  We examine the dipole approximated Pauli-Fierz Hamiltonians of the
nonrelativistic QED. We assume that the Coulomb potential of the nuclei
together with the Coulomb interaction between the electrons can be approximated
by harmonic potentials. By an exact diagonalization method, we prove that the
binding energy of the two hydrogen atoms behaves as $R^{-7}$, provided that the
distance between atoms $R$ is sufficiently large. We employ the Feynman's
representation of the quantized radiation fields which enables us to
diagonalize Hamiltonians, rigorously. Our result supports the famous conjecture
by Casimir and Polder.
