equation}\else \Alph{section}\arabic{equation}\fi} % Comment A\arabic{equation} maybe \def\thetable{\ifnumbysec % better? 15/4/95 \Alph{section}\arabic{table}\else A\arabic{table}\fi} \def\thefigure{\ifnumbysec \Alph{section}\arabic{figure}\else A\arabic{figure}\fi}} \def\noappendix{\setcounter{figure}{0} \setcounter{table}{0} \def\thetable{\arabic{table}} \def\thefigure{\arabic{figure}}} \setlength\arraycolsep{5\p@} \setlength\tabcolsep{6\p@} \setlength\arrayrulewidth{.4\p@} \setlength\doublerulesep{2\p@} \setlength\tabbingsep{\labelsep} \skip\@mpfootins = \skip\footins \setlength\fboxsep{3\p@} \setlength\fboxrule{.4\p@} \renewcommand\theequation{\arabic{equation}} \newcounter{figure} \renewcommand\thefigure{\@arabic\c@figure} \def\fps@figure{tbp} \def\ftype@figure{1} \def\ext@figure{lof} \def\fnum@figure{\figurename~\thefigure} \newenvironment{figure}{\footnotesize\rm\@float{figure}}% {\end@float\normalsize\rm} \newenvironment{figure*}{\footnotesize\rm\@dblfloat{figure}}{\end@dblfloat} \newcounter{table} \renewcommand\thetable{\@arabic\c@table} \def\fps@table{tbp} \def\ftype@table{2} \def\ext@table{lot} \def\fnum@table{\tablename~\thetable} \newenvironment{table}{\footnotesize\rm\@float{table}}% {\end@float\normalsize\rm} \newenvironment{table*}{\footnotesize\rm\@dblfloat{table}}% {\end@dblfloat\normalsize\rm} \newlength\abovecaptionskip \newlength\belowcaptionskip \setlength\abovecaptionskip{10\p@} \setlength\belowcaptionskip{0\p@} % % Added redefinition of \@caption so captions are not written to % aux file therefore less need to \protect fragile commands % \long\def\@caption#1[#2]#3{\par\begingroup \@parboxrestore \normalsize \@makecaption{\csname fnum@#1\endcsname}{\ignorespaces #3}\par \endgroup} % \long\def\@makecaption#1#2{\vskip \abovecaptionskip \begin{indented} \item[]{\bf #1.} #2 \end{indented}\vskip\belowcaptionskip} \let\@portraitcaption=\@makecaption \DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm} \DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf} \DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt} \DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf} \DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit} \DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl} \DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc} \ifiopams \renewcommand{\cal}{\protect\pcal} \else \newcommand{\cal}{\protect\pcal} \fi \newcommand{\pcal}{\@fontswitch{\relax}{\mathcal}} \ifiopams \renewcommand{\mit}{\protect\pmit} \else \newcommand{\mit}{\protect\pmit} \fi \newcommand{\pmit}{\@fontswitch{\relax}{\mathnormal}} \newcommand\@pnumwidth{1.55em} \newcommand\@tocrmarg {2.55em} \newcommand\@dotsep{4.5} \setcounter{tocdepth}{3} \newcommand\tableofcontents{% \section*{\contentsname \@mkboth{\uppercase{\contentsname}}{\uppercase{\contentsname}}}% \@starttoc{toc}% } \newcommand\l@part[2]{% \ifnum \c@tocdepth >-2\relax \addpenalty{\@secpenalty}% \addvspace{2.25em \@plus\p@}% \begingroup \setlength\@tempdima{3em}% \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth {\leavevmode \large \bfseries #1\hfil \hbox to\@pnumwidth{\hss #2}}\par \nobreak \if@compatibility \global\@nobreaktrue \everypar{\global\@nobreakfalse\everypar{}} \fi \endgroup \fi} \newcommand\l@section[2]{% \ifnum \c@tocdepth >\z@ \addpenalty{\@secpenalty}% \addvspace{1.0em \@plus\p@}% \setlength\@tempdima{1.5em}% \begingroup \parindent \z@ \rightskip \@pnumwidth \parfillskip -\@pnumwidth \leavevmode \bfseries \advance\leftskip\@tempdima \hskip -\leftskip #1\nobreak\hfil \nobreak\hbox to\@pnumwidth{\hss #2}\par \endgroup \fi} \newcommand\l@subsection {\@dottedtocline{2}{1.5em}{2.3em}} \newcommand\l@subsubsection{\@dottedtocline{3}{3.8em}{3.2em}} \newcommand\l@paragraph {\@dottedtocline{4}{7.0em}{4.1em}} \newcommand\l@subparagraph {\@dottedtocline{5}{10em}{5em}} \newcommand\listoffigures{% \section*{\listfigurename \@mkboth{\uppercase{\listfigurename}}% {\uppercase{\listfigurename}}}% \@starttoc{lof}% } \newcommand\l@figure{\@dottedtocline{1}{1.5em}{2.3em}} \newcommand\listoftables{% \section*{\listtablename \@mkboth{\uppercase{\listtablename}}{\uppercase{\listtablename}}}% \@starttoc{lot}% } \let\l@table\l@figure \newenvironment{theindex} {\if@twocolumn \@restonecolfalse \else \@restonecoltrue \fi \columnseprule \z@ \columnsep 35\p@ \twocolumn[\section*{\indexname}]% \@mkboth{\uppercase{\indexname}}% {\uppercase{\indexname}}% \thispagestyle{plain}\parindent\z@ \parskip\z@ \@plus .3\p@\relax \let\item\@idxitem} {\if@restonecol\onecolumn\else\clearpage\fi} \newcommand\@idxitem {\par\hangindent 40\p@} \newcommand\subitem {\par\hangindent 40\p@ \hspace*{20\p@}} \newcommand\subsubitem{\par\hangindent 40\p@ \hspace*{30\p@}} \newcommand\indexspace{\par \vskip 10\p@ \@plus5\p@ \@minus3\p@\relax} \newcommand\contentsname{Contents} \newcommand\listfigurename{List of Figures} \newcommand\listtablename{List of Tables} \newcommand\refname{References} \newcommand\indexname{Index} \newcommand\figurename{Figure} \newcommand\tablename{Table} \newcommand\partname{Part} \newcommand\appendixname{Appendix} \newcommand\abstractname{Abstract} \newcommand\today{\number\day\space\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\year} \setlength\columnsep{10\p@} \setlength\columnseprule{0\p@} \newcommand{\Tables}{\clearpage\section*{Tables and table