Content-Type: multipart/mixed; boundary="-------------0003281459756" This is a multi-part message in MIME format. ---------------0003281459756 Content-Type: text/plain; name="00-131.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-131.comments" This is an article for the Kluwer Encyclopedia of Mathematics, Supplement vol.II, p.455-457 (2000). ---------------0003281459756 Content-Type: text/plain; name="00-131.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-131.keywords" Lieb-Thirring inequalities ---------------0003281459756 Content-Type: application/x-tex; name="thomas-fermi7-encyclopedia.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="thomas-fermi7-encyclopedia.tex" \documentclass[twocolumn,notitlepage,]{article} \baselineskip=4.5ex %\pagestyle{empty} %\usepackage{rotating} %\setlength{\topmargin}{-1truein} %\setlength{\textwidth}{7truein} %\setlength{\textheight}{9.5truein} %\setlength{\parindent}{8mm} \frenchspacing %\setlength{\oddsidemargin}{-.25truein} %\setlength{\evensidemargin}{-.25truein} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \def\sr{{\cal R}} \def\1{{\bf 1}} %\def\d{\rm d} \def\x{{\bf x}} \def\A{{\bf A}} \def\Tr{{\rm Tr}} \def\C{{\bf C}} \def\R{{\bf R}} \def\E{{\cal E}} \def\TF{{\rm TF}} \def\MTF{{\rm MTF}} \def\mfr#1/#2{\hbox{${{#1} \over {#2}}$}} \def\const.{{\rm const.}} \def\beq{\begin{equation}} \def\eeq{\end{equation}} \def\beqa{\begin{eqnarray}} \def\eeqa{\end{eqnarray}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}{Lemma}[section] %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \setlength{\unitlength}{1.0cm} %\title{\bf \\ %A rigorous derivation %\footnotetext{\copyright 1998 by Elliott H. Lieb} \def\boxit#1{\thinspace\hbox{\vrule\vtop{\vbox{\hrule\kern1pt \hbox{\vphantom{\tt/}\thinspace{\tt#1}\thinspace}}\kern1pt\hrule}\vrule} \thinspace} \font\eightit=cmti8 \def\ve{{\varepsilon}} \def\C{{\bf C}} \def\D{{\rm D}} \def\E{{\rm E}} \def\R{{\bf R}} \def\Z{{\bf Z}} \def\L{{\Lambda}} \def\l{{\lambda}} \def\d{{\rm d}} \def\Q{{{\rm C}_{\Lambda}}} \def\X{{\underline{X}}} \def\Tr{{\rm Tr}} %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%% {\bf THOMAS-FERMI THEORY} -- Sometimes called the `statistical theory', it was invented by L. H. Thomas\cite{TH} and E. Fermi\cite{EF}, shortly after Schr\"odinger invented his quantum-mechanical wave equation, in order to approximately describe the \emph{electron density}, $\rho(x)$, $x\in \R^3$, and the \emph{ground state energy}, $E(N)$ for a large atom or molecule with a large number, $N$, of electrons. Schr\"odinger's equation, which would give the exact density and energy, cannot be easily handled when $N$ is large. A starting point for the theory is the \emph{TF energy functional}. For a molecule with $K$ nuclei of charges $Z_i >0$ and locations $ R_i\in \R^3 \ (i=1,...,K)$, it is \begin{eqnarray} %\[ {\cal E}(\rho) & := & \frac{3}{5} \gamma \int_{\R^3}\rho(x)^{5/3} \ \d x - \int_{\R^3} V (x) \rho (x) \ \d x \nonumber \\ & + & \frac{1}{2} \int_{\R^3} \int_{\R^3} ~ \frac{\rho(x) \rho (y)}{|x-y|} \ \d x \d y + U %\] \end{eqnarray} in suitable units. Here, \begin{eqnarray*} V(x) &=& \sum_{j=1}^{K} Z_j |x-r_j|^{-1}\ ,\\ U &=& \sum_{1 \leq i < j \leq K} Z_i Z_j |R_i - R_j|^{-1} \ , \end{eqnarray*} and $\gamma = (3 \pi^2)^{2/3}$. The constraint on $\rho$ is $\rho (x) \geq 0$ and $\int_{\R^3} \rho = N$. The functional $\rho \rightarrow {\cal E}(\rho)$ is convex.\\ \noindent The justification for this functional is this: \noindent $\bullet$ The first term is roughly the minimum quantum-mechanical kinetic energy of $N$ electrons needed to produce an electron density $\rho$.