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We construct a solution of an analog of the Schr\"{o}dinger equation for theHamiltonian $ H_I (z, t, q_1, q_2, p_1, p_2) $ corresponding to the secondequation $P_1^2$ in the Painleve I hierarchy. This solution is produced by anexplicit change of variables from a solution of the linear equations whosecompatibility condition is the ordinary differential equation $P_1^2$ withrespect to $z$. This solution also satisfies an analog of the Schr\"{o}dingerequation 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 $P_1^2$. Asimilar situation occurs for the $P_2^2$ equation in the Painleve II hierarchy.

"Quantization" of higher hamiltonian analogues of the Painleve I and Painleve II equations with two degrees of freedom