captions} \def\fps@table{hp}\noappendix} \newcommand{\Figures}{\clearpage\section*{Figure captions} \def\fps@figure{hp}\noappendix} % \newcommand{\Figure}[1]{\begin{figure} \caption{#1} \end{figure}} % \newcommand{\Table}[1]{\begin{table} \caption{#1} \begin{indented} \lineup \item[]\begin{tabular}{@{}l*{15}{l}}} \def\endTable{\end{tabular}\end{indented}\end{table}} \let\endtab=\endTable % \newcommand{\fulltable}[1]{\begin{table} \caption{#1} \lineup \begin{tabular*}{\textwidth}{@{}l*{15}{@{\extracolsep{0pt plus 12pt}}l}}} \def\endfulltable{\end{tabular*}\end{table}} % % \newcommand{\Bibliography}[1]{\section*{References}\par\numrefs{#1}} \newcommand{\References}{\section*{References}\par\refs} \def\thebibliography#1{\list {\hfil[\arabic{enumi}]}{\topsep=0\p@\parsep=0\p@ \partopsep=0\p@\itemsep=0\p@ \labelsep=5\p@\itemindent=-10\p@ \settowidth\labelwidth{\footnotesize[#1]}% \leftmargin\labelwidth \advance\leftmargin\labelsep \advance\leftmargin -\itemindent \usecounter{enumi}}\footnotesize \def\newblock{\ } \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} \let\endthebibliography=\endlist \def\numrefs#1{\begin{thebibliography}{#1}} \def\endnumrefs{\end{thebibliography}} \let\endbib=\endnumrefs % \def\thereferences{\list{}{\topsep=0\p@\parsep=0\p@ \partopsep=0\p@\itemsep=0\p@\labelsep=0\p@\itemindent=-18\p@ \labelwidth=0\p@\leftmargin=18\p@ }\footnotesize\rm \def\newblock{\ } \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax } % \let\endthereferences=\endlist \newlength{\indentedwidth} \newdimen\mathindent \indentedwidth=\mathindent % % Macro to used for references in the Harvard system % \newenvironment{harvard}{\list{}{\topsep=0\p@\parsep=0\p@ \partopsep=0\p@\itemsep=0\p@\labelsep=0\p@\itemindent=-18\p@ \labelwidth=0\p@\leftmargin=18\p@ }\footnotesize\rm \def\newblock{\ } \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax}{\endlist} % \def\refs{\begin{harvard}} \def\endrefs{\end{harvard}} % \newenvironment{indented}{\begin{indented}}{\end{indented}} \newenvironment{varindent}[1]{\begin{varindent}{#1}}{\end{varindent}} % \def\indented{\list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@ \labelwidth=0\p@\leftmargin=\mathindent\topsep=0\p@\partopsep=0\p@ \parsep=0\p@\listparindent=15\p@}\footnotesize\rm} \let\endindented=\endlist \def\varindent#1{\setlength{\varind}{#1}% \list{}{\itemsep=0\p@\labelsep=0\p@\itemindent=0\p@ \labelwidth=0\p@\leftmargin=\varind\topsep=0\p@\partopsep=0\p@ \parsep=0\p@\listparindent=15\p@}\footnotesize\rm} \let\endvarindent=\endlist \def\[{\relax\ifmmode\@badmath\else \begin{trivlist} \@beginparpenalty\predisplaypenalty \@endparpenalty\postdisplaypenalty \item[]\leavevmode \hbox to\linewidth\bgroup$ \displaystyle \hskip\mathindent\bgroup\fi} \def\]{\relax\ifmmode \egroup $\hfil \egroup \end{trivlist}\else \@badmath \fi} \def\equation{\@beginparpenalty\predisplaypenalty \@endparpenalty\postdisplaypenalty \refstepcounter{equation}\trivlist \item[]\leavevmode \hbox to\linewidth\bgroup $ \displaystyle \hskip\mathindent} \def\endequation{$\hfil \displaywidth\linewidth\@eqnnum\egroup \endtrivlist} % \@namedef{equation*}{\[} \@namedef{endequation*}{\]} % \def\eqnarray{\stepcounter{equation}\let\@currentlabel=\theequation \global\@eqnswtrue \global\@eqcnt\z@\tabskip\mathindent\let\\=\@eqncr \abovedisplayskip\topsep\ifvmode\advance\abovedisplayskip\partopsep\fi \belowdisplayskip\abovedisplayskip \belowdisplayshortskip\abovedisplayskip \abovedisplayshortskip\abovedisplayskip $$\halign to \linewidth\bgroup\@eqnsel$\displaystyle\tabskip\z@ {##{}}$&\global\@eqcnt\@ne $\displaystyle{{}##{}}$\hfil &\global\@eqcnt\tw@ $\displaystyle{{}##}$\hfil \tabskip\@centering&\llap{##}\tabskip\z@\cr} \def\endeqnarray{\@@eqncr\egroup \global\advance\c@equation\m@ne$$\global\@ignoretrue } \mathindent = 6pc % \def\eqalign#1{\null\vcenter{\def\\{\cr}\openup\jot\m@th \ialign{\strut$\displaystyle{##}$\hfil&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} % \def\eqalignno#1{\displ@y \tabskip\z@skip \halign to\displaywidth{\hspace{5pc}$\@lign\displaystyle{##}$% \tabskip\z@skip &$\@lign\displaystyle{{}##}$\hfill\tabskip\@centering &\llap{$\@lign\hbox{\rm##}$}\tabskip\z@skip\crcr #1\crcr}} % \newif\ifnumbysec \def\theequation{\ifnumbysec \arabic{section}.\arabic{equation}\else \arabic{equation}\fi} \def\eqnobysec{\numbysectrue\@addtoreset{equation}{section}} \newcounter{eqnval} \def\numparts{\addtocounter{equation}{1}% \setcounter{eqnval}{\value{equation}}% \setcounter{equation}{0}% \def\theequation{\ifnumbysec \arabic{section}.\arabic{eqnval}{\it\alph{equation}}% \else\arabic{eqnval}{\it\alph{equation}}\fi}} \def\endnumparts{\def\theequation{\ifnumbysec \arabic{section}.\arabic{equation}\else \arabic{equation}\fi}% \setcounter{equation}{\value{eqnval}}} % \def\cases#1{% \left\{\,\vcenter{\def\\{\cr}\normalbaselines\openup1\jot\m@th% \ialign{\strut$\displaystyle{##}\hfil$&\tqs \rm##\hfil\crcr#1\crcr}}\right.}% % \newcommand{\e}{\mathrm{e}} \newcommand{\rme}{\mathrm{e}} \newcommand{\rmi}{\mathrm{i}} \newcommand{\rmd}{\mathrm{d}} \renewcommand{\qquad}{\hspace*{25pt}} \newcommand{\tdot}[1]{\stackrel{\dots}{#1}} % Added 1/9/94 \newcommand{\tqs}{\hspace*{25pt}} \newcommand{\fl}{\hspace*{-\mathindent}} \newcommand{\Tr}{\mathop{\mathrm{Tr}}\nolimits} \newcommand{\tr}{\mathop{\mathrm{tr}}\nolimits} \newcommand{\Or}{\mathord{\mathrm{O}}} %changed from \mathop 20/1/95 \newcommand{\lshad}{[\![} \newcommand{\rshad}{]\!]