Content-Type: multipart/mixed; boundary="-------------0509100804875" This is a multi-part message in MIME format. ---------------0509100804875 Content-Type: text/plain; name="05-313.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-313.comments" Contribution to the XXV group theory colloquium, Cocoyoc (Mexico), August 2004 ---------------0509100804875 Content-Type: text/plain; name="05-313.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-313.keywords" Schrodinger and Heisenber pictures; rigged Hilbert space ---------------0509100804875 Content-Type: application/x-tex; name="04cocoyoc.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="04cocoyoc.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % IoP Publishing Ltd % % Documentation for Latex style file iopconf.sty % % for producing CRC for Conference Proceedings % % in book format: final page size 234 x 156 mm % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % Last updated 22 March 2004 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % Character check % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % ! exclamation mark " double quote % # hash ` opening quote (grave) % & ampersand ' closing quote (acute) % $ dollar % percent % ( open parenthesis ) close paren. % - hyphen = equals sign % | vertical bar ~ tilde % @ at sign _ underscore % { open curly brace } close curly % [ open square ] close square bracket % + plus sign ; semi-colon % * asterisk : colon % < open angle bracket > close angle % , comma . full stop % ? question mark / forward slash % \ backslash ^ caret (circumflex) % % ABCDEFGHIJKLMNOPQRSTUVWXYZ % abcdefghijklmnopqrstuvwxyz % 1234567890 % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} \usepackage{amsmath} \usepackage{mathptmx} \usepackage{iopconf} \usepackage{graphicx} \usepackage{bm} \usepackage{amsthm} \begin{document} \title{On the connection between the Schr\"odinger and the Heisenberg pictures for unbounded operators} \author{Rafael de la Madrid} \affil{Departamento de F\'\i sica Te\'orica, Facultad de Ciencias, Universidad del Pa\'\i s Vasco, E-48080 Bilbao, Spain \\ E-mail: {\texttt{wtbdemor@lg.ehu.es}} } %URL: {\texttt{http://www.ehu.es/$\sim$wtbdemor}}} \beginabstract It is well known that the unboundedness of operators in Hilbert space entails domain troubles. It is also well known that most domain troubles can be surmounted by extending the Hilbert space to a rigged Hilbert space. In this note, we point out another of such troubles, namely the correspondence between the Schr\"odinger and the Heisenberg pictures for unbounded operators, and sketch the solution of this problem within the rigged Hilbert space. \endabstract \section{Introduction} Quantum Mechanics textbooks show that the Schr\"odinger and the Heisenberg pictures are physically equivalent, because they yield the same probability amplitudes for measuring an observable $A$ in a state $\varphi$. Textbooks, however, usually omit the fact that for unbounded operators, the manipulations that lead from the Schr\"odinger to the Heisenberg picture must be taken with care, due to domain problems. The purpose of this note is to point out those domain problems and to sketch their solution by extending the Hilbert space to the rigged Hilbert space. For a class of potentials, we use a theorem by Hunziker to solve such problems explicitly. \section{The problem} Suppose that the algebra ${\cal A}$ of observables of a system consists of the following operators: \begin{equation} {\cal A}= \{ H, A_1, A_2, \ldots , A_N \} \, , \end{equation} where $H$ is the Hamiltonian and $A_1$, $A_2$, $\ldots$, $A_N$ are the other relevant observables of the system (e.g., position and momentum). Those operators are assumed to be self-adjoint on a Hilbert space ${\cal H}$. For simplicity, we restrict ourselves to pure states and to observables that do not depend explicitly on time. Then, in the Schr\"odinger picture, which shall be denoted by the subscript ${\rm S}$, the observables are kept fixed in time, \begin{equation} A_{\rm S}(t)=A_{\rm S}(0)=A \, , \end{equation} whereas the states evolve in time according to the Schr\"odinger equation, \begin{equation} \rmi \hbar \frac{\rmd}{\rmd t}\varphi _{\rm S}(t)= H\varphi _{\rm S}(t) \, . \end{equation} Integration of this equation leads to \begin{equation} \varphi _{\rm S}(t)=\rme ^{-\rmi H t/\hbar}\varphi _{\rm S}(0) \equiv \rme ^{-\rmi H t/\hbar}\varphi \, . \end{equation} In the Schr\"odinger picture, the expectation value of the measurement of the observable $A$ in the state $\varphi$ is given by \begin{equation} \langle A \rangle _{\rm S}(t) = \langle \varphi _{\rm S}(t)|A_{\rm S}|\varphi _{\rm S}(t) \rangle = \langle \rme ^{-\rmi H t/\hbar}\varphi |A| \rme ^{-\rmi H t/\hbar}\varphi \rangle \, . \label{exvaS} \end{equation} In the Heisenberg picture, which shall be denoted by the subscript ${\rm H}$, the states are kept fixed in time, \begin{equation} \varphi _{\rm H}(t)=\varphi _{\rm H}(0)= \varphi \, , \end{equation} whereas the observables evolve in time according to ``Heisenberg's equation of motion:'' \begin{equation} \rmi \hbar \frac{\rmd}{\rmd t}A_{\rm H}(t)= [A_{\rm H}(t), H] \, . \label{Heqmot} \end{equation} In integrated form, Eq.~(\ref{Heqmot}) reads as \begin{equation} A_{\rm H}(t)=\rme ^{\rmi Ht/\hbar}A_{\rm H}(0)\rme ^{-\rmi Ht/\hbar} = \rme ^{\rmi Ht/\hbar}A\rme ^{-\rmi Ht/\hbar} \, . \label{teob} \end{equation} In the Heisenberg picture, the expectation value of the measurement of $A$ in the state $\varphi$ is given by \begin{equation} \langle A \rangle _{\rm H}(t) = \langle \varphi _{\rm H}|A_{\rm H}(t)|\varphi _{\rm H} \rangle = \langle \varphi | \rme ^{\rmi H t/\hbar}A\rme ^{-\rmi H t/\hbar}|\varphi \rangle \, . \label{exvaH} \end{equation} The equivalence of the Schr\"odinger and Heisenberg pictures is guaranteed by the equality of the expectation values~(\ref{exvaS}) and~(\ref{exvaH}): \begin{equation} \langle A \rangle _{\rm S}(t) = \langle A \rangle _{\rm H}(t) \, , \end{equation} which follows from the unitarity of the group evolution operator $\rme ^{-\rmi H t/\hbar}$. When the operators of the algebra ${\cal A}$ are all bounded, they are defined on the whole of the Hilbert space $\cal H$, and domain troubles do not arise. But if at least one operator of the algebra, say $A_1$, is unbounded, then $A_1$ cannot be defined on the whole of the Hilbert space, but at the most on a dense subspace ${\cal D}(A_1)$ of the Hilbert space on which $A_1$ is self-adjoint. In such event, one has to specify on what states the algebraic operations involving unbounded operators are valid, since algebraic operations (e.g., sums, products and commutation relations) of unbounded operators are not defined on the whole of ${\cal H}$: If $A$ and $B$ are two unbounded operators defined on two dense subdomains ${\cal D}(A)$ and ${\cal D}(B)$ of $\cal H$, then the sum of $A$ and $B$, $A+B$, is defined only for $f \in {\cal D}(A) \cap {\cal D}(B)$; the product of $A$ by $B$, $BA$, is defined only for those $f \in {\cal D}(A)$ such that $Af \in {\cal D}(B)$; the commutation relation of $A$ and $B$, $[A,B]= AB-BA$, is defined only for those $f$ such that $f \in {\cal D}(A) \cap {\cal D}(B)$, $Af \in {\cal D}(B)$ and $Bf \in {\cal D}(A)$. Likewise algebraic operations, the time evolution~(\ref{teob}) of an unbounded operator $A$ cannot be defined on the whole of $\cal H$. Clearly, the time evolution of $A$, \begin{equation} A(t) \equiv \rme ^{\rmi Ht/\hbar}A\rme ^{-\rmi Ht/\hbar} \, , \label{teobA} \end{equation} is defined only for those $f\in {\cal H}$ such that $\rme ^{-\rmi H t/\hbar}f \in {\cal D}(A)$. Thus, the Heisenberg picture of an unbounded operator $A$ is not defined on the whole of $\cal H$. We have therefore to face the fact that algebraic operations and the Heisenberg picture of unbounded operators entail domain troubles. As we are going to see in the next section, such domain troubles can be surmounted by extending the Hilbert space to the rigged Hilbert space. \section{Sketch of a solution} The way the rigged Hilbert space surmounts the domain troubles of algebraic operations is well known (see~\cite{JPA04} for a recent, simple example). Basically, when resonances are not involved, one has to construct the maximal invariant subspace of the algebra of operators, \begin{equation} \Phi = \bigcap _{A\in {\cal A}} {\cal D}(A) \, . \end{equation} The space $\Phi$ is obviously contained in the domains of the observables of the algebra, \begin{equation} \Phi \subset {\cal D}(A) \, , \quad A \in {\cal A} \, , \end{equation} and is the largest subspace of the Hilbert space that remains invariant under the action of all the operators of the algebra: \begin{equation} A \Phi \subset \Phi \, , \qquad A \in {\cal A} \, . \end{equation} It is precisely this invariance what makes all algebraic operations (e.g., sums, multiplications and commutation relations) well defined on $\Phi$. In addition, the bras $\langle a|$ and the kets $|a\rangle$ associated with the continuous spectra of the operators belong to the dual, $\Phi ^{\prime}$, and to the antidual, $\Phi ^{\times}$, spaces, respectively: \begin{equation} \begin{array}{ll} \langle a| \in \Phi ^{\prime} \, , \\ [2ex] |a\rangle \in \Phi ^{\times} \, . \end{array} \end{equation} Now, it is clear that in order to avoid the domain troubles of the time evolution~(\ref{teobA}) of an unbounded observable $A$, we simply need to let $A(t)$ act on a subspace whose time evolution is included in ${\cal D}(A)$. Since we want this to happen for all the operators of the algebra, it is natural to demand that the space $\Phi$ be invariant under the action of the time evolution group: \begin{equation} \rme ^{-\rmi Ht/\hbar} \Phi \subset \Phi \, . \label{invofphte} \end{equation} When the invariance~(\ref{invofphte}) holds, the time evolution of all the operators of the algebra is well defined on $\Phi$, and $\Phi$ is invariant under $A(t)$: \begin{equation} A(t) \Phi \subset \Phi \, , \qquad A\in {\cal A} \, . \end{equation} This invariance makes, in particular, all algebraic operations involving the time evolution of the observables well defined. The problem is, it is not known whether the invariance~(\ref{invofphte}) holds for any Hamiltonian and for any algebra we could think of. But, as we are going to see in the next section, for some cases of interest a theorem by Hunziker provides a positive answer. To finish this section we note that, as a byproduct of the invariance~(\ref{invofphte}), one can define the time evolution of the bras and kets: \begin{equation} \begin{array}{ll} \langle a| (t) = \langle a|\rme ^{\rmi Ht/\hbar} \, , \\ [2ex] |a\rangle (t) = \rme ^{-\rmi Ht/\hbar} |a\rangle \, . \end{array} \label{tebrasandkets} \end{equation} In the rigged Hilbert space language, the precise definition of this time evolution