Exact solution of coupled equations and the hyperspherical formalism: Calculation of expectation values and wavefunctions of three Coulomb-bound particles

Journal Article · Sat Oct 01 00:00:00 EDT 1983 · Ann. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1016/0003-4916(83)90004-0· OSTI ID:5372408

Haftel, M I; Mandelzweig, V B

Exact solutions of one-dimensional coupled differential equations are developed by substituting in power series. The properties of these solutions and the possibility of their application the few-body problem in the framework of the hyperspherical method are studied. The necessity of logarithmic terms in the nonrelativistic many-body wavefunctions, as well as their absence in the relativistic case, is stressed. Explicit form of the solution of the one-dimensional hyperspherical matrix equation corresponding to the three-body Coulomb problem is found and used to obtain Schroedinger and Faddeev bound state wavefunctions, correlatin integrals and probabilities of different hyperspherical states. The results of calculations with inclusion of up to 25 hyperspherical harmonics (K/sub m/ = 16) for the ground and excited state of the helium atom, the ground state of the positronium in and the negative hydrogen in are given and compared with those obtained by the multiconfigurational Hartree-Fock and variational methods as well as with other hyperspherical calculations. We find that generally the correlation integrals converge as the energies, that is, as 1/K/sup 4//sub m/. While the method is essentially exact, computer round-off error limits the precision for K/sub m/>12 in the positronium calculations.

• Research Organization: Naval Research Laboratory, Code 6651, Washington, DC 20375
• OSTI ID: 5372408
• Journal Information: Ann. Phys. (N.Y.); (United States), Journal Name: Ann. Phys. (N.Y.); (United States) Vol. 150:1; ISSN APNYA
• Country of Publication: United States
• Language: English

Related Subjects

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640303 -- Atomic
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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
ANALYTICAL SOLUTION
ANIONS
BINDING ENERGY
BOUND STATE
BOUNDARY CONDITIONS
CHARGED PARTICLES
COULOMB FIELD
DIFFERENTIAL EQUATIONS
ELECTRIC FIELDS
ELEMENTS
ENERGY
EQUATIONS
EXPECTATION VALUE
FADDEEV EQUATIONS
FLUIDS
FUNCTIONS
GASES
HELIUM
HYDROGEN IONS
HYDROGEN IONS 1 MINUS
IONS
MANY-BODY PROBLEM
NONMETALS
ONE-DIMENSIONAL CALCULATIONS
POSITRONIUM
POWER SERIES
RARE GASES
RECURSION RELATIONS
SERIES EXPANSION
THREE-BODY PROBLEM
WAVE FUNCTIONS