} \newcommand{\case}[2]{{\textstyle\frac{#1}{#2}}} \def\pt(#1){({\it #1\/})} \newcommand{\dsty}{\displaystyle} \newcommand{\tsty}{\textstyle} \newcommand{\ssty}{\scriptstyle} \newcommand{\sssty}{\scriptscriptstyle} \def\lo#1{\llap{${}#1{}$}} \def\eql{\llap{${}={}$}} \def\lsim{\llap{${}\sim{}$}} \def\lsimeq{\llap{${}\simeq{}$}} \def\lequiv{\llap{${}\equiv{}$}} % \newcommand{\eref}[1]{(\ref{#1})} \newcommand{\sref}[1]{section~\ref{#1}} \newcommand{\fref}[1]{figure~\ref{#1}} \newcommand{\tref}[1]{table~\ref{#1}} \newcommand{\Eref}[1]{Equation (\ref{#1})} \newcommand{\Sref}[1]{Section~\ref{#1}} \newcommand{\Fref}[1]{Figure~\ref{#1}} \newcommand{\Tref}[1]{Table~\ref{#1}} \newcommand{\opencircle}{\mbox{\Large$\circ\,$}} % moved Large outside maths \newcommand{\opensquare}{\mbox{$\rlap{$\sqcap$}\sqcup$}} \newcommand{\opentriangle}{\mbox{$\triangle$}} \newcommand{\opentriangledown}{\mbox{$\bigtriangledown$}} \newcommand{\opendiamond}{\mbox{$\diamondsuit$}} \newcommand{\fullcircle}{\mbox{{\Large$\bullet\,$}}} % moved Large outside maths \newcommand{\fullsquare}{\,\vrule height5pt depth0pt width5pt} \newcommand{\dotted}{\protect\mbox{${\mathinner{\cdotp\cdotp\cdotp\cdotp\cdotp\cdotp}}$}} \newcommand{\dashed}{\protect\mbox{-\; -\; -\; -}} \newcommand{\broken}{\protect\mbox{-- -- --}} \newcommand{\longbroken}{\protect\mbox{--- --- ---}} \newcommand{\chain}{\protect\mbox{--- $\cdot$ ---}} \newcommand{\dashddot}{\protect\mbox{--- $\cdot$ $\cdot$ ---}} \newcommand{\full}{\protect\mbox{------}} \def\;{\protect\psemicolon} \def\psemicolon{\relax\ifmmode\mskip\thickmuskip\else\kern .3333em\fi} \def\lineup{\def\0{\hbox{\phantom{\footnotesize\rm 0}}}% \def\m{\hbox{$\phantom{-}$}}% \def\-{\llap{$-$}}} % %%%%%%%%%%%%%%%%%%%%% % Tables rules % %%%%%%%%%%%%%%%%%%%%% \newcommand{\boldarrayrulewidth}{1\p@} % Width of bold rule in tabular environment. \def\bhline{\noalign{\ifnum0=`}\fi\hrule \@height \boldarrayrulewidth \futurelet \@tempa\@xhline} \def\@xhline{\ifx\@tempa\hline\vskip \doublerulesep\fi \ifnum0=`{\fi}} % % Rules for tables with extra space around % \newcommand{\br}{\ms\bhline\ms} \newcommand{\mr}{\ms\hline\ms} % \newcommand{\centre}[2]{\multispan{#1}{\hfill #2\hfill}} \newcommand{\crule}[1]{\multispan{#1}{\hspace*{\tabcolsep}\hrulefill \hspace*{\tabcolsep}}} \newcommand{\fcrule}[1]{\ifnum\thetabtype=1\multispan{#1}{\hrulefill \hspace*{\tabcolsep}}\else\multispan{#1}{\hrulefill}\fi} % % Extra spaces for tables and displayed equations % \newcommand{\ms}{\noalign{\vspace{3\p@ plus2\p@ minus1\p@}}} \newcommand{\bs}{\noalign{\vspace{6\p@ plus2\p@ minus2\p@}}} \newcommand{\ns}{\noalign{\vspace{-3\p@ plus-1\p@ minus-1\p@}}} \newcommand{\es}{\noalign{\vspace{6\p@ plus2\p@ minus2\p@}}\displaystyle}% % \newcommand{\etal}{{\it et al\/}\ } \newcommand{\dash}{------} \newcommand{\nonum}{\par\item[]} %\par added 1/9/93 \newcommand{\mat}[1]{\underline{\underline{#1}}} % % abbreviations for IOPP journals % \newcommand{\CQG}{{\it Class. Quantum Grav.} } \newcommand{\CTM}{{\it Combust. Theory Modelling\/} } \newcommand{\DSE}{{\it Distrib. Syst. Engng\/} } \newcommand{\EJP}{{\it Eur. J. Phys.} } \newcommand{\HPP}{{\it High Perform. Polym.} } % added 4/5/93 \newcommand{\IP}{{\it Inverse Problems\/} } \newcommand{\JHM}{{\it J. Hard Mater.} } % added 4/5/93 \newcommand{\JO}{{\it J. Opt.} } \newcommand{\JOA}{{\it J. Opt. A: Pure Appl. Opt.} } \newcommand{\JOB}{{\it J. Opt. B: Quantum Semiclass. Opt.} } \newcommand{\JPA}{{\it J. Phys. A: Math. Gen.} } \newcommand{\JPB}{{\it J. Phys. B: At. Mol. Phys.} } %1968-87 \newcommand{\jpb}{{\it J. Phys. B: At. Mol. Opt. Phys.} } %1988 and onwards \newcommand{\JPC}{{\it J. Phys. C: Solid State Phys.} } %1968--1988 \newcommand{\JPCM}{{\it J. Phys.: Condens. Matter\/} } %1989 and onwards \newcommand{\JPD}{{\it J. Phys. D: Appl. Phys.} } \newcommand{\JPE}{{\it J. Phys. E: Sci. Instrum.} } \newcommand{\JPF}{{\it J. Phys. F: Met. Phys.} } \newcommand{\JPG}{{\it J. Phys. G: Nucl. Phys.} } %1975--1988 \newcommand{\jpg}{{\it J. Phys. G: Nucl. Part. Phys.} } %1989 and onwards \newcommand{\MSMSE}{{\it Modelling Simulation Mater. Sci. Eng.} } \newcommand{\MST}{{\it Meas. Sci. Technol.} } %1990 and onwards \newcommand{\NET}{{\it Network: Comput. Neural Syst.} } \newcommand{\NJP}{{\it New J. Phys.} } \newcommand{\NL}{{\it Nonlinearity\/} } \newcommand{\NT}{{\it Nanotechnology} } \newcommand{\PAO}{{\it Pure Appl. Optics\/} } \newcommand{\PM}{{\it Physiol. Meas.} } % added 4/5/93 \newcommand{\PMB}{{\it Phys. Med. Biol.} } \newcommand{\PPCF}{{\it Plasma Phys. Control. Fusion\/} } % added 4/5/93 \newcommand{\PSST}{{\it Plasma Sources Sci. Technol.} } \newcommand{\PUS}{{\it Public Understand. Sci.} } \newcommand{\QO}{{\it Quantum Opt.} } \newcommand{\QSO}{{\em Quantum Semiclass. Opt.} } \newcommand{\RPP}{{\it Rep. Prog. Phys.} } \newcommand{\SLC}{{\it Sov. Lightwave Commun.} } % added 4/5/93 \newcommand{\SST}{{\it Semicond. Sci. Technol.} } \newcommand{\SUST}{{\it Supercond. Sci. Technol.} } \newcommand{\WRM}{{\it Waves Random Media\/} } \newcommand{\JMM}{{\it J. Micromech. Microeng.\/} } % % Other commonly quoted journals % \newcommand{\AC}{{\it Acta Crystallogr.} } \newcommand{\AM}{{\it Acta Metall.} } \newcommand{\AP}{{\it Ann. Phys., Lpz.