Accurate analytic presentation of solution of the Schroedinger equation with arbitrary physical potential

Journal Article · Sat Dec 15 04:00:00 EST 2007 · Annals of Physics (New York)

DOI:https://doi.org/10.1016/j.aop.2007.02.003· OSTI ID:21077669

Drukarev, E G ^[1]; Mandelzweig, V B ^[2]

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1. Petersburg Nuclear Physics Institute, Gatchina, St. Petersburg 188300 (Russian Federation)
2. Racah Institute of Physics, Hebrew University, Jerusalem 91904 (Israel)

High precision approximate analytic expressions of the ground state energies and wave functions for the arbitrary physical potentials are found by first casting the Schroedinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms by a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. The approach is illustrated on the examples of the Yukawa, Woods-Saxon and funnel potentials. For the latter potential, solutions describing charmonium, bottonium and topponium are analyzed. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of physical parameters. The accuracy ranging between 10{sup -4} and 10{sup -8} for the energies and, correspondingly, 10{sup -2} and 10{sup -4} for the wave functions is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects correspondent physical systems.

• OSTI ID: 21077669
• Journal Information: Annals of Physics (New York), Journal Name: Annals of Physics (New York) Journal Issue: 12 Vol. 322; ISSN 0003-4916; ISSN APNYA6
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BINDING ENERGY
CHARMONIUM
EXACT SOLUTIONS
GROUND STATES
ITERATIVE METHODS
NONLINEAR PROBLEMS
NUMERICAL SOLUTION
SCHROEDINGER EQUATION
WAVE FUNCTIONS
WOODS-SAXON POTENTIAL