pset4

# pset4 - 1 Physics 219 Problem Set 4 Due Date March 4 2008 1...

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1 Physics 219 - Problem Set 4 Due Date: March 4, 2008 1. Bose Condensation of Relativistic Particles. Consider a 3 D system of massless relativistic particles with energy ε ( k ) = ck What is the Bose-Einstein condensation temperature for such particles in terms of the speed of light c , Planck’s constant h , Boltzmann’s constant k B , and the particle density N / V . 2. Density Matrices. (a) Consider a system + reservoir in a pure state | ψ i , as in class: | ψ i = r , s c r , s | r i| s i where | r i describes the state of the reservoir and | s i describes the state of the system. The system by itself will be described by a density matrix ρ . Show that ρ satisﬁes tr ( ρ ) = 1. (b) Following the logic which we used in the case of the canonical ensemble, show that the von Neumann entropy of a system described by density matrix ρ is S = - k B tr ( ρ ln ρ ) Show that this result is basis-independent. 3. One-particle Density Matrix.

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pset4 - 1 Physics 219 Problem Set 4 Due Date March 4 2008 1...

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