Valence-1 spectrum from RealQM empty-core + Neumann

Valence-1 spectrum — empty-core + Neumann (RealQM-native)

Solve −½ u″ + [ℓ(ℓ+1)/(2r²) − Z/r] u = E u on r ∈ [r_c, R_max]
with Neumann ψ′(r_c) = 0 (i.e. u′(r_c) = u(r_c)/r_c) at the inner boundary and u(R_max) = 0 at the outer. The interior r < r_c is empty (no wave function); inner shells are conceptually absorbed into the bare Coulomb kernel −Z/r outside r_c. This is the proper RealQM treatment for a single valence electron over a closed inner core.

Parameters

Atom

Z_kernel

r_c (a.u.)

R_max (a.u.)

Grid points

Eigenvalues E_{n,ℓ}

n \ ℓ	0 (s)	1 (p)	2 (d)
click Compute

Reference: hydrogen E_n = −Z²/(2n²) = −0.500, −0.125, −0.0556, −0.0312 Ha for n=1,2,3,4.

Quantum defect δ_ℓ

Computed from E_{n,ℓ} = −1/(2(n−δ)²): δ_ℓ = n − 1/√(−2E).
Reference defects (NIST): Li 0.40/0.05/0.00; Na 1.35/0.86/0.01; K 2.18/1.71/0.28.

n \ ℓ	0 (s)	1 (p)	2 (d)

Model vs observed (Ha)

n,ℓ	model	obs	Δ

RMS error — Ha over fitted levels.