Momentum density and spatial form of correlated density matrix in model two-electron atoms with harmonic confinement

摘要

The detailed nature of the correlated first-order density matrix for the model atoms in the title for arbitrary interparticle interaction u(r12) is studied. One representation with contracted information is first explored by constructing the momentum density ρ(p) in terms of the wave function of the relative motion, say ΨR(r12), which naturally depends on the choice of u(r12). For u(r12)=e2∕r12, the so-called Hookean atom, and for the inverse square law u(r12)=λ∕r122, plots are presented of the above density ρ(p) in momentum space. The correlated kinetic energy is recovered from averaging p2∕2m, m denoting the electron mass, with respect to ρ(p). The second method developed is in coordinate space and expands the density matrix γ(r1,r2) in Legendre polynomials, using relative coordinate r1−r2, center-of-mass coordinate (r1+r2)∕2 and the angle, θ say, between these two vectors. For the Moshinsky atom in which u(r12)=12kr122 only the s term (l=0) contributes to the Legendre polynomial expansion. The specific example we present of the inverse square law model is shown to be characterized by the low-order terms (s+d) of the Legendre expansion. The Wigner function is finally calculated analytically for both Moshinsky and inverse square law models.