  The dynamical equations describing the evolution of a self-gravitating fluid
can be rewritten in the form of a Schrodinger equation coupled to a Poisson
equation determining the gravitational potential. This wave-mechanical
representation allows an approach to cosmological gravitational instability
that has numerous advantages over standard fluid-based methods. We explore the
usefulness of the Schrodinger approach by applying it to a number of simple
examples of self-gravitating systems in the weakly non-linear regime. We show
that consistent description of a cold self-gravitating fluid requires an extra
"quantum pressure" term to be added to the usual Schrodinger equation and we
give examples of the effect of this term on the development of gravitational
instability. We also show how the simple wave equation can be modified by the
addition of a non-linear term to incorporate the effects of gas pressure
described by a polytropic equation-of-state.