\\ \noindent $\bullet$ The second term is the attractive interaction of the $N$ electrons with the $K$ nuclei, via the {\it Coulomb potential} $V$.\\ \noindent $\bullet$ The third is approximately the electron-electron repulsive energy.\\ \noindent $\bullet$ $U$ is the nuclear-nuclear repulsion and is an important constant.\\ The {\it TF energy} is defined to be \[ E^{\rm{TF}}(N) = \inf \{ {\cal E}(\rho) : \rho \in L^{5/3}, \int \rho = N, \rho \geq 0 \} \ , \] i.e., the TF energy and density is obtained by minimizing ${\cal E}(\rho)$ with $\rho \in L^{5/3} (\R^3)$ and $\int \rho = N$. The {\it Euler-Lagrange equation}, called the {\it Thomas-Fermi equation}, is \begin{equation} \gamma \rho (x)^{2/3} = \left[ \Phi (x) - \mu \right]_+, \end{equation} where $[a]_+$ = $\max \{0,a\}$, $\mu$ is some constant ({\it Lagrange multiplier}) and $\Phi$ is the {\it TF potential}: \begin{equation} \Phi (x) = V(x) - \int_{\R^3} |x-y|^{-1} \rho(y) \ \d y. \end{equation} The following essential mathematical facts about the TF equation were established by E.H. Lieb and B. Simon~\cite{LS} ({\it cf.} the review article~\cite{EL}). \begin{enumerate} \item There is a density $\rho^{\rm{TF}}_{N}$ that minimizes ${\cal E}(\rho)$ if and only if $N \leq Z : = \sum_{j=1}^{K} Z_j$. This $\rho^{\rm{TF}}_{N}$ is unique and it satisfies the TF equation (2) for some $\mu \geq 0$. Every positive solution, $\rho$, of (2) is a minimizer of (1) for $N = \int \rho$. If $N > Z$ then $E^{\rm{TF}}(N) = E^{\rm{TF}}(Z)$ and any minimizing sequence converges weakly in $L^{5/3}(\R^3)$ to $\rho^{\rm{TF}}_Z$. \item $\Phi (x) \geq 0$ for all $x$. (This need not be so for the real Schr\"odinger $\rho$.) \item $\mu = \mu(N)$ is a strictly monotonically decreasing function of $N$ and $\mu(Z)=0$ (the {\it neutral case}). $\mu$ is the {\it chemical potential}, namely \[ \mu (N) = -\frac{\partial E^{\rm{TF}}(N)}{\partial N} \ . \] $E^{\rm{TF}}(N)$ is a strictly convex, decreasing function of $N$ for $N \leq Z$ and $E^{\rm{TF}}(N) = E^{\rm{TF}}(Z)$ for $N \geq Z$. If $N < Z$, $\rho^{\rm{TF}}_N$ has compact support. \end{enumerate} When $N=Z$, (2) becomes $\gamma \rho^{2/3} = \Phi$. By applying the Laplacian $\Delta$ to both sides we obtain $$ - \Delta \Phi (x) + \ 4 \pi \gamma^{-3/2} \Phi (x)^{3/2} =4\pi \sum_{j=1}^{K} \ Z_j \ \delta (x-R_j) \ , $$ which is the form in which the TF equation is usually stated (but it is valid only for $N=Z$). An important property of the solution is {\it Teller's theorem}~\cite{ET} (proved rigorously in \cite{LS}) which implies that the TF molecule is always unstable, i.e., for each $N \leq Z$ there are $K$ numbers $N_j \in (0,Z_j)$ with $\sum_j N_j = N$ such that \begin{equation} E^{\rm{TF}}(N) > \sum^{K}_{j=1} \ E^{\rm{TF}}_{\rm{atom}} (N_j, Z_j) \ , \end{equation} where $E_{\rm{atom}}^{\rm{TF}} (N_j, Z_j)$ is the TF energy with $K=1, Z=Z_j$ and $N=N_j$. The presence of $U$ in (1) is crucial for this result. The inequality is strict. Not only does $E^{\rm{TF}}$ decrease when the nuclei are pulled infinitely far apart (which is what (4) says) but any dilation of the nuclear coordinates $(R_j \rightarrow \ell R_j, \ell > 1)$ will decrease $E^{\rm{TF}}$ in the neutral case ({\it positivity of the pressure})~\cite{EL},~\cite{BL}. This theorem plays an important role in the {\it stability of matter}. An important question concerns the connection between $E^{\rm{TF}}(N)$ and $E^{\rm{Q}}(N)$, the ground state energy (= infimum of the spectrum) of the Schr\"odinger operator, $H$, it was meant to approximate. \[ H = - \sum^{N}_{i=1} \left[ \Delta_i + V(x_i) \right] + \sum_{1 \leq i < j \leq N} |x_i - x_j|^{-1} + U \ , \] which acts on the antisymmetric functions $\wedge^{N} L^2 (\R^3; \C^2)$ (i.e., functions of space and spin). It used to be believed that $E^{\rm{TF}}$ is asymptotically exact as $N \rightarrow \infty$ but this is not quite right; $Z \rightarrow \infty$ is also needed. Lieb and Simon~\cite{LS} proved that if we fix $K$ and $Z_j/Z$ and we set $R_j = Z^{-1/3} R_j^0$, with fixed $R_j^0 \in \R^3$, and set $N = \lambda Z$, with $0 \leq \lambda < 1$ then \begin{equation} \lim_{Z \rightarrow \infty} \ \ E^{\rm{TF}}(\lambda Z)/E^{\rm{Q}}(\lambda Z) = 1 \ . \end{equation} In particular, a simple change of variables shows that $E^{\rm{TF}}_{\rm{atom}} (\lambda, Z) = Z^{7/3} E^{\rm{TF}}_{\rm{atom}} (\lambda, 1)$ and hence the true energy of a large atom is asymptotically proportional to $Z^{7/3}$. Likewise, there is a well-defined sense in which the quantum mechanical density converges to $\rho^{\rm{TF}}_N$ ({\it cf.}~\cite{LS}). The TF density for an atom located at $R=0$, which is spherically symmetric, scales as \begin{eqnarray*} \rho^{\rm{TF}}_{\rm{atom}} (x; N = \lambda Z, Z) &=&\\ Z^2\rho^{\rm{TF}}_{\rm{atom}} (&Z^{1/3} x;& N = \lambda, Z = 1) \ . \end{eqnarray*} Thus, a large atom (i.e., large $Z$) is {\it smaller} than a $Z = 1$ atom by a factor $Z^{-1/3}$ in radius. Despite this seeming paradox, TF theory gives the correct electron density in a real atom --- so far as the bulk of the electrons is concerned --- as $Z \rightarrow \infty$ Another important fact is the large $|x|$ asymptotics of $\rho^{\rm{TF}}_{\rm{atom}}$ for a neutral atom. As $|x| \rightarrow \infty$, $$ \rho^{\rm{TF}}_{\rm{atom}} (x, N = Z, Z) \sim \gamma^3 (3/\pi)^3 |x|^{-6} \ , $$ {\it independent} of $Z$. Again, this behavior agrees with quantum mechanics --- on a length scale $Z^{-1/3}$, which is where the bulk of the electrons are to be found. In light of the limit theorem (5), Teller's theorem can be understood as saying that as $Z \rightarrow \infty$ the quantum mechanical binding energy of a molecule is of lower order in $Z$ than the total ground state energy. Thus, Teller's theorem is not a defect of TF theory (although it is sometimes interpreted that way) but an important statement about the true quantum mechanical situation. For finite $Z$ one can show, using the Lieb-Thirring inequality~\cite{LT} and the Lieb-Oxford inequality~\cite{LO}, that $E^{\rm{TF}}(N)$, with a modified $\gamma$, gives a lower bound to $E^{\rm{Q}}(N)$. Several `improvements' to Thomas-Fermi theory have been proposed, but none have a fundamental significance in the sense of being `exact' in the $Z \rightarrow \infty$ limit. The von Weizs\"acker correction consists in adding a term \[ (\rm{const.}) \int_{\R^3} | \nabla \sqrt{\rho (x)} |^2 \ \d x \] to ${\cal E} (\rho)$. This preserves the convexity of ${\cal E}(\rho)$ and adds (const.)$Z^2$ to $E^{\rm{TF}}(N)$ when $Z$ is large. It also has the effect that the range of $N$ for which there is a minimizing $\rho$ is extend from [0,Z] to [0,Z + (const.) K]. Another correction, the {\it Dirac exchange energy}, is to add \[ -({\rm{const.}}) \int_{\R^3} \rho (x)^{4/3} \ \d x \] to ${\cal E}(\rho)$. This spoils the convexity but not the range [0,Z] for which a minimizing $\rho$ exists {\it cf.