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The quantum Liouville equation describes the time evolution of the density operator (\hat{\rho}) in quantum mechanics. It is the quantum analog of the classical Liouville equation and governs the dynamics of statistical ensembles of quantum systems. The density operator (\hat{\rho}) is defined for a system in a mixed or pure state as [ \hat{\rho} = \sum_i p_i , |\psi_i\rangle \langle __________ ________ __________ _____ _______ _____ ___ ________ _____.
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