The Separation of the Rotational Coordinates from the N-Particle Schroedinger Equation

Abstract

By group theoretical arguments, it can be shown that a wave function, ψ_μ^L, for a system of N particles corresponding to a total angular momentum quantum number, L, and a quantum number, μ, referring to the z component of angular momentum may be written as a sum of terms: $${\psi }_{\mu }^L{\mathrm{=\Sigma }}_s{D}^L{\mathrm{(R)}}_{\mathrm{\mu s}}^{*}{\chi }_s^L.$$ The D^L(R)_μs are the representation coefficients for the Lth irreducible representation of the three‐dimensional rotation group, and are functions of the three coordinates specifying the orientation of the system of particles in space. The χ_μ^L are functions of the 3N−6 coordinates specifying the relative configuration of the N‐particle system. The set of coupled differential equations for the functions, χ_μ^L, is obtained explicitly. The special case of the three‐particle system is discussed in detail. The present treatment is more directly usable than the previous discussions since the basic equations do not involve implicit relationships between the variables.