} } \newcommand{\APNY}{{\it Ann. Phys., NY\/} } \newcommand{\APP}{{\it Ann. Phys., Paris\/} } \newcommand{\CJP}{{\it Can. J. Phys.} } \newcommand{\JAP}{{\it J. Appl. Phys.} } \newcommand{\JCP}{{\it J. Chem. Phys.} } \newcommand{\JJAP}{{\it Japan. J. Appl. Phys.} } \newcommand{\JP}{{\it J. Physique\/} } \newcommand{\JPhCh}{{\it J. Phys. Chem.} } \newcommand{\JMMM}{{\it J. Magn. Magn. Mater.} } \newcommand{\JMP}{{\it J. Math. Phys.} } \newcommand{\JOSA}{{\it J. Opt. Soc. Am.} } \newcommand{\JPSJ}{{\it J. Phys. Soc. Japan\/} } \newcommand{\JQSRT}{{\it J. Quant. Spectrosc. Radiat. Transfer\/} } \newcommand{\NC}{{\it Nuovo Cimento\/} } \newcommand{\NIM}{{\it Nucl. Instrum. Methods\/} } \newcommand{\NP}{{\it Nucl. Phys.} } \newcommand{\PL}{{\it Phys. Lett.} } \newcommand{\PR}{{\it Phys. Rev.} } \newcommand{\PRL}{{\it Phys. Rev. Lett.} } \newcommand{\PRS}{{\it Proc. R. Soc.} } \newcommand{\PS}{{\it Phys. Scr.} } \newcommand{\PSS}{{\it Phys. Status Solidi\/} } \newcommand{\PTRS}{{\it Phil. Trans. R. Soc.} } \newcommand{\RMP}{{\it Rev. Mod. Phys.} } \newcommand{\RSI}{{\it Rev. Sci. Instrum.} } \newcommand{\SSC}{{\it Solid State Commun.} } \newcommand{\ZP}{{\it Z. Phys.} } % \pagestyle{headings} \pagenumbering{arabic} % Arabic page numbers \raggedbottom \onecolumn \endinput %% %% End of file `iopart.cls'. ---------------0502100318172 Content-Type: application/x-tex; name="iopart10.clo" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="iopart10.clo" %% %% This is file `iopart10.clo' %% %% This file is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \ProvidesFile{iopart10.clo}[1997/01/13 v1.0 IOP Book file (size option)] \renewcommand\normalsize{% \@setfontsize\normalsize\@xpt\@xiipt \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ \belowdisplayskip \abovedisplayskip \let\@listi\@listI} \normalsize \newcommand\small{% \@setfontsize\small\@ixpt{11}% \abovedisplayskip 8.5\p@ \@plus3\p@ \@minus4\p@ \abovedisplayshortskip \z@ \@plus2\p@ \belowdisplayshortskip 4\p@ \@plus2\p@ \@minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 4\p@ \@plus2\p@ \@minus2\p@ \parsep 2\p@ \@plus\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\footnotesize{% \@setfontsize\footnotesize\@viiipt{9.5}% \abovedisplayskip 6\p@ \@plus2\p@ \@minus4\p@ \abovedisplayshortskip \z@ \@plus\p@ \belowdisplayshortskip 3\p@ \@plus\p@ \@minus2\p@ \def\@listi{\leftmargin\leftmargini \topsep 3\p@ \@plus\p@ \@minus\p@ \parsep 2\p@ \@plus\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\scriptsize{\@setfontsize\scriptsize\@viipt\@viiipt} \newcommand\tiny{\@setfontsize\tiny\@vpt\@vipt} \newcommand\large{\@setfontsize\large\@xiipt{14}} \newcommand\Large{\@setfontsize\Large\@xivpt{18}} \newcommand\LARGE{\@setfontsize\LARGE\@xviipt{22}} \newcommand\huge{\@setfontsize\huge\@xxpt{25}} \newcommand\Huge{\@setfontsize\Huge\@xxvpt{30}} \if@twocolumn \setlength\parindent{12\p@} \else \setlength\parindent{15\p@} \fi \setlength\headheight{12\p@} \setlength\headsep {12\p@} \setlength\topskip {10\p@} \setlength\footskip{20\p@} \setlength\maxdepth{.5\topskip} \setlength\@maxdepth\maxdepth \setlength\textwidth{31pc} \setlength\textheight{49pc} \setlength\oddsidemargin {24\p@} \setlength\evensidemargin {24\p@} \setlength\marginparwidth {72\p@} \setlength\marginparsep {10\p@} \setlength\marginparpush{5\p@} \setlength\topmargin{\z@} \setlength\footnotesep{6.65\p@} \setlength{\skip\footins} {9\p@ \@plus 4\p@ \@minus 2\p@} \setlength\floatsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\textfloatsep {20\p@ \@plus 2\p@ \@minus 4\p@} \setlength\intextsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\dblfloatsep {12\p@ \@plus 2\p@ \@minus 2\p@} \setlength\dbltextfloatsep{20\p@ \@plus 2\p@ \@minus 4\p@} \setlength\@fptop{0\p@} \setlength\@fpsep{8\p@ \@plus 2fil} \setlength\@fpbot{0\p@} \setlength\@dblfptop{0\p@} \setlength\@dblfpsep{8\p@ \@plus 2fil} \setlength\@dblfpbot{0\p@} \setlength\partopsep{2\p@ \@plus 1\p@ \@minus 1\p@} \def\@listI{\leftmargin\leftmargini \parsep=\z@ \topsep=5\p@ \@plus3\p@ \@minus3\p@ \itemsep=3\p@ \@plus2\p@ \@minus\p@} \let\@listi\@listI \@listi \def\@listii {\leftmargin\leftmarginii \labelwidth\leftmarginii \advance\labelwidth-\labelsep \topsep=2\p@ \@plus2\p@ \@minus\p@ \parsep=\z@ \itemsep=\parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii \advance\labelwidth-\labelsep \topsep=\z@ \parsep=\z@ \partopsep=\z@ \itemsep=\z@} \def\@listiv {\leftmargin\leftmarginiv \labelwidth\leftmarginiv \advance\labelwidth-\labelsep} \def\@listv {\leftmargin\leftmarginv \labelwidth\leftmarginv \advance\labelwidth-\labelsep} \def\@listvi {\leftmargin\leftmarginvi \labelwidth\leftmarginvi \advance\labelwidth-\labelsep} \endinput %% %% End of file `iopart.clo'. ---------------0502100318172 Content-Type: application/x-tex; name="iopart12.clo" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="iopart12.clo" %% %% This is file `iopart12.clo' %% %% This file is distributed in the hope that it will be useful, %% but WITHOUT ANY WARRANTY; without even the implied warranty of %% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. %% %% \CharacterTable %% {Upper-case \A\B\C\D\E\F\G\H\I\J\K\L\M\N\O\P\Q\R\S\T\U\V\W\X\Y\Z %% Lower-case \a\b\c\d\e\f\g\h\i\j\k\l\m\n\o\p\q\r\s\t\u\v\w\x\y\z %% Digits \0\1\2\3\4\5\6\7\8\9 %% Exclamation \! Double