  For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space
$\mathcal{F}$ the operator $\exp(-it\mathcal{H})$ is the evolution operator for
the Schr\"odinger equation $i\psi'_t=\mathcal{H}\psi$, i.e. if
$\psi(0,x)=\psi_0(x)$ then $\psi(t,x)=(\exp(-it\mathcal{H})\psi_0)(x)$ for
$x\in Q.$ The space $\mathcal{F}$ here is the space of wave functions $\psi$
defined on an abstract space $Q$, the configuration space of a quantum system,
and $\mathcal{H}$ is the Hamiltonian of the system. In this paper the operator
$\exp(-it\mathcal{H})$ for all real values of $t$ is expressed in terms of the
family of self-adjoint bounded operators $S(t), t\geq 0$, which is
Chernoff-tangent to the operator $-\mathcal{H}$. One can take
$S(t)=\exp(-t\mathcal{H})$, or use other, simple families $S$ that are listed
in the paper. The main theorem is proven on the level of semigroups of bounded
operators in $\mathcal{F}$ so it can be used in a wider context due to its
generality. Two examples of application are provided.
