Solutions10

Solutions10 - PHYS 333 Winter 2008 Assignment 10 1....

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PHYS 333 Winter 2008 Assignment 10 1. Schroeder 7.16 A system of N (spinless) identical fermions has equally spaced energy levels which are all non-degenerate. (That is, each level holds only one fermion.) At low temperatures, it is useful to represent each state of the system by a column of dots, where a filled dot represents an occupied level, and an open dot represents an un-occupied level. At low temperatures, all states below a certain level are occupied, and are not shown on the dot-diagram. Suppose that the spacing between the levels is η , and that q is the number of energy units over and above the ground-state energy U 0 . That is, the (internal) energy is U = U 0 + . The figure shows all possible states for 0 q 3. r r r r r r r r r r r r r r r r r r r r r r r r r r r r b b b b b b b b b b b b b b b b b b b b b b b b b b b b p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p q = 0 1 2 3 (a) Draw dot-diagrams for q = 4 , 5 and 6. r r r r r b b b b r b b r r r r b r b b r b b b r r r r b b r r b b b b r r r b r r b r b b b b r r b r r r r b b b b b r r r r r b b b b b r b r r r r b r b b b r b b r r r r b b r b r b b b r r r b r r b b r b b b r r r b r b r r b b b b r r b r r r b r b b b b r b r r r r r b b b b b r r r r r b b b b b b r r r r r b r b b b b r b r r r r b b r b b r b b r r r b r r b b b r b b r r r r b b b r r b b b r r r b r b r b r b b b r r b r r r b b r b b b r r r b b r r r b b b b r r b r r b r r b b b b r b r r r r b r b b b b b r r r r r r b b b b b q = 4 5 6 (b) Compute, therefore, the probability of each level being occupied when q = 6. Plot a graph of this probability as a function of the energy of each level, and from the graph, 1
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estimate μ and T by “fitting” to the Fermi-Dirac distribution. The probabilities for each level shown on the diagram are thus 10 11 , 10 11 , 9 11 , 8 11 , 7 11 , 6 11 , 5 11 , 4 11 , 3 11 , 2 11 , 1 11 , 1 11 , The levels not shown either have probabilities equal to zero or unity: three of each of these are shown in the following diagram. The “fit” has μ = 9 . 5 η and T = 2 . 1 η/k B . 5
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Solutions10 - PHYS 333 Winter 2008 Assignment 10 1....

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