  We present a phase-space method for the Bose-Hubbard model based on the
Q-function representation. In particular, we consider two model Hamiltonians in
the mean-field approximation; the first is the standard "one site" model where
quantum tunneling is approximated entirely using mean-field terms; the second
"two site" model explicitly includes tunneling between two adjacent sites while
treating tunneling with other neighbouring sites using the mean-field
approximation. The ground state is determined by minimizing the classical
energy functional subject to quantum mechanical constraints, which take the
form of uncertainty relations. For each model Hamiltonian we compare the ground
state results from the Q-function method with the exact numerical solution. The
results from the Q-function method, which are easy to compute, give a good
qualitative description of the main features of the Bose-Hubbard model
including the superfluid to Mott insulator. We find the quantum mechanical
constraints dominate the problem and show there are some limitations of the
method particularly in the weak lattice regime.