quote \" Hash (number) \# %% Dollar \$ Percent \% Ampersand \& %% Acute accent \' Left paren \( Right paren \) %% Asterisk \* Plus \+ Comma \, %% Minus \- Point \. Solidus \/ %% Colon \: Semicolon \; Less than \< %% Equals \= Greater than \> Question mark \? %% Commercial at \@ Left bracket \[ Backslash \\ %% Right bracket \] Circumflex \^ Underscore \_ %% Grave accent \` Left brace \{ Vertical bar \| %% Right brace \} Tilde \~} \ProvidesFile{iopart12.clo}[1997/01/15 v1.0 LaTeX2e file (size option)] \renewcommand\normalsize{% \@setfontsize\normalsize\@xiipt{16}% \abovedisplayskip 12\p@ \@plus3\p@ \@minus7\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ \belowdisplayskip \abovedisplayskip \let\@listi\@listI} \normalsize \newcommand\small{% \@setfontsize\small\@xipt{14}% \abovedisplayskip 11\p@ \@plus3\p@ \@minus6\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6.5\p@ \@plus3.5\p@ \@minus3\p@ \def\@listi{\leftmargin\leftmargini \topsep 9\p@ \@plus3\p@ \@minus5\p@ \parsep 4.5\p@ \@plus2\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\footnotesize{% % \@setfontsize\footnotesize\@xpt\@xiipt \@setfontsize\footnotesize\@xpt{13}% \abovedisplayskip 10\p@ \@plus2\p@ \@minus5\p@ \abovedisplayshortskip \z@ \@plus3\p@ \belowdisplayshortskip 6\p@ \@plus3\p@ \@minus3\p@ \def\@listi{\leftmargin\leftmargini \topsep 6\p@ \@plus2\p@ \@minus2\p@ \parsep 3\p@ \@plus2\p@ \@minus\p@ \itemsep \parsep}% \belowdisplayskip \abovedisplayskip } \newcommand\scriptsize{\@setfontsize\scriptsize\@viiipt{9.5}} \newcommand\tiny{\@setfontsize\tiny\@vipt\@viipt} \newcommand\large{\@setfontsize\large\@xivpt{18}} \newcommand\Large{\@setfontsize\Large\@xviipt{22}} \newcommand\LARGE{\@setfontsize\LARGE\@xxpt{25}} \newcommand\huge{\@setfontsize\huge\@xxvpt{30}} \let\Huge=\huge \if@twocolumn \setlength\parindent{14\p@} \else \setlength\parindent{18\p@} \fi \setlength\headheight{14\p@} \setlength\headsep{14\p@} \setlength\topskip{12\p@} \setlength\footskip{24\p@} \setlength\maxdepth{.5\topskip} \setlength\@maxdepth\maxdepth \setlength\textwidth{37.2pc} \setlength\textheight{56pc} \setlength\oddsidemargin {\z@} \setlength\evensidemargin {\z@} \setlength\marginparwidth {72\p@} \setlength\marginparsep{10\p@} \setlength\marginparpush{5\p@} \setlength\topmargin{-12pt} \setlength\footnotesep{8.4\p@} \setlength{\skip\footins} {10.8\p@ \@plus 4\p@ \@minus 2\p@} \setlength\floatsep {14\p@ \@plus 2\p@ \@minus 4\p@} \setlength\textfloatsep {24\p@ \@plus 2\p@ \@minus 4\p@} \setlength\intextsep {16\p@ \@plus 4\p@ \@minus 4\p@} \setlength\dblfloatsep {16\p@ \@plus 2\p@ \@minus 4\p@} \setlength\dbltextfloatsep{24\p@ \@plus 2\p@ \@minus 4\p@} \setlength\@fptop{0\p@} \setlength\@fpsep{10\p@ \@plus 1fil} \setlength\@fpbot{0\p@} \setlength\@dblfptop{0\p@} \setlength\@dblfpsep{10\p@ \@plus 1fil} \setlength\@dblfpbot{0\p@} \setlength\partopsep{3\p@ \@plus 2\p@ \@minus 2\p@} \def\@listI{\leftmargin\leftmargini \parsep=\z@ \topsep=6\p@ \@plus3\p@ \@minus3\p@ \itemsep=3\p@ \@plus2\p@ \@minus1\p@} \let\@listi\@listI \@listi \def\@listii {\leftmargin\leftmarginii \labelwidth\leftmarginii \advance\labelwidth-\labelsep \topsep=3\p@ \@plus2\p@ \@minus\p@ \parsep=\z@ \itemsep=\parsep} \def\@listiii{\leftmargin\leftmarginiii \labelwidth\leftmarginiii \advance\labelwidth-\labelsep \topsep=\z@ \parsep=\z@ \partopsep=\z@ \itemsep=\z@} \def\@listiv {\leftmargin\leftmarginiv \labelwidth\leftmarginiv \advance\labelwidth-\labelsep} \def\@listv{\leftmargin\leftmarginv \labelwidth\leftmarginv \advance\labelwidth-\labelsep} \def\@listvi {\leftmargin\leftmarginvi \labelwidth\leftmarginvi \advance\labelwidth-\labelsep} \endinput %% %% End of file `iopart12.clo'. ---------------0502100318172 Content-Type: application/x-tex; name="setstack.sty" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="setstack.sty" %% %% This is file `setstack.sty', created by VIK 15 Dec 1998 %% Reproduces useful macros from amsmath.sty, thus avoiding the need to load the entire %% amsmath.sty and run into conflicts %% Adds definitions for \overset, \underset, \sideset, \substack, \boxed, \leftroot, %% \uproot, \dddot, \ddddot, \varrow, \harrow (see The LateX Companion, pp 225-227) \NeedsTeXFormat{LaTeX2e}% LaTeX 2.09 can't be used (nor non-LaTeX) [1998/12/15]% LaTeX date must December 1998 or later \ProvidesPackage{setstack} \DeclareRobustCommand{\text}{% \ifmmode\expandafter\text@\else\expandafter\mbox\fi} \let\nfss@text\text \def\text@#1{\mathchoice {\textdef@\displaystyle\f@size{#1}}% {\textdef@\textstyle\tf@size{\firstchoice@false #1}}% {\textdef@\textstyle\sf@size{\firstchoice@false #1}}% {\textdef@\textstyle \ssf@size{\firstchoice@false #1}}% \check@mathfonts } \def\textdef@#1#2#3{\hbox{{% \everymath{#1}% \let\f@size#2\selectfont #3}}} % adds underset, overset, sideset and % substack features from amsmath.sty (Companion p. 226) \def\invalid@tag#1{\@amsmath@err{#1}{\the\tag@help}\gobble@tag} \def\dft@tag{\invalid@tag{\string\tag\space not allowed here}} \def\default@tag{\let\tag\dft@tag} \def\Let@{\let\\\math@cr} \def\restore@math@cr{\def\math@cr@@@{\cr}} \def\overset#1#2{\binrel@{#2}% \binrel@@{\mathop{\kern\z@#2}\limits^{#1}}} \def\underset#1#2{\binrel@{#2}% \binrel@@{\mathop{\kern\z@#2}\limits_{#1}}} \def\sideset#1#2#3{% \@mathmeasure\z@\displaystyle{#3}% \global\setbox\@ne\vbox to\ht\z@{}\dp\@ne\dp\z@ \setbox\tw@\box\@ne \@mathmeasure4\displaystyle{\copy\tw@#1}% \@mathmeasure6\displaystyle{#3\nolimits#2}% \dimen@-\wd6 \advance\dimen@\wd4 \advance\dimen@\wd\z@ \hbox to\dimen@{}\mathop{\kern-\dimen@\box4\box6}% } \newenvironment{subarray}[1]{% \vcenter\bgroup \Let@ \restore@math@cr \default@tag \baselineskip\fontdimen10 \scriptfont\tw@ \advance\baselineskip\fontdimen12 \scriptfont\tw@ \lineskip\thr@@\fontdimen8 \scriptfont\thr@@ \lineskiplimit\lineskip \ialign\bgroup\ifx c#1\hfil\fi $\m@th\scriptstyle##$\hfil\crcr }{% \crcr\egroup\egroup } \newcommand{\substack}[1]{\subarray{c}#1\endsubarray} % definitions of dddot and ddddot(p. 225) \def\dddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ \hbox{\normalfont ...