# 11. Schrodinger as Narrow-Band Maxwell The Schrodinger equation appears here as a controlled envelope limit of Maxwell transport. Chapters 8 and 9 already gave two things needed for that limit: discrete stable modes and an emergent mass scale. The remaining task is to describe slow modulation of one such 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 difference between exact Maxwell transport and the Schrodinger limit. 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, to leading order in the narrow-band parameter, $$ i\partial_t\psi = -\frac{c^2}{2\omega_0}\nabla^2\psi +O(\varepsilon^2). $$ 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 rest 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 arises as the narrow-band envelope equation of a stable Maxwell mode. 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. Quantum mechanics is the effective theory of slowly varying Maxwell envelopes.