is as follows: \begin{equation} \begin{array}{ll} \langle a|\rme ^{\rmi Ht/\hbar}|\varphi \rangle \equiv \langle a|\rme ^{-\rmi Ht/\hbar}\varphi \rangle \, , \quad \varphi \in \Phi \, , \\ [2ex] \langle \varphi| \rme ^{-\rmi Ht/\hbar} |a\rangle \equiv \langle \rme ^{\rmi Ht/\hbar} \varphi| a\rangle \, , \quad \varphi \in \Phi \, . \end{array} \label{tebrasandketsrhs} \end{equation} Because $\langle a|$ and $|a\rangle$ are defined only when they act on $\Phi$, definitions~(\ref{tebrasandketsrhs}) make sense only when $\rme ^{-\rmi Ht/\hbar}\varphi$ and $\rme ^{\rmi Ht/\hbar}\varphi$ belong to $\Phi$; that is, definitions~(\ref{tebrasandketsrhs}) make sense only when $\Phi$ is invariant under the time evolution group. Thus, the invariance of $\Phi$ under the time evolution group, which guarantees that $A(t)$ is well defined, also guarantees that the time evolution of the bras and kets is well defined. \section{Example} In practical applications, the most important unbounded observables we encounter are the position, the momentum and the energy operators. If, for simplicity, we restrict ourselves to one dimension, the position operator $Q$ is defined as multiplication by the position coordinate: \begin{equation} Qf(x)=xf(x) \, ; \end{equation} the momentum operator $P$ is defined as differentiation with respect to the position coordinate: \begin{equation} Pf(x)=- \rmi \hbar \frac{\rmd f(x)}{\rmd x} \, ; \end{equation} and the energy operator, or Hamiltonian, $H$ is the sum of the kinetic energy operator and the potential $V(x)$: \begin{equation} Hf(x)= -\frac{\hbar ^2}{2m}\frac{\rmd ^2f(x)}{\rmd ^2x}+ V(x)f(x) \, . \end{equation} As we explained in the previous section, we have to construct the maximal invariant subspace $\Phi$ of the algebra $\{ H, Q, P \}$, and then see whether $\Phi$ is invariant under $\rme ^{-\rmi Ht/\hbar}$. The form of $\Phi$ and the invariance of $\Phi$ under $\rme ^{-\rmi Ht/\hbar}$ depend on the form of $V(x)$. In this note, we shall only consider potentials $V(x)$ that are bounded $C^{\infty}$-functions with bounded derivatives. For such potentials, the maximal invariant subspace of the algebra $\{ H, Q, P \}$ is the Schwartz space: \begin{equation} \Phi = {\cal S}({R}) \, . \end{equation} This space is indeed invariant under $\rme ^{-\rmi Ht/\hbar}$, as the following theorem states for any dimension $n$~\cite{HUNZIKER}: \vskip0.3cm {\bf Theorem} (Hunziker) \ {\it If $V(x)$ is a bounded $C^{\infty}$-function on ${R}^n$ with bounded derivatives, then ${\cal S}({R}^n)$ is invariant under the unitary group $\rme ^{-\rmi Ht/\hbar}$ and the mapping $(\varphi , t)\to \rme ^{-\rmi Ht/\hbar}\varphi$ of ${\cal S}({R}^n)\times {R}$ onto ${\cal S}({R}^n)$ is continuous (in the sense of the conventional topology of ${\cal S}({R}^n)$).} \vskip0.3cm Thus, when $V(x)$ is a bounded $C^{\infty}$-function with bounded derivatives, the time evolution (and therefore the Heisenberg picture) of the algebra $\{ H, Q, P \}$ is well defined on $\Phi = {\cal S}({R})$. As a byproduct, the time evolution of the bras $\langle E|$, $\langle x|$, $\langle p|$ and kets $|E\rangle$, $|x\rangle$, $|p\rangle$ of $H$, $Q$, $P$ is also well defined, since those bras and kets act as functionals over $\Phi = {\cal S}({R})$. When $V(x)$ is not continuous but only piece-wise continuous (e.g., $V(x)$ is a rectangular barrier potential), the discontinuities of the potential have to be taken into consideration. For example, if $V(x)$ is continuous everywhere except at, say, $x=a,b$, then the maximal invariant subspace of $\{ H, Q, P \}$ is the space ${\cal S}({R}-\{ a, b \})$ constructed in~\cite{JPA04}. In this case, one has to prove that ${\cal S}({R}-\{ a, b \})$ is invariant under $\rme ^{-\rmi Ht/\hbar}$. Although the invariance of ${\cal S}({R}-\{ a, b \})$ under $\rme ^{-\rmi Ht/\hbar}$ is still to be proven, an examination of the proof of Hunziker's theorem~\cite{HUNZIKER} suggests that such invariance should hold. For other ``reasonable'' potentials the invariance of the corresponding $\Phi$ under time evolution is to be expected, too. By ``reasonable'' we mean that the potential can be considered as a small perturbation to the kinetic energy, in the sense of Kato~\cite{KATO}. \section{Conclusion} We have discussed the problems in defining the Heisenberg picture of an algebra of unbounded observables. We have seen that the Heisenberg picture is well defined when the maximal invariant subspace of the algebra remains invariant under the time evolution group. Such invariance should hold in general, although we have shown it only for potentials that are smooth. More precisely, when $V(x)$ is a bounded $C^{\infty}$-function with bounded derivatives, Hunziker's theorem has been used to show that the Heisenberg picture of the algebra $\{ H, Q, P \}$ is well defined on the Schwartz space. \section*{Acknowledgments} The author is indebted to E.~Galapon for enlightening discussions. Financial support from the Basque Government through reintegration fellowship No.~BCI03.96 is gratefully acknowledged. {\it Note:} The version of Hunziker's theorem I provided in this paper is not the most general one. In fact, after this paper was already in press, I realized that a theorem by Roberts~\cite{ROBERTS} combined with the general version of Hunziker's theorem~\cite{HUNZIKER} guarantees that, when the potential is infinitely differentiable on some open set of ${R}^n$ whose complement has zero Lebesgue measure, and when the potential satisfies the Kato condition~\cite{KATO}, then the maximal invariant subspace of the algebra $\{ H, Q, P \}$ is dense and invariant under $\rme ^{-\rmi Ht/\hbar}$. In particular, the space ${\cal S}({R}-\{ a, b \})$ of~\cite{JPA04} is indeed invariant under the time evolution of the 1D rectangular barrier Hamiltonian. \begin{thebibliography}{9} \bibitem{JPA04} R.~de la Madrid, \JPA~{\bf 37}, 8129-8157 (2004); {\sf quant-ph/0407195}. \bibitem{HUNZIKER} W.~Hunziker, \JMP~{\bf 7}, 300-304 (1966). \bibitem{KATO} T.~Kato, \emph{Perturbation Theory for Linear Operators} (Springer Verlag: New York) 1966. \bibitem{ROBERTS} J.~E.~Roberts, J.~Math.~Phys.~{\bf 7}, 1097 (1966); J.~E.~Roberts, Commun.~Math.~Phys.~{\bf 3}, 98 (1966). \end{thebibliography} \end{document} ---------------0509100804875 Content-Type: application/x-tex; name="iopconf.sty" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="iopconf.sty" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % IOP Publishing Ltd and Adam Hilger Ltd % % Latex style file for producing CRC for % % Conference Proceedings % % % 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{\hfil[\arabic{enumi}]}{\itemsep=0pt\labelsep=7pt\itemindent=-14pt \settowidth\labelwidth{[#1]} \leftmargin\labelwidth \advance\leftmargin\labelsep \advance\leftmargin -\itemindent \usecounter{enumi}}\setb@se{12pt}\small \def\newblock{\hskip .11em plus .33em minus .07em} \sloppy\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} % % \newref used to separate references under single number % \newcommand{\newref}{\\\hspace*{-14pt}} % % Macro to used for references in the Harvard system % \def\references{\section*{References\@mkboth {REFERENCES}{REFERENCES}}\begin{list}{}{\itemsep=0pt\labelsep=0pt\itemindent=-28pt \labelwidth=0pt\leftmargin=28pt}\setb@se{12pt}\small \def\newblock{\hskip .11em plus .33em minus .07em} \begin{sloppy}\clubpenalty4000\widowpenalty4000 \sfcode`\.