}~\cite{LS} for both of these corrections. When a uniform external magnetic field $B$ is present, the operator $- \Delta$ in $H$ is replaced by $$ |i\nabla + A (x)|^2 + \sigma \cdot B(x) \ , $$ with curl $A=B$ and $\sigma$ denoting the Pauli spin matrices. This leads to a modified TF theory that is asymptotically exact as $Z \rightarrow \infty$, but the theory depends on the manner in which $B$ varies with $Z$. There are five distinct regimes and theories: $B \ll Z^{4/3}, B \sim Z^{4/3}, Z^{4/3} \ll B \ll Z^3, B \sim Z^3, \gg Z^3$. These theories~\cite{LSY1},~\cite{LSY2} are relevant for neutron stars. Another class of TF theories with magnetic fields is relevant for electrons confined to two-dimensional geometries (quantum dots)~\cite{LSY3}. In this case there are three regimes. A convenient review is~\cite{LSY4}. Still another modification of TF theory is its extension from a theory of the ground states of atoms and molecules (which corresponds to zero temperature) to a theory of positive temperature states of large systems such as stars (cf.~\cite{JM},~\cite{WT}). \begin{thebibliography}{99} \bibitem[BL]{BL} BENGURIA, R. AND LIEB, E.H.: `The positivity of the pressure in Thomas-Fermi theory', {\it Commun. Math. Phys.} {\bf 63} (1978), 193-218. {\it Errata} {\bf 71}, (1980), 94. \bibitem[EF]{EF} FERMI, E.: `Un metodo statistico per la determinazione di alcune priorieta dell'atome', {\it Rend. Accad. Naz. Lincei} {\bf 6} (1927), 602-607. \bibitem[EL]{EL} LIEB, E.H.: `Thomas-Fermi and related theories of atoms and molecules', {\it Rev. Mod. Phys.} {\bf 53} (1981), 603-641. Errata {\bf 54} (1982), 311. \bibitem[LO]{LO} LIEB, E.H. AND OXFORD, S.: `An improved lower bound on the indirect coulomb energy', {\it Int. J. Quant. Chem.} {\bf 19} (1981), 427-439. \bibitem[LS]{LS} LIEB, E.H. AND SIMON, B.: `The Thomas-Fermi theory of atoms, molecules and solids', {\it Adv. in Math} {\bf 23} (1977), 22-116. \bibitem[LSY1]{LSY1} LIEB, E.H., SOLOVEJ, J.P., AND YNGVASON, J.: `Asymptotics of heavy atoms in high magnetic fields: I. lowest Landau band region', {\it Commun. Pure Appl. Math.} {\bf 47} (1994), 513-591. \bibitem[LSY2]{LSY2} LIEB, E.H., SOLOVEJ, J.P., AND YNGVASON, J.: `Asymptotics of heavy atoms in high magnetic fields: II. semiclassical regions', {\it Commun. Math. Phys.} {\bf 161} (1994), 77-124. \bibitem[LSY3]{LSY3} LIEB, E.H., SOLOVEJ, J.P., AND YNGVASON, J.: `Ground states of large quantum dots in magnetic fields', {\it Phys. Rev. B} {\bf 51} (1995) 10646-10665. \bibitem[LSY4]{LSY4} LIEB, E.H., SOLOVEJ, J.P., AND YNGVASON, J.: `Asymptotics of natural and artificial atoms in strong magnetic fields, in W. THIRRING (ed.): {\it The stability of matter: from atoms to stars, selecta of E. H. Lieb}, second edition, Springer, 1997, pp. 145-167. \bibitem[JM]{JM} MESSER, J.: `Temperature dependent Thomas-Fermi theory': Vol. 147 of {\it Lecture Notes in Physics}, Springer, 1981. \bibitem[ET]{ET} TELLER, E.,: `On the stability of molecules in Thomas-Fermi theory', {\it Rev. Mod. Phys.} {\bf 34} (1962), 627-631. \bibitem[LT]{LT} LIEB, E.H. AND THIRRING W.: `Inequalities for the moments of the eigenvalues of the Schr\"odinger Hamiltonian and their relation to Sobolev inequalities', in E. LIEB, B. SIMON, A. WIGHTMAN (eds.): {\it `Studies in Mathematical Physics}', Princeton University Press, 1976, pp. 269-303. \bibitem[WT]{WT} THIRRING, W.: `A course in mathematical physics': Vol. 4, Springer, 1983, pp. 209-277. \bibitem[TH]{TH} THOMAS, L.H.: `The calculation of atomic fields', {\it Proc. Camb. Phil. Soc.} {\bf 23} (1927), 542-548. \end{thebibliography} \bigskip \rightline{\it Elliott H. Lieb} \rightline{\it Departments of Mathematics and Physics} \rightline{\it Princeton University} \bigskip \footnoterule \bigskip \noindent {\copyright 1998 by Elliott H. Lieb} \end{document} ---------------0003281459756--