  The dynamical equations describing the evolution of a self-gravitating fluid
of cold dark matter (CDM) can be written in the form of a Schrodinger equation
coupled to a Poisson equation describing Newtonian gravity. It has recently
been shown that, in the quasi-linear regime, the Schrodinger equation can be
reduced to the exactly solvable free-particle Schrodinger equation. The
free-particle Schrodinger equation forms the basis of a new approximation
scheme -the free-particle approximation - that is capable of evolving
cosmological density perturbations into the quasi-linear regime. The
free-particle approximation is essentially an alternative to the adhesion model
in which the artificial viscosity term in Burgers' equation is replaced by a
non-linear term known as the quantum pressure. Simple one-dimensional tests of
the free-particle method have yielded encouraging results. In this paper we
comprehensively test the free-particle approximation in a more cosmologically
relevant scenario by appealing to an N-body simulation. We compare our results
with those obtained from two established methods: the linearized fluid approach
and the Zeldovich approximation. We find that the free-particle approximation
comprehensively out-performs both of these approximation schemes in all tests
carried out and thus provides another useful analytical tool for studying
structure formation on cosmological scales.
