# 9a. Interaction as Phase Accumulation Chapter 9 derived the free Schrodinger equation as the effective narrow-band sector of a stable Maxwellian mode. Interaction enters when that envelope propagates through a structured background. In the envelope description, such a background appears as a local potential: $$ i\hbar\,\partial_t\psi = \left( -\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t) \right)\psi. $$ $V$ is the compact interaction writing for how the background changes the phase accumulated by the envelope during propagation. Once the background is given, this term is fully writable; what varies from case to case is only how much of that background structure is already known. The propagator makes this explicit: $$ K(x,T;x_0,0) = \int\mathcal Dq\; \exp\!\left[ \frac{i}{\hbar}\int_0^T dt \left( \frac{1}{2}m\dot q^2 - V(q,t) \right) \right]. $$ All interaction enters through the action in the exponential. The background rotates the phase of the wave. This is especially clear in the double-slit case. If two spatial channels experience different interaction backgrounds, represented by $V_1$ and $V_2$, the physically relevant quantity is the action difference between them: $$ \Delta\phi = \frac{1}{\hbar}\int_0^T dt\,(V_1-V_2). $$ The total amplitude at the screen is then $$ \psi(x) = \psi_1^{(0)}(x) + e^{i\Delta\phi}\psi_2^{(0)}(x), $$ where $\psi_j^{(0)}$ is the reference amplitude for free or symmetric propagation. If $V_1 = V_2$, then $\Delta\phi = 0$ and the full interference pattern is recovered. If the two backgrounds differ, the relative phase changes and the pattern changes with it. What disappears is coherent phase relation. This closes the loop left open in chapter 9. The potential term in Schrodinger's equation is the compact phase-writing of background interaction inside the same wave-envelope dynamics.