  In the present work we consider a time-dependent Schr\"odinger equation for
systems invariant under the reparametrization of time. We develop the two-stage
procedure of construction such systems from a given initial ones, which is not
invariant under the time reparametrization. One of the first-class constraints
of the systems in such description becomes the time-dependent Schr\"odinger
equation. The procedure is applicable in the supersymmetric theories as well.
The $n=2$ supersymmetric quantum mechanics is coupled to world-line
supergravity, and the local supersymmetric action is constructed leading to the
square root representation of the time-dependent Schr\"odinger equation.
