Exact classical and quantum mechanics of a particle coupled to a membrane

The exact dynamical solution is given for a harmonically bound particle of finite size (the oscillator) coupled rigidly to a finitely extended two-dimensional mechanical continuum transmission line (the membrane). A quantum mechanical ultraviolet divergence in the particle's momentum fluctuations is discussed. In the limit of a very large membrane and a very small particle the oscillator's classical friction coefficient is obtained explicitly.