Casimir Effect: Zero-Point Modes and the d^-4 Pressure

An interactive Casimir effect between two perfect parallel plates. Between the plates only the standing modes with $k_n = n \pi / d$ survive; the long-wavelength modes that do not fit ($\lambda > 2 d$) are excluded, and the regularised difference from the free-space continuum is the attractive Casimir energy $E/A = -\pi^2 \hbar c / (720 d^3)$ with pressure $P = -dE/dd / A = \pi^2 \hbar c / (240 d^4) \sim 1.30 \, \mathrm{mPa}$ at $d = 1 \, \mu\mathrm{m}$. The plate panel draws the allowed standing modes (cyan), the excluded long modes (red) and the inward vacuum pressure, intensifying as the plates close; the law panel is the $d^{-4}$ pressure on log-log (slope $-4$); the energy panel contrasts $P \sim d^{-4}$ and $|E/A| \sim d^{-3}$. The pressure is about $1.3 \, \mathrm{mPa}$ at $d = 1 \, \mu\mathrm{m}$ and scales as $d^{-4}$ (the energy as $d^{-3}$), the force is attractive and equals $-dE/dd$, and it rises steeply as the plates close.