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2002AugQualSTAT - Ben Sauerwine Practice for Qualifying...

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Unformatted text preview: Ben Sauerwine Practice for Qualifying Exams Problem Source: CMU August 2002 Qualifying Exam Consider N identical but distinguishable very weakly interacting five-state quantum subsystems with energy levels shown as in figure 1. Figure 1: Energy level diagram of a five-state quantum system. (a) What is the ratio of the fraction of systems in state 4 ε to those in state 2 ε for 2 ε = T k B ? A numerical answer is required. Throughout these solutions, let T k B 1 = β Using the canonical ensemble, I have (for one particle) ( ) 14 8 6 2 7 4 3 1 2 − − − − − − − − + + + + = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + + + = e e e e Z e e e e e Z ε β βε βε βε βε However, the proportion in state 4 ε is ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − ε β ε 2 , 8 4 Z e P and the proportion in 2 ε is ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = − ε β ε 2 , 2 2 Z e P so that ( ) ( ) 6 2 8 2 4 , , − − − = = e e e P P β ε β ε (b) How high, in units of B k ε , must the temperature T be so that the minimum...
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2002AugQualSTAT - Ben Sauerwine Practice for Qualifying...

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