  We propose a new fully-discretized finite difference scheme for a quantum
diffusion equation, in both one and two dimensions. This is the first
fully-discretized scheme with proven positivity-preserving and energy stable
properties using only standard finite difference discretization. The difficulty
in proving the positivity-preserving property lies in the lack of a maximum
principle for fourth order PDEs. To overcome this difficulty, we reformulate
the scheme as an optimization problem based on variational structure and use
the singular nature of the energy functional near the boundary values to
exclude the possibility of non-positive solutions. The scheme is also shown to
be mass conservative and consistent.
