  Quantum Mechanics places limits on achievable transverse beam spot sizes of
particle accelerators. We estimate this limit for a linear collider to be
\Delta x > \hbar c f/E\delta_0 where f is the final focal length, E the beam
energy, and \delta_0 the intrinsic transverse Gaussian width of the electron
wave-function. \delta_0 is determined in the phase space damping rings, and we
find \delta_0 \approx \sqrt{\hbar c /eB} where B is the typical wiggler
magnetic field strength in this system. For the NLC \delta_0 = 25 nm, and
\Delta x = 0.06 nm, about two orders of magnitude smaller than the design goal.
We can recover a crude estimate of the classical result when we include
radiative relaxation effects. We also consider a synchrotron and obtain \Delta
x > \sqrt{\hbar c f/E} = O(1.0) nm. We discuss formulation of quantum beam
optics relevant to these issues.
