The procedure of the quantum linearization of the Hamiltonian ordinary differential equations with one degree of freedom is introduced. It is offered to be used for the classification of integrable equations of the Painleve type. By this procedure an
d all natural numbers $n$ we construct the solutions $Psi(hbar,t,x,n)$ to the non-stationary Shr{o}dinger equation with the Hamiltonian $H = (p^2+q^2)/2$ which tend to zero as $xtopminfty$. On the curves $x=q_n (hbar, t) $ defined by the old Bohr-Sommerfeld quantization rule the solutions satisfy the relation $ihbar Psi _xequiv p_n (hbar, t) Psi $, where $p_n (hbar, t) = (q_n (hbar, t)) _t $ is the classical momentum corresponding to the harmonic $q_n (hbar, t) $.
We construct a solution of an analog of the Schr{o}dinger equation for the Hamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the second equation $P_1^2$ in the Painleve I hierarchy. This solution is produced by an explicit change of var
iables from a solution of the linear equations whose compatibility condition is the ordinary differential equation $P_1^2$ with respect to $z$. This solution also satisfies an analog of the Schr{o}dinger equation corresponding to the Hamiltonian $ H_{II} (z, t, q_1, q_2, p_1, p_2) $ of Hamiltonian system with respect to $t$ which is compatible with $P_1^2$. A similar situation occurs for the $P_2^2$ equation in the Painleve II hierarchy.
