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

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