  Quantum mechanics has been formulated in phase space, with the Wigner
function as the representative of the quantum density operator, and classical
mechanics has been formulated in Hilbert space, with the Groenewold operator as
the representative of the classical Liouville density function. Semiclassical
approximations to the quantum evolution of the Wigner function have been
defined, enabling the quantum evolution to be approached from a classical
starting point. Now analogous semiquantum approximations to the classical
evolution of the Groenewold operator are defined, enabling the classical
evolution to be approached from a quantum starting point. Simple nonlinear
systems with one degree of freedom are considered, whose Hamiltonians are
polynomials in the Hamiltonian of the simple harmonic oscillator. The behaviour
of expectation values of simple observables and of eigenvalues of the
Groenewold operator, are calculated numerically and compared for the various
semiclassical and semiquantum approximations.