}\vss}}}} \def\ddddot#1{{\mathop{#1}\limits^{\vbox to-1.4\ex@{\kern-\tw@\ex@ \hbox{\normalfont....}\vss}}}} % definitions of leftroot, uproot (p.225) \begingroup \catcode`\"=12 \gdef\@@sqrt#1{\radical"270370 {#1}} \endgroup \def\leftroot{\@amsmath@err{\Invalid@@\leftroot}\@eha} \def\uproot{\@amsmath@err{\Invalid@@\uproot}\@eha} \newcount\uproot@ \newcount\leftroot@ \def\root{\relaxnext@ \DN@{\ifx\@let@token\uproot\let\next@\nextii@\else \ifx\@let@token\leftroot\let\next@\nextiii@\else \let\next@\plainroot@\fi\fi\next@}% \def\nextii@\uproot##1{\uproot@##1\relax\FN@\nextiv@}% \def\nextiv@{\ifx\@let@token\@sptoken\DN@. {\FN@\nextv@}\else \DN@.{\FN@\nextv@}\fi\next@.}% \def\nextv@{\ifx\@let@token\leftroot\let\next@\nextvi@\else \let\next@\plainroot@\fi\next@}% \def\nextvi@\leftroot##1{\leftroot@##1\relax\plainroot@}% \def\nextiii@\leftroot##1{\leftroot@##1\relax\FN@\nextvii@}% \def\nextvii@{\ifx\@let@token\@sptoken \DN@. {\FN@\nextviii@}\else \DN@.{\FN@\nextviii@}\fi\next@.}% \def\nextviii@{\ifx\@let@token\uproot\let\next@\nextix@\else \let\next@\plainroot@\fi\next@}% \def\nextix@\uproot##1{\uproot@##1\relax\plainroot@}% \bgroup\uproot@\z@\leftroot@\z@\FN@\next@} \def\plainroot@#1\of#2{\setbox\rootbox\hbox{% $\m@th\scriptscriptstyle{#1}$}% \mathchoice{\r@@t\displaystyle{#2}}{\r@@t\textstyle{#2}} {\r@@t\scriptstyle{#2}}{\r@@t\scriptscriptstyle{#2}}\egroup} \def\r@@t#1#2{\setboxz@h{$\m@th#1\@@sqrt{#2}$}% \dimen@\ht\z@\advance\dimen@-\dp\z@ \setbox\@ne\hbox{$\m@th#1\mskip\uproot@ mu$}% \advance\dimen@ by1.667\wd\@ne \mkern-\leftroot@ mu\mkern5mu\raise.6\dimen@\copy\rootbox \mkern-10mu\mkern\leftroot@ mu\boxz@} % definition of \boxed (math in frame, no dollars) p 225 \def\boxed#1{\fbox{\m@th$\displaystyle#1$}} % definition of \smash for top and bottom parts of expression in braces (p.227) \renewcommand{\smash}[2][tb]{% \def\smash@{#1}% \ifmmode\@xp\mathpalette\@xp\mathsm@sh\else \@xp\makesm@sh\fi{#2}} % additional difinitions for arrows % zero mm wide #2 mm long vertical arrow shifted 1 mm to left, 1 mm up \newcommand{\varrow}[2]{% \unitlength=1mm \begin{picture}(0,6) % (0,0) % - for #1 to point arrow down,+ to point arrow up \end{picture}% \put(0,6){\vector(0,#1 3){#2}} % 6,- for down, 0, + for up } % 1 mm high #2 mm long horizontal arrow shifted 1 mm up (0,-1) \newcommand{\harrow}[2]{% \unitlength=1mm \begin{picture}(8,1)(0,-1) % % 1 mm distance between arrow and stackreled object over it (8,1) \put(0,0){\vector(#1 2,0){#2}} % - to point arrow left, + right \end{picture}% } \endinput %% end of setstack.sty %% Corrections history: %% ---------------0502100318172 Content-Type: application/postscript; name="pwm.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pwm.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Title: pwm.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Mon Jun 21 11:31:53 2004 %%For: rafa@linux6 (Rafael de la Madrid) %%BoundingBox: 0 0 434 346 %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 346 moveto 0 0 lineto 434 0 lineto 434 346 lineto closepath clip newpath -215.3 436.5 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#055 /minus 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /Times-Italic /Times-Italic-iso isovec ReEncode /Times-Roman /Times-Roman-iso isovec ReEncode /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % /Times-Roman-iso ff 360.00 scf sf 5325 5100 m gs 1 -1 sc (R e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5475 5175 m gs 1 -1 sc ( l) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5475 4875 m gs 1 -1 sc ( * ikx) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 5250 1875 m gs 1 -1 sc ( T e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5400 1650 m gs 1 -1 sc ( * ikx) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 8925 1875 m gs 1 -1 sc (R e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 1950 m gs 1 -1 sc ( r) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 1650 m gs 1 -1 sc ( * -ikx) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 8925 3225 m gs 1 -1 sc ( e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 3000 m gs 1 -1 sc ( ikx) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 5175 6450 m gs 1 -1 sc ( e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5325 6225 m gs 1 -1 sc ( -ikx) col0 sh gr % Polyline 7.500 slw n 6000 3525 m 6000 3675 l gs col0 s gr % Polyline n 7800 3600 m 10800 3600 l gs col0 s gr % Polyline n 6000 6750 m 6000 6900 l gs col0 s gr % Polyline n 7800 6825 m 10800 6825 l gs col0 s gr % Polyline n 3600 6825 m 7200 6825 l gs col0 s gr % Polyline gs clippath 6015 2130 m 6015 2070 