  At one loop, quantum kinks are described by a sum of quantum harmonic
oscillator Hamiltonians, and so their spectra are known exactly. We find the
first correction beyond one loop to the quantum states corresponding to kinks
with an excited bound or unbound normal mode, and also the corresponding
two-loop correction to the energy cost of exciting the normal mode. In the case
of unbound normal modes, this correction is equal to sum of the corresponding
nonrelativistic kinetic energy plus the usual one-loop correction to the mass
of the corresponding plane wave in the absence of a kink. We also sketch a
diagrammatic method for such calculations.
