# Hydrogen Matrix Lab 0.6.0 — conserved flow and observable harmonics

## Bounded question

The lab already evolved a coherent hydrogen wavefunction with the Schrödinger equation, but it displayed only the position density. What equation converts that interference into conserved spatial flow, and does every density beat become an electric-dipole/light harmonic?

The missing bridge is not a new extra-dimensional law. For a particle of charge \\(q\\), electromagnetic minimal coupling is

\[
i\hbar\partial_t\Psi=
\left[\frac{(-i\hbar\nabla-q\mathbf A)^2}{2m}+q\phi\right]\Psi,
\]

with conserved probability

\[
\partial_t\rho+\nabla\!\cdot\mathbf j=0,
\qquad
\mathbf j=\frac{\hbar}{2mi}
(\Psi^*\nabla\Psi-\Psi\nabla\Psi^*)-
\frac{q}{m}\mathbf A|\Psi|^2.
\]

Release 0.6 implements the field-free \\(\\mathbf A=0\\) current for the existing fixed-proton Coulomb `1s + 2p_z` packet. The Coulomb scalar potential was already present in earlier releases.

## Exact result and finite checks

For

\[
\Psi=(\phi_{1s}+e^{-i\theta}\phi_{2p_z})/\sqrt2,
\qquad \theta=(E_2-E_1)t/\hbar,
\]

the engine evaluates the full vector field

\[
\mathbf j=\frac{\sin\theta}{2}
(\phi_{2p_z}\nabla\phi_{1s}-\phi_{1s}\nabla\phi_{2p_z}).
\]

The written eigenvalue-equation proof gives `partial_t rho + div(j) = 0` identically. On the released finite grid, the exact dipole matrix element

\[
\langle1s|z|2p_z\rangle=\frac{256}{243\sqrt2}a_0
=0.744935539028a_0
\]

is reproduced as `0.744932784051 a0` (`3.70e-6` relative error). The integrated current agrees with the time derivative of the dipole within `5.35e-4` relative error on the production grid.

An implementation that imports none of the production module independently used radial quadrature and fourth-order Cartesian finite differences. It reproduced the dipole matrix element and ideal spontaneous rate, and obtained a continuity residual of `0.0128041` relative L2 outside the declared `r < 0.5 a0` nuclear-cusp region. The full-interior residual `0.116031` is preserved because a Cartesian finite-difference stencil converges slowly across the Coulomb cusp; it is not hidden as a pass.

The corresponding ideal electric-dipole numbers are:

- dipole amplitude: `6.3158269e-30 C m = 1.8934378 D`;
- spontaneous `2p_z -> 1s` rate: `6.26831503e8 s^-1`;
- ideal lifetime: `1.59532505 ns`.

These are standard fixed-proton, nonrelativistic electric-dipole calculations. They are not a precision-QED lifetime prediction or a new measurement.

## More harmonics, with an observation gate

An equal `1s + 2p_z + 3s` packet contains three pairwise density gaps. The electric-dipole observable sees only two:

| Pair | Gap (Hartree) | Ideal wavelength (nm) | Density beat | z-dipole beat |
|---|---:|---:|---|---|
| `2p_z <-> 3s` | 0.0694444444 | 656.112276 | yes | yes |
| `1s <-> 2p_z` | 0.375 | 121.502273 | yes | yes |
| `1s <-> 3s` | 0.4444444444 | 102.517543 | yes | **no** |

The `1s <-> 3s` matrix element vanishes because both states have even parity. The separate `1s + 2s` control also has a changing local density but exactly zero one-photon electric-dipole signal. This is the important counterexample: adding energy-gap harmonics without an observable matrix element overpredicts what light can reveal. NIST's atomic-spectroscopy framework likewise defines transition strength through the relevant multipole matrix element, not through density motion alone.

## Decision and limits

The answer to “did we forget a quantum equation?” is now precise: Schrödinger evolution was already present; probability current, electromagnetic minimal coupling, and the observable/selection-rule gate were not explicit in the engine. Release 0.6 adds the field-free current and dipole gate without inventing an extra-dimensional correction.

The arrows in the visualization are probability flow, not paths followed by a localized electron. The coherent packet must be prepared and is not the stationary ground state. The model is fixed-proton, nonrelativistic, spinless, and field-free. Experiments can image ultrafast charge motion in prepared wave packets, but that does not turn this computed hydrogen slice into measured electron footage. Nothing here establishes a fourth or fifth spatial dimension, a universal grid/tick, or a theological cause.

All 42 package tests pass. The independent audit passes all seven frozen checks.

## Next test

Drive the atom with a frozen weak electromagnetic pulse and solve the time-dependent Schrödinger equation in two gauges. Require length-gauge and velocity-gauge agreement, predict the allowed `1s -> 2p` response and the one-photon-suppressed `1s -> 2s` control before comparing transition probabilities or lifetimes with compiled spectroscopy.

## Sources

- NIST, *Atomic Spectroscopy — Spectral Lines*: https://physics.nist.gov/Pubs/AtSpec/node17.html
- NIST, *Energy Levels of Neutral Hydrogen*: https://physics.nist.gov/PhysRefData/Handbook/Tables/hydrogentable5.htm
- Kübel et al., *Spatiotemporal imaging of valence electron motion* (primary experiment): https://arxiv.org/abs/1903.03739
- Rossi et al., *Preparation and control of electronic wave packets in neutral molecules via attosecond x-ray processes* (2026 primary method): https://arxiv.org/abs/2606.22102