Homework06_solution - Homework # 6 - Solution Problem # 1:...

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Homework # 6 - Solution Problem # 1: Show that the kinetic energy of a three-dimensional electron gas of N electrons at zero temperature is U =3/5 NE F . For free electrons the density of states is given by 3/2 1/2 22 2 () 2 Vm DE E π ⎛⎞ = ⎜⎟ ⎝⎠ = . Using the formula for the Fermi energy 2/3 3 2 F N E mV = = , we can express it as follows 3 2 F N E E = . The kinetic energy of N free electrons at zero temperature is given by 00 33 25 FF EE F F N UE D E d E E d EN E E == = ∫∫ . Problem # 2: Show that the density of states of a free-electron gas in two dimensions is independent of energy. In two dimensions there is one allowed wavevector per area 2 2/ L in k space. Since the area of 2 2 F k is occupied by N electrons, the total number of states is given by 2 2 2 (2 / ) 2 kL k N L ππ . where a factor of 2 comes from the spin degeneracy. The Fermi energy is therefore given by 2 2 2 F F k N E mm L = = . Generalizing this result we find the total number of orbitals of energy < E : 2 2 mL E N = = .
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This note was uploaded on 09/21/2011 for the course PHYSICS 101 taught by Professor Wormer during the Spring '08 term at Aarhus Universitet.

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Homework06_solution - Homework # 6 - Solution Problem # 1:...

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