The Evaluation of Quantum Integrals by Epstein P. S.

By Epstein P. S.

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Foundations of Quantum Mechanics (Reading, Addison-Wesley). Landau, L. D. and Lifshitz, E. M. (1970). Statistical Mechanics (Reading, Addison-Wesley). Mackey, G. W. (1963). Mathematical Foundations of Quantum Mechanics (Reading, Benjamin). Piron, C. (1976). Foundations of Quantum Physics (Reading, Benjamin). Tolman, R. C. (1938). The Principles of Statistical Mechanics (London, Oxford). von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics (Princeton, Princeton University Press).

10). The projection operator method is quite general, and with it we may obtain an “intermediate” equation, the generalized master equation. From Eq. 13) i(1 − P)ρ˙ = (1 − P)L(Pρ + (1 − P)ρ). 14) Writing a formal solution to Eq. 14), we have t (1 − P)ρ = − i dt [exp(−i(1 − P)L(1 − P)t )(1 − P) × L Pρ(t − t )] 0 + exp(−i(1 − P)L(1 − P)t)(1 − P)ρ(0); t > 0. 15) Here a time initial value, ρ(0), has been assumed, with 0 ≤ t ≤ ∞. Thus, Eq. 15) is not equivalent to the von Neumann equation, where −∞ ≤ t ≤ ∞.

Finally, using the fact that pq has length (squared) λ, the length of pq1 , which we call z, is 1 1 z = λ − y = λ − (1 + λ) = (λ − 1) . 2 2 We now rewrite the relation previously obtained, 2w (q) = w (q1 ), as 2w (λ) = w 1 (λ − 1) 2 16 Foundations of quantum statistical mechanics for λ > 1. Since by our construction, r ⊥ q, 1 λ w (λ) + w = w ( p) = 1, we have that 1 − w (λ) = w 1 . λ If we now define x = (1 + λ)−1 = cos2 θ, the rest of the demonstration follows by simple algebra. e. λ > 1), and for a second relation, 1 − f (x) = f (1 − x) .

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