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Homework06_solution

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

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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 2 2 2 ( ) 2 V m D E E π = = . Using the formula for the Fermi energy 2/3 2 2 3 2 F N E m V π = = , we can express it as follows 1/2 3/ 2 3 ( ) 2 F N D E E E = . The kinetic energy of N free electrons at zero temperature is given by 3/ 2 3/2 0 0 3 3 ( ) 2 5 F F E E F F N U ED E dE E dE NE 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 π in k space. Since the area of 2 2 F k π is occupied by N electrons, the total number of states is given by 2 2 2 2 2 (2 / ) 2 F F k L k N L π π π = = . where a factor of 2 comes from the spin degeneracy. The Fermi energy is therefore given by 2 2 2 2 2 F F k N E m mL π = = = = .

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Homework06_solution - Homework 6 Solution Problem 1 Show...

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