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ph2b_quiz4_soln

# ph2b_quiz4_soln - Ph2b Quiz 4 Solutions Stephen Privitera...

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Ph2b Quiz 4 Solutions Stephen Privitera [email protected] March 9, 2009 1 2D Fermi Gas 1. Determine the Fermi energy F of the 2D degenerate electron gas (1pt). Determine also the total energy of the system (1pt). The Fermi energy is the energy of the highest filled orbital at τ = 0. For a 2D electron gas, the energy and the quantum number n are related according to = 2 π 2 2 mA n 2 (1) where n 2 = n 2 x + n 2 y and A is the area in which the gas is enclosed. Using this connection, we can find the highest filled energy level by finding the highest filled quantum number. To find the highest filled quantum number n F , we fill each successive quantum state with electrons, starting with the ground state and ending with n F until the total number of particles is equal to N . In 2D, this gives N = 2 1 4 π n 2 F (2) The factor of 2 takes into account electron spin, while the factor of 1 / 4 takes into ac- count that the quantum numbers must be positive. Solving (2) for n F and inserting the result into the dispersion relation (1), we find for the Fermi energy F = N π 2 mA (3) 1

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To find the total energy of the system in the ground state, it helps (but is not necessary) to have the density of states D ( ), which is found in part (b) of this problem. The total energy will be given by U = F 0 D ( ) d (4) At τ = 0 all states with < F are fully occupied, while states with > F are fully unoccupied. In part (b), we find that D ( ) = mA/ 2 π is a constant (independent of ) and therefore the energy must be given by: U = 1 2 D ( ) 2 F = 1 2 mA 2 π N π 2 mA 2 (5) = 1 2 N F (6) 2. Determine the density of states D ( ) (1pt). Show that the integration of this density up to the Fermi energy is equal to the total number N of electrons in the system. Give a short physical explanation of this expected result (1pt).
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