  We have studied the effects of the Standard-Model Extension (SME) on hydrogen
as a realization of new physics effects that incorporate Lorentz and CPT
violation. Specifically, we calculated the SME-induced energy level shifts at
order $\alpha^2$ times the SME parameters. We obtained contributions at this
order both from the non-relativistic effective Hamiltonian for motion of a
spin-1/2 particle in the presence of SME interactions and also from SME
corrections to the propagator for exchange photons. We applied our result to
the $2S-1S$ transition in hydrogen, which has been measured with extremely high
precision. The results obtained in this work give the leading SME corrections
for this transition.