l 5864 2070 l 5984 2100 l 5864 2130 l cp eoclip n 5400 2100 m 6000 2100 l gs col0 s gr gr % arrowhead n 5864 2130 m 5984 2100 l 5864 2070 l 5864 2130 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 8985 5295 m 8985 5355 l 9136 5355 l 9016 5325 l 9136 5295 l cp eoclip n 9000 5325 m 9600 5325 l gs col0 s gr gr % arrowhead n 9136 5295 m 9016 5325 l 9136 5355 l 9136 5295 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 6015 5355 m 6015 5295 l 5864 5295 l 5984 5325 l 5864 5355 l cp eoclip n 5400 5325 m 6000 5325 l gs col0 s gr gr % arrowhead n 5864 5355 m 5984 5325 l 5864 5295 l 5864 5355 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 5385 5895 m 5385 5955 l 5536 5955 l 5416 5925 l 5536 5895 l cp eoclip n 5400 5925 m 6000 5925 l gs col0 s gr gr % arrowhead n 5536 5895 m 5416 5925 l 5536 5955 l 5536 5895 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 8985 2070 m 8985 2130 l 9136 2130 l 9016 2100 l 9136 2070 l cp eoclip n 9000 2100 m 9600 2100 l gs col0 s gr gr % arrowhead n 9136 2070 m 9016 2100 l 9136 2130 l 9136 2070 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 9615 2730 m 9615 2670 l 9464 2670 l 9584 2700 l 9464 2730 l cp eoclip n 9000 2700 m 9600 2700 l gs col0 s gr gr % arrowhead n 9464 2730 m 9584 2700 l 9464 2670 l 9464 2730 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw n 7200 6825 m 7200 5025 l gs col0 s gr % Polyline n 7200 5025 m 7800 5025 l gs col0 s gr % Polyline n 7800 5025 m 7800 6825 l gs col0 s gr % Polyline n 7200 3600 m 7200 1800 l gs col0 s gr % Polyline n 7200 1800 m 7800 1800 l gs col0 s gr % Polyline n 7800 1800 m 7800 3600 l gs col0 s gr % Polyline 7.500 slw n 3600 3600 m 7200 3600 l gs col0 s gr /Times-Roman-iso ff 180.00 scf sf 9075 1950 m gs 1 -1 sc ( ) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5475 1950 m gs 1 -1 sc ( ) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 8925 5100 m gs 1 -1 sc (T e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 4875 m gs 1 -1 sc ( * -ikx) col0 sh gr /Times-Italic-iso ff 180.00 scf sf 5925 7275 m gs 1 -1 sc ( 0 a b) col0 sh gr /Times-Italic-iso ff 180.00 scf sf 5925 4050 m gs 1 -1 sc ( 0 a b) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3675 2775 m gs 1 -1 sc (\(2a\)) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3675 6000 m gs 1 -1 sc (\(2b\)) col0 sh gr $F2psEnd rs ---------------0502100318172 Content-Type: application/postscript; name="pwp.eps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="pwp.eps" %!PS-Adobe-2.0 EPSF-2.0 %%Title: pwp.eps %%Creator: fig2dev Version 3.2 Patchlevel 3d %%CreationDate: Mon Jun 21 11:31:17 2004 %%For: rafa@linux6 (Rafael de la Madrid) %%BoundingBox: 0 0 434 346 %%Magnification: 1.0000 %%EndComments /$F2psDict 200 dict def $F2psDict begin $F2psDict /mtrx matrix put /col-1 {0 setgray} bind def /col0 {0.000 0.000 0.000 srgb} bind def /col1 {0.000 0.000 1.000 srgb} bind def /col2 {0.000 1.000 0.000 srgb} bind def /col3 {0.000 1.000 1.000 srgb} bind def /col4 {1.000 0.000 0.000 srgb} bind def /col5 {1.000 0.000 1.000 srgb} bind def /col6 {1.000 1.000 0.000 srgb} bind def /col7 {1.000 1.000 1.000 srgb} bind def /col8 {0.000 0.000 0.560 srgb} bind def /col9 {0.000 0.000 0.690 srgb} bind def /col10 {0.000 0.000 0.820 srgb} bind def /col11 {0.530 0.810 1.000 srgb} bind def /col12 {0.000 0.560 0.000 srgb} bind def /col13 {0.000 0.690 0.000 srgb} bind def /col14 {0.000 0.820 0.000 srgb} bind def /col15 {0.000 0.560 0.560 srgb} bind def /col16 {0.000 0.690 0.690 srgb} bind def /col17 {0.000 0.820 0.820 srgb} bind def /col18 {0.560 0.000 0.000 srgb} bind def /col19 {0.690 0.000 0.000 srgb} bind def /col20 {0.820 0.000 0.000 srgb} bind def /col21 {0.560 0.000 0.560 srgb} bind def /col22 {0.690 0.000 0.690 srgb} bind def /col23 {0.820 0.000 0.820 srgb} bind def /col24 {0.500 0.190 0.000 srgb} bind def /col25 {0.630 0.250 0.000 srgb} bind def /col26 {0.750 0.380 0.000 srgb} bind def /col27 {1.000 0.500 0.500 srgb} bind def /col28 {1.000 0.630 0.630 srgb} bind def /col29 {1.000 0.750 0.750 srgb} bind def /col30 {1.000 0.880 0.880 srgb} bind def /col31 {1.000 0.840 0.000 srgb} bind def end save newpath 0 346 moveto 0 0 lineto 434 0 lineto 434 346 lineto closepath clip newpath -215.3 436.5 translate 1 -1 scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /reencdict 12 dict def /ReEncode { reencdict begin /newcodesandnames exch def /newfontname exch def /basefontname exch def /basefontdict basefontname findfont def /newfont basefontdict maxlength dict def basefontdict { exch dup /FID ne { dup /Encoding eq { exch dup length array copy newfont 3 1 roll put } { exch newfont 3 1 roll put } ifelse } { pop pop } ifelse } forall newfont /FontName newfontname put newcodesandnames aload pop 128 1 255 { newfont /Encoding get exch /.notdef put } for newcodesandnames length 2 idiv { newfont /Encoding get 3 1 roll put } repeat newfontname newfont definefont pop end } def /isovec [ 8#055 /minus 8#200 /grave 8#201 /acute 8#202 /circumflex 8#203 /tilde 8#204 /macron 8#205 /breve 8#206 /dotaccent 8#207 /dieresis 8#210 /ring 8#211 /cedilla 8#212 /hungarumlaut 8#213 /ogonek 8#214 /caron 8#220 /dotlessi 8#230 /oe 8#231 /OE 8#240 /space 8#241 /exclamdown 8#242 /cent 8#243 /sterling 8#244 /currency 8#245 /yen 8#246 /brokenbar 8#247 /section 8#250 /dieresis 8#251 /copyright 8#252 /ordfeminine 