An economical method to calculate eigenvalues of the Schroedinger Equation

Abstract

The method is an extension to negative energies of a spectral integral equation method to solve the Schroedinger equation, developed previously for scattering applications. One important innovation is a re-scaling procedure in order to compensate for the exponential behaviour of the negative energy Green's function. Another is the need to find approximate energy eigenvalues, to serve as starting values for a subsequent iteration procedure. In order to illustrate the new method, the binding energy of the He-He dimer is calculated, using the He-He TTY potential. In view of the small value of the binding energy, the wave function has to be calculated out to a distance of 3000 a.u. Two hundred mesh points were sufficient to obtain an accuracy of three significant figures for the binding energy, and with 320 mesh points the accuracy increased to six significant figures. An application to a potential with two wells separated by a barrier, is also made.

19 pages, 3 figures, submitted to Eur. J. Phys

References in corpus (8)

• Universality in Few-body Systems with Large Scattering Length
• Matter Wave Diffraction from an Inclined Transmission Grating: Searching for the Elusive He-4 Trimer Efimov State
• Binding Energies and Scattering Observables in the 4^He_3 Atomic System
• Comparison of numerical methods for the calculation of cold atom collisions
• Manifestations of the Efimov Effect for Three Identical Bosons
• Scattering length of the helium atom - helium dimer collision
• A Novel Method for the Solution of the Schroedinger Eq. in the Presence of Exchange Terms
• A Selective Relaxation Method for Numerical Solution of Schrödinger Problems

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• Noncommutative sedeons and their application in field theory
• The vibrating inhomogeneous string: a topic for a course in Computational Physics