‘Kramer’ Type Expressions, The Virial Theorem and Generalizations

Abstract

Consider a particle moving in a central potential V(r) = Ar^p. The (radial) differential equation for u_nl (r) = r R_nl(r), (where the complete wave-function ψ(r, θ, ɸ) = R_nl(r) Y ^l_m (θ, ɸ)) is

$$ \frac{{d^2 }} {{dr^2 }}u_{n\ell } (r) = \left\{ {\frac{{2m}} {{\hbar ^2 }}(V(r) - E) + \frac{{\ell (\ell + 1)}} {{r^2 }}} \right\}u_{n\ell } (r),$$

((9.1))

with boundary conditions that u_nl(0) = 0, and u_nl(r) → 0.