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For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space$\mathcal{F}$ the operator $\exp(-it\mathcal{H})$ is the evolution operator forthe 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 thefamily of self-adjoint bounded operators $S(t), t\geq 0$, which isChernoff-tangent to the operator $-\mathcal{H}$. One can take$S(t)=\exp(-t\mathcal{H})$, or use other, simple families $S$ that are listedin the paper. The main theorem is proven on the level of semigroups of boundedoperators in $\mathcal{F}$ so it can be used in a wider context due to itsgenerality. Two examples of application are provided.

Quasi-Feynman formulas -- a method of obtaining the evolution operator for the Schroedinger equation