of the initial-value problem (\[2.6\]); see also ).[^7] From here follows that the Hermiticity of ${\mathcal{H}}$, ${\mathcal{H}}^\dag={\mathcal{H}}$, is equivalent to the unitarity of ${\mathcal{U}}$ (see (\[2.2\])).

Let us note that for mathematically rigorous understanding of the derivations in (\[2.5\]), (\[2.6\]), and (\[2.7\]), as well as of the chronological (path-ordered) exponent in (\[2.8\]), one has to apply the developed in [@Prugovecki-QMinHS] mathematical apparatus, but this is out of the subject of the present work.

If ${\mathcal{A}}(t)\colon {\mathcal{F}}\to{\mathcal{F}}$ is the (linear) operator corresponding to a dynamical variable ${\pmb{\mathbb{A}}}$ at the moment $t$, then the mean value (= the mathematical expectation) which it assumes at a state described by a state vector $\psi(t)$ with a finite norm is $$\label{2.9}
\langle{\mathcal{A}}(t)\rangle_\psi^t :=
\frac{\langle\psi(t) | {\mathcal{A}}(t)\psi(t)\rangle}
{\langle\psi(t) | \psi(t)\rangle}.$$

Often the operator ${\mathcal{A}}$ can be chosen to be independent of the time $t$. (This is possible, e.g., if ${\mathcal{A}}$ does not depend on $t$ explicitly [@Messiah-QM chapter VII, § 9] or if the spectrum of ${\mathcal{A}}$ does not change in time [@Fock-FQM chapter III, sect. 13].) If this is the case, it is said that the system’s evolution is depicted in the Schrödinger picture of motion [@Dirac-PQM § 28], [@Messiah-QM chapter VII, § 9].

Linear 