=1000\relax} % \def\endreferences{\end{sloppy}\end{list}\sfcode`\.=3000\clubpenalty=150 \widowpenalty=150\setb@se{14pt}\normalsize\rm} % %%%%%%%%%%%%%%%%%%%%%% % Useful definitions % %%%%%%%%%%%%%%%%%%%%%% % %\newcommand{\mbold}[1]{\mbox{\boldmath$#1$}} % Redefinition of \boldmath to work in maths: \newcommand{\etal}{\emph{et al}\ } \newcommand{\dash}{------} \newcommand{\rmd}{{\rm d}} \newcommand{\rme}{{\rm e}} \newcommand{\rmi}{{\rm i}} \newcommand{\case}[2]{{\textstyle{#1\over#2}}} \newcommand{\dotted}{$\cdot\;\cdot\;\cdot\;\cdot\;\cdot$} \newcommand{\dashed}{-\, -\, -\, -} \newcommand{\broken}{-- -- --} \newcommand{\longbroken}{--- --- ---} \newcommand{\chain}{--- $\cdot$ ---} \newcommand{\dashddot}{--- $\cdot$ $\cdot$ ---} \newcommand{\full}{------} \let\lc=\lowercase \let\uc=\uppercase \newcommand{\Or}{\mathop{\rm O}\nolimits} \newcommand{\Tr}{\mathop{\rm Tr}\nolimits} \newcommand{\tr}{\mathop{\rm tr}\nolimits} \newcommand{\bs}{\noalign{\vspace{7pt plus2pt minus2pt}}} \newcommand{\0}{\phantom{0}} \newcommand{\m}{\llap{$-$}}% % abbreviations for IOPP journals % \newcommand{\CQG}{\emph{Class. Quantum Grav.} } \newcommand{\IP}{\emph{Inverse Problems} } \newcommand{\JPA}{\emph{J. Phys. A: Math. Gen.} } \newcommand{\JPB}{\emph{J. Phys. B: At. Mol. Phys.} } %1968-87 \newcommand{\jpb}{\emph{J. Phys. B: At. Mol. Opt. Phys.} } %1988 and onwards \newcommand{\JPC}{\emph{J. Phys. C: Solid State Phys.} } %1968--1988 \newcommand{\JPCM}{\emph{J. Phys.: Condens. Matter} } %1989 and onwards \newcommand{\JPD}{\emph{J. Phys. D: Appl. Phys.} } \newcommand{\JPE}{\emph{J. Phys. E: Sci. Instrum.} } \newcommand{\JPF}{\emph{J. Phys. F: Met. Phys.} } \newcommand{\JPG}{\emph{J. Phys. G: Nucl. Phys.} } %1975--1988 \newcommand{\jpg}{\emph{J. Phys. G: Nucl. Part. Phys.} } %1989 and onwards \newcommand{\NL}{\emph{Nonlinearity} } \newcommand{\PMB}{\emph{Phys. Med. Biol.} } \newcommand{\RPP}{\emph{Rep. Prog. Phys.} } \newcommand{\SST}{\emph{Semicond. Sci. Technol.} } \newcommand{\SUST}{\emph{Supercond. Sci. Technol.} } % % Other commonly quoted journals % \newcommand{\AC}{\emph{Acta Crystallogr.} } \newcommand{\AM}{\emph{Acta Metall.} } \newcommand{\AP}{\emph{Ann. Phys., Lpz.} } \newcommand{\APNY}{\emph{Ann. Phys., NY} } \newcommand{\APP}{\emph{Ann. Phys., Paris} } \newcommand{\CJP}{\emph{Can. J. Phys.} } \newcommand{\JAP}{\emph{J. Appl. Phys.} } \newcommand{\JCP}{\emph{J. Chem. Phys.} } \newcommand{\JJAP}{\emph{Japan. J. Appl. Phys.} } \newcommand{\JP}{\emph{J. Physique} } \newcommand{\JPhCh}{\emph{J. Phys. Chem.} } \newcommand{\JMMM}{\emph{J. Magn. Magn. Mater.} } \newcommand{\JMP}{\emph{J. Math. Phys.} } \newcommand{\JOSA}{\emph{J. Opt. Soc. Am.} } \newcommand{\JPSJ}{\emph{J. Phys. Soc. Japan} } \newcommand{\JQSRT}{\emph{J. Quant. Spectrosc. Radiat. Transfer} } \newcommand{\NC}{\emph{Nuovo Cimento} } \newcommand{\NIM}{\emph{Nucl. Instrum. Methods} } \newcommand{\NP}{\emph{Nucl. Phys.} } \newcommand{\PL}{\emph{Phys. Lett.} } \newcommand{\PR}{\emph{Phys. Rev.} } \newcommand{\PRL}{\emph{Phys. Rev. Lett.} } \newcommand{\PRS}{\emph{Proc. R. Soc.} } \newcommand{\PS}{\emph{Phys. Scr.} } \newcommand{\PSS}{\emph{Phys. Status Solidi} } \newcommand{\PTRS}{\emph{Phil. Trans. R. Soc.} } \newcommand{\RMP}{\emph{Rev. Mod. Phys.} } \newcommand{\RSI}{\emph{Rev. Sci. Instrum.} } \newcommand{\SSC}{\emph{Solid State Commun.} } \newcommand{\ZP}{\emph{Z. Phys.} } % \endinput %  ---------------0509100804875--