The authors consider the Schrödinger equation \[ i\hslash \psi_ t=- (\hslash^ 2/2m) \phi_{xx}+V(x)\psi,\quad \psi |_{t=0}=\psi_ 0(x),\quad x\in {\mathbb{R}} \] with continuous lower bounded and increased at infinity potential V(x). For this case it is possible to build an ”improved” WKB approximation \(\psi_ c\) (see D. Elworthy and A. Truman [J. Math. Phys. 22, 2144-2166 (1981; Zbl 0485.70024)] which is for sufficient small time localized in some compact region of the space. The function \(\psi_ c(x,t)\) as any WKB solution behaves singularly as \(\hslash \downarrow 0\) since it includes the fast oscillating phase factor. But it is natural to think that the quantum mechanical expectation \(<\psi F>_ Q\) of the quantum observable F in the state defined by a wave function \(\psi\) has much more regular behavior. Thus the main result of the paper is Theorem 3.1. Let \(\psi_ c(x,t)\) be a stationary ”improved” WKB solution corresponding to a given sufficiently large energy E, \(\Delta_ E=[a(E),b(E)]\), \(V(a(E))=V(b(E))=E\), V(x)\(\in E\), \(x\in \Delta_ E\). Then for any function \(f(x,p)\in L_ 1({\mathbb{R}}\times {\mathbb{R}})\) on the classical phase space whose support in configurational space is contained in \(\Delta_ E\) independently of momentum and for sufficiently small time t \[ \lim_{\hslash \downarrow 0}<\psi_ c F_ f>_ Q=<f>_{microcan}\equiv \int_{{\mathbb{R}}^ 2}f(x,p)\quad \delta (H(x,p)-E) dx dp/\int_{{\mathbb{R}}^ 2}\theta (x)\delta (x,p)-E)dx dp \] where \(F_ f\) is the quantum observable corresponding to f, \(\theta (x)\in C_ 0^{\infty}({\mathbb{R}})\), supp f(\(\cdot,p)\subset \sup p \theta (x)\), \(\theta |_{\sup p f}=1\). In Appendices A and B it is shown how to generalize this one-dimensional result to the case of spherically symmetric potential and completely integrable systems, in Appendix C a corresponding result on the eigenstates of harmonic oscillator is derived.