  Some of the most enduring questions in physics--including the quantum
measurement problem and the quantization of gravity--involve the interaction of
a quantum system with a classical environment. Two linearly coupled harmonic
oscillators provide a simple, exactly soluble model for exploring such
interaction. Even the ground state of a pair of identical oscillators exhibits
effects on the quantum nature of one oscillator, e.g., a diminution of position
uncertainty, and an increase in momentum uncertainty and uncertainty product,
from their unperturbed values. Interaction between quantum and classical
oscillators is simulated by constructing a quantum state with one oscillator
initially in its ground state, the other in a coherent or Glauber state. The
subsequent wave function for this state is calculated exactly, both for
identical and distinct oscillators. The reduced probability distribution for
the quantum oscillator, and its position and momentum expectation values and
uncertainties, are obtained from this wave function. The oscillator acquires an
oscillation amplitude corresponding to a beating between the normal modes of
the system; the behavior of the position and momentum uncertainties can become
quite complicated. For oscillators with equal unperturbed frequencies, i.e., at
resonance, the uncertainties exhibit a time-dependent quantum squeezing which
can be extreme.
