  High temperature and white noise approximations are frequently invoked when
deriving the quantum Brownian equation for an oscillator. Even if this white
noise approximation is avoided, it is shown that if the zero point energies of
the environment are neglected, as they often are, the resultant equation will
violate not only the basic tenet of quantum mechanics that requires the density
operator to be positive, but also the uncertainty principle. When the
zero-point energies are included, asymptotic results describing the evolution
of the oscillator are obtained that preserve positivity and, therefore, the
uncertainty principle.
