  We show that one dimensional non-stationary Schr\"odi-nger equation with a
specific choice of potential reduces to the quantum Painlev\'e II equation and
the solution of its Riccati form appears as a dominant term of that potential.
Further, we show that Painlev\'e II Riccati solution is an equivalent
representation of centrifugal expression of radial Schr\"odinger potential.
This expression is used to derive the approximated form of the Yukawa potential
of radial Schr\"odinger equation which can be solved by applying the
Nikiforov-Uvarov method. Finally, we express the approximated form of Yukawa
potential explicitly in terms of qunatume Painlev\'e II solution.
