Quantum electrodynamics of spinless particles between conducting plates

The energy of any system containing charged particles arises partly from its coupling to the quantized electromagnetic field; it changes on inserting the system between conducting plates, because these alter the normal modes of the field relative to free space. We calculate to order e$^{2}$ such energy shifts for a free electron and for a hydrogen atom, neglecting spin, but allowing in full for electrostatic and retardation corrections as well as for changes in the Lamb shift proper. The potential between a particle and a single plate follows as a limiting case. An appendix details the distinction between classical and quantum effects. For free electrons there is a quantum shift of order [Note: See the image of page 251 for this formatted text] e$^{2}$h/mcL$^{2}$, and classical corrections of order e$^{2}$v$^{2}$/c$^{2}$L (L = plate separation). For atoms, for L $\ll $ $\Lambda $ = (typical absorption wavelength), the shift is of order e$^{2}$a$^{2}$/L$^{3}$ (a = Bohr radius). For L $\ll $ $\Lambda $, the ground state shift is of order [Note: See the image of page 251 for this formatted text] hcL$^{-4}$ x atomic polarizability, and excited state shifts are of order [Note: See the image of page 251 for this formatted text] e$^{2}$h$^{2}$/a$^{2}$m$^{2}$c$^{2}$L = (e$^{2}$hc)$^{2}$e$^{2}$/L. The latter dominate the effect on all electric dipole (optical-frequency) lines. Comparison with experiments to date must await extension of the theory to particles with spin.