On the Perturbation Series in Large O rder of Anharmonic Oscillators

2000 ◽  
Vol 1 (1) ◽  
pp. 193-200 ◽  
Author(s):  
T. Koike
Keyword(s):  
Perturbation Series ◽  
Anharmonic Oscillators

Analytical expressions for the energies of anharmonic oscillators

2000 ◽  
Vol 78 (9) ◽  
pp. 845-850
Author(s):  
F M Fernández ◽  
R H Tipping
Keyword(s):  
Perturbation Parameter ◽  
Bound State ◽  
Perturbation Series ◽  
Simple Expression ◽  
Branch Points ◽  
Anharmonic Oscillators ◽  
Large Coupling ◽  
Analytical Behavior ◽  
Schrödinger Perturbation ◽  
Analytical Expressions

We propose a systematic construction of algebraic approximants for the bound-state energies of anharmonic oscillators. The approximants are based on the Rayleigh-Schrödinger perturbation series and take into account the analytical behavior of the energies at large values of the perturbation parameter. A simple expression obtained from a low-order perturbation series compares favorably with alternative approximants. Present approximants converge in the large-coupling limit and are suitable for the calculation of the energy of highly excited states. Moreover, we obtain some branch points of the eigenvalues of the anharmonic oscillator as functions of the coupling constant. PACS No.: 03.65Ge

Classical anharmonic oscillators: Rescaling the perturbation series

1984 ◽  
Vol 30 (2) ◽  
pp. 1118-1119 ◽  
Author(s):  
Kalyan Banerjee ◽  
Jayanta K. Bhattacharjee ◽  
H. S. Mani
Keyword(s):  
Perturbation Series ◽  
Anharmonic Oscillators

ON THE USE OF THE SYMBOLIC LANGUAGE MAPLE IN PHYSICS AND CHEMISTRY: SEVERAL EXAMPLES

1993 ◽  
Vol 04 (02) ◽  
pp. 257-270 ◽  
Author(s):  
J. Čížek ◽  
F. Vinette ◽  
E. J. Weniger
Keyword(s):  
Symbolic Computation ◽  
Applied Mathematics ◽  
Nonlinear Boundary Conditions ◽  
Perturbation Series ◽  
Symbolic Language ◽  
Perturbation Expansions ◽  
Anharmonic Oscillators ◽  
Summation Of Divergent Series ◽  
Quantum Theory Group ◽  
The University

The purpose of this paper is to present several physical and chemical applications of the symbolic computation language Maple which has been developed by the Symbolic Computation Group at the University of Waterloo. The paper will highlight the activity of the Quantum Theory Group of the Department of Applied Mathematics at the University of Waterloo during the years 1985–1992, which has already led to more than 20 articles on applications of Maple. Special attention will be given to examples from quantum mechanics and quantum chemistry: the application of the inner projection technique to anharmonic oscillators yielding explicit expressions for Löwdin's approximants as well as to simple many-electron problem, the use of hypervirial and Hellmann-Feynman theorems for the computation of perturbation series coefficients in rational arithmetics, the creation of coupled-cluster equations by the computer, the use of several techniques including the method of Weniger for the summation of divergent series as they occur in perturbation expansions for anharmonic oscillators or in heat conduction with nonlinear boundary conditions. In all these examples, the symbolic computation language Maple proved to be very useful, and in many cases, it was actually indispensable for obtaining a solution or for keeping control of the accuracy.

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