Quantum Painlev\'e II Lax Pair and Quantum (Matrix) Analogues of Classical Painlev\'e II equation

In this article, we present a new quantum Painlev\'e II Lax pair which explicitly involves the Planck constant $\hbar$ and an arbitrary field variable $v$ so these two objects make this new pair different from Flaschka-Newell Painlev\'e II Lax pair and that pair appears as particular case of our's pair which consolidates the Painlev\'e II equation from quantum mechanical point of view. It is shown that the compatibility of quantum Painlev\'e II Lax pair simultaneously yields a quantum Painlev\'e II equation and a quantum commutation relation between field variable $v$ and independent variable $z$. We manifest that with the different choices of arbitrary field variable system reduces to its classical version, matrix Painlev\'e II equation and derivative matrix Painlev\'e II equation. Further, we construct the gauge equivalence of quantum Painlev\'e II Lax pair whose compatibility condition gives rise to quantum p34 equation that involves $\hbar$ with power $+1$ which makes our system more quantized as compared to the existed one that carries $\hbar$ with power $+2$.