8#253 /guillemotleft 8#254 /logicalnot 8#255 /hyphen 8#256 /registered 8#257 /macron 8#260 /degree 8#261 /plusminus 8#262 /twosuperior 8#263 /threesuperior 8#264 /acute 8#265 /mu 8#266 /paragraph 8#267 /periodcentered 8#270 /cedilla 8#271 /onesuperior 8#272 /ordmasculine 8#273 /guillemotright 8#274 /onequarter 8#275 /onehalf 8#276 /threequarters 8#277 /questiondown 8#300 /Agrave 8#301 /Aacute 8#302 /Acircumflex 8#303 /Atilde 8#304 /Adieresis 8#305 /Aring 8#306 /AE 8#307 /Ccedilla 8#310 /Egrave 8#311 /Eacute 8#312 /Ecircumflex 8#313 /Edieresis 8#314 /Igrave 8#315 /Iacute 8#316 /Icircumflex 8#317 /Idieresis 8#320 /Eth 8#321 /Ntilde 8#322 /Ograve 8#323 /Oacute 8#324 /Ocircumflex 8#325 /Otilde 8#326 /Odieresis 8#327 /multiply 8#330 /Oslash 8#331 /Ugrave 8#332 /Uacute 8#333 /Ucircumflex 8#334 /Udieresis 8#335 /Yacute 8#336 /Thorn 8#337 /germandbls 8#340 /agrave 8#341 /aacute 8#342 /acircumflex 8#343 /atilde 8#344 /adieresis 8#345 /aring 8#346 /ae 8#347 /ccedilla 8#350 /egrave 8#351 /eacute 8#352 /ecircumflex 8#353 /edieresis 8#354 /igrave 8#355 /iacute 8#356 /icircumflex 8#357 /idieresis 8#360 /eth 8#361 /ntilde 8#362 /ograve 8#363 /oacute 8#364 /ocircumflex 8#365 /otilde 8#366 /odieresis 8#367 /divide 8#370 /oslash 8#371 /ugrave 8#372 /uacute 8#373 /ucircumflex 8#374 /udieresis 8#375 /yacute 8#376 /thorn 8#377 /ydieresis] def /Times-Roman /Times-Roman-iso isovec ReEncode /Times-Italic /Times-Italic-iso isovec ReEncode /Times-Roman /Times-Roman-iso isovec ReEncode /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def $F2psBegin 10 setmiterlimit 0.06000 0.06000 sc % % Fig objects follow % % Polyline 7.500 slw n 6000 3525 m 6000 3675 l gs col0 s gr % Polyline n 7800 3600 m 10800 3600 l gs col0 s gr % Polyline n 3600 3600 m 7200 3600 l gs col0 s gr % Polyline gs clippath 5385 2070 m 5385 2130 l 5536 2130 l 5416 2100 l 5536 2070 l cp eoclip n 5400 2100 m 6000 2100 l gs col0 s gr gr % arrowhead n 5536 2070 m 5416 2100 l 5536 2130 l 5536 2070 l cp gs 0.00 setgray ef gr col0 s % Polyline n 6000 6750 m 6000 6900 l gs col0 s gr % Polyline gs clippath 6015 5355 m 6015 5295 l 5864 5295 l 5984 5325 l 5864 5355 l cp eoclip n 5400 5325 m 6000 5325 l gs col0 s gr gr % arrowhead n 5864 5355 m 5984 5325 l 5864 5295 l 5864 5355 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 5385 5895 m 5385 5955 l 5536 5955 l 5416 5925 l 5536 5895 l cp eoclip n 5400 5925 m 6000 5925 l gs col0 s gr gr % arrowhead n 5536 5895 m 5416 5925 l 5536 5955 l 5536 5895 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 9615 5355 m 9615 5295 l 9464 5295 l 9584 5325 l 9464 5355 l cp eoclip n 9000 5325 m 9075 5325 l 9150 5325 l 9225 5325 l 9300 5325 l 9375 5325 l 9450 5325 l 9525 5325 l 9600 5325 l gs col0 s gr gr % arrowhead n 9464 5355 m 9584 5325 l 9464 5295 l 9464 5355 l cp gs 0.00 setgray ef gr col0 s % Polyline n 7800 6825 m 10800 6825 l gs col0 s gr % Polyline n 3600 6825 m 7200 6825 l gs col0 s gr % Polyline gs clippath 9615 2730 m 9615 2670 l 9464 2670 l 9584 2700 l 9464 2730 l cp eoclip n 9000 2700 m 9600 2700 l gs col0 s gr gr % arrowhead n 9464 2730 m 9584 2700 l 9464 2670 l 9464 2730 l cp gs 0.00 setgray ef gr col0 s % Polyline gs clippath 8985 2070 m 8985 2130 l 9136 2130 l 9016 2100 l 9136 2070 l cp eoclip n 9000 2100 m 9600 2100 l gs col0 s gr gr % arrowhead n 9136 2070 m 9016 2100 l 9136 2130 l 9136 2070 l cp gs 0.00 setgray ef gr col0 s % Polyline 30.000 slw n 7200 3600 m 7200 1800 l gs col0 s gr % Polyline n 7200 1800 m 7800 1800 l gs col0 s gr % Polyline n 7800 1800 m 7800 3600 l gs col0 s gr % Polyline n 7200 6825 m 7200 5025 l gs col0 s gr % Polyline n 7200 5025 m 7800 5025 l gs col0 s gr % Polyline n 7800 5025 m 7800 6825 l gs col0 s gr /Times-Roman-iso ff 180.00 scf sf 9075 1950 m gs 1 -1 sc ( ) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5475 1650 m gs 1 -1 sc ( -ikx) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5475 1950 m gs 1 -1 sc ( ) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 3000 m gs 1 -1 sc ( ikx) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 8925 3225 m gs 1 -1 sc (R e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 3300 m gs 1 -1 sc ( r) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 8925 1875 m gs 1 -1 sc ( e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 4875 m gs 1 -1 sc ( ikx) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 5325 6450 m gs 1 -1 sc (R e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5475 6525 m gs 1 -1 sc ( l) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 5250 5100 m gs 1 -1 sc ( e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5400 4875 m gs 1 -1 sc ( ikx) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 5325 1875 m gs 1 -1 sc ( Te) col0 sh gr /Times-Roman-iso ff 360.00 scf sf 8925 5100 m gs 1 -1 sc (T e) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 5475 6225 m gs 1 -1 sc ( -ikx) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 9075 1650 m gs 1 -1 sc ( -ikx) col0 sh gr /Times-Italic-iso ff 180.00 scf sf 5925 7275 m gs 1 -1 sc ( 0 a b) col0 sh gr /Times-Italic-iso ff 180.00 scf sf 5925 4050 m gs 1 -1 sc ( 0 a b) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3675 2775 m gs 1 -1 sc (\(1a\)) col0 sh gr /Times-Roman-iso ff 180.00 scf sf 3675 6000 m gs 1 -1 sc (\(1b\)) col0 sh gr $F2psEnd rs ---------------0502100318172--