# 9. Schrodinger as Narrow-Band Maxwell The Schrodinger equation appears here as the precisely identified effective sector of double-curl or Maxwellian transport. Chapter 8 already gave the needed bounded discrete modes. The remaining task is to describe slow modulation of one such stable mode. Each Cartesian component $f(\mathbf{r},t)$ of $\mathbf{E}$ or $\mathbf{B}$ satisfies the vacuum wave equation: $$ \left(\nabla^2-\frac{1}{c^2}\partial_t^2\right)f=0. $$ Select the positive-frequency part of the field near a stable carrier frequency $\omega_0$, and demodulate the carrier: $$ \psi(\mathbf{r},t)=e^{i\omega_0 t}f^{(+)}(\mathbf{r},t). $$ The field is narrow-band when $$ \varepsilon = \frac{\Delta\omega}{\omega_0}\ll 1, $$ so the envelope $\psi$ varies slowly compared with the carrier. After separating the carrier and the base-mode contribution, and using the fact that the carrier already satisfies the dispersion relation of the underlying stable mode, the exact envelope identity is $$ i\partial_t\psi = -\frac{c^2}{2\omega_0}\nabla^2\psi +\frac{1}{2\omega_0 c^2}\partial_t^2\psi. $$ The last term is the retained difference between the exact Maxwellian envelope identity and the standard Schrodinger sector. For spectral width $\Delta\omega$, it is controlled by $$ \left\|\frac{1}{2\omega_0 c^2}\partial_t^2\psi\right\| \le \frac{\Delta\omega^2}{2\omega_0 c^2}\|\psi\| = O(\varepsilon^2)\|\psi\|. $$ So the leading effective sector is $$ i\partial_t\psi = -\frac{c^2}{2\omega_0}\nabla^2\psi +O(\varepsilon^2). $$ Now define the emergent constants from the carrier mode itself: $$ \hbar=\frac{E_0}{\omega_0},\qquad m=\frac{E_0}{c^2}, $$ where $E_0$ is the total energy of the underlying stable mode. Then $$ \frac{c^2}{2\omega_0}=\frac{\hbar}{2m}. $$ Multiplying by $\hbar$ gives $$ i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m}\nabla^2\psi +O(\varepsilon^2). $$ This is the free Schrodinger equation. It is not a rival starting point to the transport theory. It is the dominant narrow-band sector of the exact Maxwellian envelope identity for a stable mode. The retained term $$ \frac{1}{2\omega_0 c^2}\partial_t^2\psi $$ is therefore not a defect in the derivation. It is the explicit post- Schrodinger remainder carried by the deeper transport theory. The interaction case uses the same ontology. In structured backgrounds, the envelope accumulates additional region-dependent phase. The double-slit treatment later represents such interaction regions by localized potentials $V_j$ that rotate the relative phase of the propagation channels. The potential term is therefore a summary of background interaction in the same envelope dynamics. This chapter, however, derives only the free narrow-band case. Superposition, interference, and uncertainty enter because the envelope remains a wave field. Standard quantum mechanics is the effective theory of slowly varying Maxwellian envelopes. Any experimentally accessible effect carried by the retained remainder would be new physics beyond the standard Schrodinger sector, not a failure of the derivation.