In statistical QM, the density operator for a pure states is defined as: p(t) =

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In statistical QM, the density operator for a pure states is defined as: p(t) = |t> is the state ket of the quantum system at time t. Moreover, the density operator for mixed states is defined as: , where pi is the probability of state |t>i. Please derive the Liouville-von Neumann equation for the mixed states, in dp / dt = [H(t), p(t)], based on the Schrodinger equation. The propagator U is defined as U(t,0) = exp(-iHt / h). Feynman's "quantum path integral" was developed to effectively calculate the propagator as U(t,0) = , where |phi j> and Ej are eigenkets and eigenvalues of the Hamiltonian H and beta = it/h. Please derive this equation.

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