A Schrodingcr formulation of a free scalar field propagating in Robertson-Walker fiat spacctimcs is presented. Gaussian wave functional which describe homogeneous, isotropic quantum states are considered. Several identities relating the real and imaginary parts of the Gaussian width are derived. Covariant conservation of the stress tensor is demonstrated using these relations. The formalism is applied in detail to reproduce known results in a de Sitter background.