Casimir's spheres near the Coulomb limit: energy density, pressures and radiative effects

Abstract

We study the quantum mechanics of an infinitesimally thin spherical fluid shell of radius R, having continuous charge and mass densities e/a2 and m/a2 per unit area, with a R, and subject to a Debye-type cut-off on surface-parallel wave numbers. Attention is confined to the regime ? ? 4?e2R/mc2a2 1, where nonretarded (NR) Coulomb forces dominate, and the coupling to the quantized Maxwell field is only a weak perturbation. The unperturbed ground-state energy BNR is of order (e2/ma7)1/2R2. Half of BNR is kinetic energy, localized on the shell; the other half is Coulomb energy, with a density u appreciable only within distances of order a from the shell. The pressure P = 3BNR/8?R3 follows directly from the principle of virtual work: more detailed analysis shows that P/3 comes from Coulomb forces, and 2P/3 from the zero-point motion of the fluid. It seems likely that u and P behave in much the same way also for large ?. The purely Coulombic system has stable discrete-frequency excitations (plasmons); to leading order the perturbation displaces the frequencies and allows a plasmon to decay into a photon. The displacements and decay rates tally with what one infers from the exact classical multipole phase shifts, and from the already-known energy for arbitrary values of ?.