  We investigate the ground state energy of an electron coupled to a photon
field. First, we regard the self-energy of a free electron, which we describe
by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of
the coupling constant $\alpha$, the leading order term is represented by
$2\pi^{-1} \alpha (\Lambda - \ln[1 + \Lambda])$.
  Next we put the electron in the field of an arbitrary external potential $V$,
such that the corresponding Schr\"odinger operator $p^2 + V$ has at least one
eigenvalue, and show that by coupling to the radiation field the binding energy
increases, at least for small enough values of the coupling constant $\alpha$.
Moreover, we provide concrete numbers for $\alpha$, the ultraviolet cut-off
$\Lambda$, and the radiative correction for which our procedure works.
