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The probability operator is derived from first principles for an equilibriumquantum system. It is also shown that the superposition states collapse into amixture of states giving the conventional von Neumann trace form for thequantum average. The mechanism for the collapse is found to be quite general:it results from the conservation law for a conserved, exchangeable variable(such as energy) and the entanglement of the total system wave function thatnecessarily follows. The relevance of the present results to the einselectionmechanism for decoherence, to the quantum measurement problem, and to theclassical nature of the macroscopic world are discussed.

Quantum Statistical Mechanics. I. Decoherence, Wave Function Collapse, and the von Neumann Density Matrix