Hydrogen Radial Wavefunctions

Norman Henry March

Michel de Celles

Devdatta Meghasham Mankame

Abstract It is often assumed that hydrogenic radial wavefunctions R nℓ ( r ) inherit universal orthogonality from the total wavefunctions ψ nℓm . This leads to the inference that all R nℓ are eigenfunctions of a single universal radial operator and thus form an orthonormal basis for L 2 ([0, ∞), r 2 d r ). We demonstrate this is false: orthogonality holds only within fixed- ℓ subspaces, where all R nℓ share the same self-adjoint operator h ˆ ℓ . For ℓ ≠ ℓ ′ , radial functions belong to distinct Sturm–Liouville families with operators differing in their centrifugal terms ∝ ℓ ( ℓ + 1)/ r 2 . This structural difference—not numerical artifact—explains the non-zero radial overlap (e.g. the 2 s –2 p overlap equals − 3 / 2 ). We classify all radial overlap integrals into four cases based on ( n , n ′ , ℓ , ℓ ′ ) , demonstrate non-orthogonality with explicit counterexamples, and explain the operator-family origin of this behavior. This perspective prevents subtle errors in basis expansions and perturbation theory for hydrogenic systems.

Owen L. de Lange, R. E. Raab

The radial form of Hylleraas’ equation for the hydrogen atom, Λl‖El〉=4ℏ4a−2‖El〉 (a=Bohr radius), is considered and it is shown that the operator Λl can be factorized. Hence ladder operators P±l are derived that are linear in the position operator r and are nonlinear functions of the momentum operator p. It is proven that P±l‖El〉 =2ℏ2a−1[1+(l+ (1)/(2) ± (1)/(2) )2 ×(2Mℏ−2a2E)]1/2‖E,l±1〉. In the momentum representation of wave mechanics the solutions to these equations are the radial momentum-space wavefunctions for the hydrogen atom. Thus a simple method of calculating these wavefunctions is obtained. The results complement the familiar operator solution for the hydrogen atom that is based on factorization of the radial Hamiltonian and yields operators that are linear in p and are nonlinear functions of r.