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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 ($\lambda \gt 2 d$) 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.

Figure 1. Two perfectly conducting plates exclude the long-wavelength vacuum modes that do not fit between them; the surviving zero-point pressure imbalance pulls the plates together with P = pi^2 hbar c / 240 d^4, about 1.3 mPa at 1 micron. Method: mode counting plus the zeta-regularised Casimir energy and pressure; Canvas2D, deterministic.
plate separation d (nm)1000
mode cutoff index8

WHAT TO TRY

  • Shrink the plate separation d: the Casimir energy E/A = -pi^2 hbar c / (720 d^3) and the pressure P ~ d^-4 climb steeply. Halving d multiplies the attractive force by sixteen.
  • Tune the mode cutoff index: only standing modes with k_n = n pi / d fit between the plates, and long-wavelength modes (lambda > 2d) are excluded. That missing vacuum energy is the entire effect.
  • Read the pressure readout: it is the regularized difference between the mode sum inside and the free-space continuum outside, a measurable force from zero-point energy.