Formula omitted] for asymptotic coefficients of radial wave functions in quantum mechanics

Abstract

Using the 1 n - expansion , we obtain analytic formulae for the bound state radial wave functions, including its asymptotic coefficients at r → 0 and r → ∞, for an arbitrary smooth potential V( r). The formulae are asymptotically exact in the limit n r → ∞ ( n = n r + l + 1 is the principal quantum number and the expansion parameter is 1 n ). Comparison with exact solutions and numerical calculations for the power-law and short-range potentials show that the applicability region of these formulae is usually prolonged up to small quantum numbers, n ∼ 1. With growing n r, the accuracy of the formulae decreases, but the WKB method becomes applicable in this case.