P D S.E. II1 3D Schr. Eqn.:Radial Eqn. For V funtion of radius only. Look at radial equation often rewritten as note l(l+1) term. Angular momentum. Acts like repulsive potential (ala classical mechanics) energy eigenvalues typically depend on 2 quantum numbers (n and l). Only 1/r potentials depend only on n (and true for hydrogen atom only in first order. After adding perturbations due to spin and relativity, depends on n and j=l+s).

P D S.E. II2 Particle in spherical Box Griffiths Example 4.1. E&R section Good first model for nuclei plug into radial equation look first at l=0 boundary conditions. R=u/r and must be finite at r=0. Gives B=0. For continuity, must have R=u=0 at r=a. gives sin(ka)=0 and note plane wave solution are

P D S.E. II3 Particle in spherical Box For l>0 solutions are Bessel functions (see Griffiths). Often arises in scattering off spherically symmetric potentials (like nuclei…..) energy will depend on both quantum numbers and so 1s 1p 1d 2s 2p 2d 3s 3d …………….and ordering (except higher E for higher n,l) depending on details gives what nucleii (what Z or N) have filled (sub)shells being different than what atoms have filled electronic shells. In atoms: in nuclei (with j subshells)