Quantum three-body problems

Abstract

An improved hyperspherical harmonic method for the quantum threebody problem is presented to separate three rotational degrees of freedom completely from the internal ones. In this method, the Schrödinger equation of three-body problem is reduced to a system of linear algebraic equations in terms of the orthogonal bases of functions. As an important example in quantum mechanics, the energies and the eigenfunctions of some states of the helium atom and the helium-like ions are calculated. PACS number(s): 03.65. Ge, and 02.30.Gp

Quantum three-body system in D dimensions

“…The formula for Q µν q (x, y) holds for D = 3 (x 4 = y 4 = 0, ν = 0 or 1) [35,29] and D > 4. When D = 4 we denote the highest weight states by Q (S)µν q (x, y) and Q (A)µν q (x, y) for the selfdual representations and the antiselfdual representations, respectively:…”

“…The generalized radial equations satisfied by the functions are established explicitly [29]. For the typical three-body system in the real three dimensional space [30,31], such as a helium atom [32,33] and a positronium negative ion [34], the generalized radial equations [35] have been solved numerically with high precision.…”

Independent eigenstates of angular momentum in a quantumN-body system

“…Substituting Eq. ( 20) into the Schrödinger equations ( 6) and ( 9), one is able to easily derive the generalized radial equations for the generalized radial functions ψ ℓλ q (ξ 1 , ξ 2 , η 2 ) [3,27]:…”

“…It is due to the existence of ζ j that Q ℓ0 q (R 1 , R 2 ) and Q ℓ1 q (R 1 , R 2 ) appear together in the expansion of the wave function. By comparison, all internal variables in a quantum three-body system have even parity (ζ j = 0) so that in the expansion (20) of a wave function with a given parity only the base functions with the same parity appear [3,4,27,28].…”

“…One describes the motion of the center of mass as a free particle, and the other describes the motion of the system in the center-of-mass frame. It is no loss of generality to suppose the center of mass of the system to be at rest, so that the configuration is completely specified by (N − 1) vectors r cj , 1 ≤ j ≤ N − 1, which are usually chosen as the Jacobi coordinate vectors R j for simplicity [1][2][3] (see Sec. II).…”