# Hydrogen 1S→2P electric-dipole benchmark

Fixed Grid String Lab 0.3.0 / Theory of Reality 1.78.

This derivation fixes the ordinary nonrelativistic quantum benchmark used by
`DIPOLE_PROTOCOL_0.3.json`. It establishes conditional mathematics for the
declared model. It does not derive electromagnetism from higher-dimensional
strings or validate a physical spatial grid.

## Radial operators

In atomic units, separate

\[
\psi_{lm}(r,\theta,\phi)=\frac{u_l(r)}{r}Y_{lm}(\theta,\phi),
\qquad \int_0^\infty |u_l(r)|^2dr=1.
\]

The Coulomb radial operator is

\[
h_l=-\frac12\frac{d^2}{dr^2}+\frac{l(l+1)}{2r^2}-\frac1r,
\qquad u_l(0)=0.
\]

On sites \(r_j=ja\), with outer Dirichlet wall \(u_M=0\), the declared matrix
has

\[
(H_{a,l})_{jj}=\frac1{a^2}-\frac1{ja}+\frac{l(l+1)}{2(ja)^2},
\qquad
(H_{a,l})_{j,j+1}=-\frac1{2a^2}.
\]

The mesh is a numerical radial coordinate. It is not a cubic physical lattice.

## Continuum targets

For the infinite-mass Coulomb problem,

\[
E_{1S}=-\frac12,\qquad E_{2P}=-\frac18,
\qquad \Delta E=\frac38.
\]

The normalized reduced radial functions are

\[
u_{1S}(r)=2re^{-r},
\qquad
u_{2P}(r)=\frac{r^2}{2\sqrt6}e^{-r/2}.
\]

Therefore the radial length-gauge dipole integral is

\[
I_r=\int_0^\infty u_{2P}(r)\,r\,u_{1S}(r)\,dr
=\frac1{\sqrt6}\int_0^\infty r^4e^{-3r/2}dr
=\boxed{\frac{256}{81\sqrt6}}.
\]

For linearly polarized light along \(z\),

\[
\langle Y_{10}|\cos\theta|Y_{00}\rangle=\frac1{\sqrt3},
\]

so

\[
\boxed{\langle2P,0|z|1S,0\rangle
=\frac{256}{243\sqrt2}}.
\]

The directional oscillator strength is

\[
f_z=2\Delta E\,|\langle2P,0|z|1S,0\rangle|^2
=\boxed{\frac{8192}{19683}}
=0.4161967179799827\ldots.
\]

For an initial S state, this equals the usual multiplet-summed isotropic line
oscillator strength: the factor \(1/3\) from isotropic averaging is canceled by
the three degenerate 2P magnetic substates. This equality depends on the
nonrelativistic degeneracy and conventions just stated.

## Angular selection rules

For \(z=r\cos\theta\),

\[
\langle l'm'|\cos\theta|lm\rangle=0
\]

unless \(m'=m\) and \(l'=l\pm1\). Thus the implemented one-photon E1 model
allows 1S→2P with \(m'=0\), while 1S→2S, 1S→3D and the z-polarized
1S→2P \(m'=1\) matrix elements vanish from the angular factor. These are
selection rules inside the imported electric-dipole model, not explanations of
why nature has that interaction.

## Discrete matrix element and error separation

An eigenvector normalized in the discrete Euclidean norm represents
\(u_j\simeq\sqrt a\,u(ja)\). Hence the Riemann approximation is

\[
I_r(a,R)=\sum_{j=1}^{M-1}u^{(2P)}_j(ja)u^{(1S)}_j,
\]

with no additional factor of \(a\). Multiplying again by \(a\) would double
count the quadrature weight implicit in the discrete normalization.

The protocol varies \(a\) and \(R\) separately. Changes under \(R=80\to160\)
diagnose the outer wall at fixed spacing; changes under grid refinement at
\(R=160\) diagnose spatial discretization. Neither numerical convergence nor
agreement with an equation supplied to the program counts as evidence that
space itself is discrete.

The finite model omits reduced proton mass, fine structure, Lamb shift, recoil,
finite nuclear size, radiative corrections, natural width, external fields,
line shape and detector response. It therefore does not predict the precision
components of the physical Lyman-α doublet. NIST's critically evaluated atomic
spectra resources provide spectroscopy reference context, but their wavelengths
are not used to fit or score this release.