Symbolic computation of integrals by recurrence

Barnett, Mike

doi:10.1145/944567.944571

Cited by 2 publications

(1 citation statement)

References 14 publications

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“…This article is first and foremost an ode to the vision and ingenuity of computing pioneers, but it also makes the point that there are lots of hidden treasures in the "old" scientific literature, that can be revisited with today's powerful symbolic computation software. We are not the first to advocate using symbolic computations in scientific computing, see for example [3] (unfortunately he was unaware of [13]), and the current impressive application to high-energy physics [4], but we believe that there is a huge potential for exploiting symbolic computation on problems that previously seemed intractable. This would complement the extensive use (and according to Nobelist Philip Anderson, excessive and sometimes abusive use [1]) of Monte Carlo methods.…”

Section: Discussionmentioning

confidence: 99%

The 1958 Pekeris-Accad-WEIZAC Ground-Breaking Collaboration that Computed Ground States of Two-Electron Atoms (and its 2010 Redux)

Koutschan

Zeilberger

2011

Math Intelligencer

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We have come a long wayIn order to appreciate how well off we mathematicians and scientists are today, with extremely fast hardware and lots and lots of memory, as well as with powerful software, both for numeric and symbolic computation, it may be a good idea to go back to the early days of electronic computers and compare how things went then. We have chosen, as a case study, a problem that was considered a huge challenge at the time. Namely, we looked at C.L. Pekeris's [9] seminal 1958 work 4 on the ground state energies of two-electron atoms. We went through all the computations ab initio with today's software and hardware. SchrödingerLet's recall the (time-independent) Schrödinger equation for the state function (alias wave function) ψ(x, y, z) of a one-electron atom with a stationary nucleus (see, for example, [8] Eq. (30-1) with N = 1), in atomic units:where Z denotes the nuclear charge, E the energy of the system, and r = x 2 + y 2 + z 2 the distance of the electron to the nucleus. Schrödinger's solution of this eigenvalue problem is one of the greatest classics of modern physics, familiar to all physics students (and chemistry students, but unfortunately not math), using separation of (dependent) variables, and getting explicit and exact results for the eigenvalues (the possible energy levels E) and even for the corresponding eigenfunctions ψ. Because the eigenfunctions (or more precisely their squares) are interpreted as probability distributions, certain restrictions have to be imposed on ψ; in particular, the integral of |ψ| 2 over the whole domain must be finite. The eigenvalues then are exactly those values of E for which the Schrödinger equation admits such a solution. It turns out that these eigenfunctions are expressible in terms of the venerable special functions of mathematical physics, namely (associated) Legendre and (associated) Laguerre polynomials. But exactly the same predictions (about the energy levels) were already made by the "old", ad hoc, Bohr-Sommerfeld quantum mechanics; the "new" wave-and matrix-quantum theories needed to predict facts that were beyond the scope of the old theory, thereby offering a crucial confirmation. That's why Schrödinger himself, Hylleraas, and many other physicists tried to derive the energy levels (alias eigenvalues) for two-electron atoms, whose Schrödinger equation, for the wave function ψ = ψ(x 1 , y 1 , z 1 , x 2 , y 2 , z 2 ), is

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Section: Discussionmentioning

confidence: 99%