Reduction of the Breit Coulomb equation to an equivalent Schroedinger equation, and investigation of the behavior of the wave function near the origin

Journal Article · Mon Nov 14 23:00:00 EST 1988 · Phys. Rev. D; (United States)

DOI:https://doi.org/10.1103/PhysRevD.38.3295· OSTI ID:6696769

Malenfant, J

The Breit equation for two equal-mass spin-1/2 particles interacting through an attractive Coulomb potential is separated into its angular and radial parts, obtaining coupled sets of first-order differential equations for the radial wave functions. The radial equations for the /sup 1/J/sub J/, /sup 3/J/sub J/, and /sup 3/P/sub 0/ states are further reduced to a single, one-dimensional Schroedinger equation with a relatively simple effective potential. No approximations, other than the initial one of an instantaneous Coulomb interaction, are made in deriving this equation; it accounts for all relativistic effects, as well as for mixing between different components of the wave function. Approximate solutions are derived for this Schroedinger equation, which gives the correct O(..cap alpha../sup 4/) term for the 1 /sup 1/S/sub 0/ energy and for the n/sup 1/J/sub J/ energies, for J>0. The radial equations for the /sup 3/(J +- 1)/sub J/ states are reduced to two second-order coupled equations. At small r, the Breit Coulomb wave functions behave as r/sup ..nu..//sup -1/, where ..nu.. is either ..sqrt..J(J+1)+1-..cap alpha../sup 2//4 or ..sqrt..J(J+1)-..cap alpha../sup 2//4 . The /sup 1/S/sub 0/ and /sup 3/P/sub 0/ wave functions therefore diverge at the origin as r/sup //sup ..sqrt..//sup 1-//sup ..cap alpha..//sup <2//4 -1$. This divergence of the J = 0 states, however, does not occur when the spin-spin interaction, -(..cap alpha../r)..cap alpha..x..cap alpha.., is added to the Coulomb potential.

• Research Organization: Department of Physics, University of California, Los Angeles, California 90024
• OSTI ID: 6696769
• Journal Information: Phys. Rev. D; (United States), Journal Name: Phys. Rev. D; (United States) Vol. 38:10; ISSN PRVDA
• Country of Publication: United States
• Language: English

Related Subjects

657002* -- Theoretical &amp; Mathematical Physics-- Classical &amp; Quantum Mechanics
657003 -- Theoretical &amp; Mathematical Physics-- Relativity &amp; Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
BOUND STATE
COULOMB FIELD
COUPLING
DIFFERENTIAL EQUATIONS
DIRAC EQUATION
ELECTRIC FIELDS
ENERGY LEVELS
EQUATIONS
FIELD THEORIES
FINE STRUCTURE
FUNCTIONS
GENERAL RELATIVITY THEORY
INTERACTIONS
INTERMEDIATE COUPLING
J-J COUPLING
ONE-DIMENSIONAL CALCULATIONS
PARTIAL DIFFERENTIAL EQUATIONS
PARTICLE INTERACTIONS
RELATIVITY THEORY
RENORMALIZATION
SCHROEDINGER EQUATION
WAVE EQUATIONS
WAVE FUNCTIONS