From: Paul Abbott Subject: Re: elementary solutions of Schr.eq. Date: Mon, 18 Jan 1999 15:39:53 +0800 Newsgroups: sci.math,sci.chem Pertti Lounesto wrote: > > Allen Adler wrote: > > > > How does one prove that the time-independent Schroedinger equation > > for a helium atom doesn't have any nonzero elementary solutions? And > > how generally does the proof apply to other atoms and molecules? > > Here, I'm willing to pretend that the nuclei are at rest. > > > > Allan Adler > > Allan (posting as Allen), you probably want to read: > > H.A. Bethe, E.E. Salpeter: Quantum Mechanics of One- and Two-Electron > Atoms, Springer, Berlin, 1957. Actually, a little bit more has happened since 1957! For the two-electron problem, the solutions are easily shown to be non- analytic throughout configuration space -- but this does not imply that exact solution is impossible. The original proof of non-analyticity was by Bartlett (1935). Using interparticle coordinates (IC) -- r1, r2 and r12 -- he expanded the helium eigenfunction as a power series in IC and showed that such an expansion leads to an inconsistent recurrence relation. The proof is simple and is easy to generalize to other atoms and molecules. The formal solution for Helium -- the Fock Expansion (FE) -- includes powers of Log[r1^2+r2^2] and was found by Fock (1954, 1958). This can be viewed as a generalization of Fuch's Theorem to partial differential equations. This logarithmic behaviour comes into play at the triple-point singularity. Introducing k=i+j+m (where i, j, and m are the powers of r1, r2, and r12 respectively), the expansion for the ground state of helium is known for k=0,1, and 2, and methods for extending the derivation to higher orders has been studied in some detail (Abbott and Maslen 1987, Gottschalk, Abbott and Maslen 1987). For a formal solution to become a physical solution it must be capable of satisfying the boundary conditions. Proof of the convergence of the Fock expansion for the Helium ground state was obtained by Leray (1982, 83, 84) and Morgan (1986). References Abbott P C and Maslen E N 1987 J. Phys. A, 20 2043-75 Bartlett J H 1937 Phys. Rev. 51 661-9 Fock V A 1954 Izv. Akad. Nauk. 18 161 ____ 1958 Kgl. Norske Vidensk. Sels. Forh. 31 138-52 Gottschalk J E, Abbott P C and Maslen E N 1987 J. Phys. A, 20 2077-2104 Leray J 1982a Actes du 6�me Congres du Groupement des Mathematiciens d'Expression Latine (Paris: Gauthier-Villars) pp179-87 ____ 1982b Methods of Functional analysis and Theory of Elliptic Operators (Naples) pp165-77 ____ 1983 in Bifurcation Theory, Mechanics and Physics C P Bruter et al (eds) (Dordrecht: Reidel) pp99-108 ____ 1984 Lecture Notes in Physics vol 195 (Berlin: Springer) pp235-47 Morgan J D 1986 Theor. Chim. Acta 69 181-223 Cheers, Paul ____________________________________________________________________ Paul Abbott Phone: +61-8-9380-2734 Department of Physics Fax: +61-8-9380-1014 The University of Western Australia Nedlands WA 6907 mailto:paul@physics.uwa.edu.au AUSTRALIA http://www.physics.uwa.edu.au/~paul God IS a weakly left-handed dice player ____________________________________________________________________