3. von Neumann entropy. Let H be the Hamiltonian, ??(t) be the density operator, and T be the temperature. The von Neumann entropy is defined by S[??] = ?Tr?? ln ??. (a) Calculate dS/dt (using the von Neumann equation for ??). (b) Using the Lagrange multiplier method, find the density operator ??eq that maximizes S[??] under the constraint Tr(??H) = E with E fixed. (c) Show that d??eq / dt = 0.
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Let S[Ļ] = āTr(Ļ ln Ļ). The von Neumann equation is iħ dĻ/dt = [H, Ļ], so dĻ/dt = (1/(iħ))[H, Ļ]. Show moreā¦
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