  On the basis of extensive numerical studies it is argued that there are
strong analogies between the probabilistic behavior of quantum systems defined
by Hermitian Hamiltonians and the deterministic behavior of classical
mechanical systems extended into the complex domain. Three models are examined:
the quartic double-well potential $V(x)=x^4-5x^2$, the cubic potential
$V(x)=frac{1}{2}x^2-gx^3$, and the periodic potential $V(x)=-\cos x$. For the
quartic potential a wave packet that is initially localized in one side of the
double-well can tunnel to the other side. Complex solutions to the classical
equations of motion exhibit a remarkably analogous behavior. Furthermore,
classical solutions come in two varieties, which resemble the even-parity and
odd-parity quantum-mechanical bound states. For the cubic potential, a quantum
wave packet that is initially in the quadratic portion of the potential near
the origin will tunnel through the barrier and give rise to a probability
current that flows out to infinity. The complex solutions to the corresponding
classical equations of motion exhibit strongly analogous behavior. For the
periodic potential a quantum particle whose energy lies between -1 and 1 can
tunnel repeatedly between adjacent classically allowed regions and thus execute
a localized random walk as it hops from region to region. Furthermore, if the
energy of the quantum particle lies in a conduction band, then the particle
delocalizes and drifts freely through the periodic potential. A classical
particle having complex energy executes a qualitatively analogous local random
walk, and there exists a narrow energy band for which the classical particle
becomes delocalized and moves freely through the potential.
