PS8Sol_f10 - Physics 112 Fall 2010 Professor William...

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Physics 112 Fall 2010 Professor William Holzapfel Homework 8 Solutions Problem 1, Kittel 7.1: Density of Orbitals in one and two dimensions In one dimension the orbitals are of the form ψ n ( x ) = A sin( nπx/L ) where n is a positive integer. The energy is ± n = ~ 2 2 m ± πn L ² 2 (1) and the density in n space is simply 2 dn , since there is one state per unit interval (the 2 comes from the spin degeneracy). We can invert (1) to get n ( ± ) = L π ~ 2 = dn = L 2 π ~ r 2 m ± d±. Since the density of states D ( ± ) is defined by D ( ± ) 2 dn , we have D ( ± ) = L π ~ r 2 m ± . (b) Now the orbitals are of the form ψ n ( x ) = A sin( n x πx/L ) sin( n y πy/L ), where n x ,n y > 0 and the energy has the same form (1), but with n = q n 2 x + n 2 y . The density of states in n -space is 1 4 · 2 · 2 πndn = πndn (2) where the 1/4 is because we’re only interested in the first quadrant, the 1/2 is for spin and the 2 πndn is just the area element in 2D n -space in polar coordinates. As above, we have D ( ± ) πndn = π ³ L π ~ 2 ´ · 1 2 µ L π ~ r 2 m ± ! = mL 2 π ~ 2 = D ( ± ) = mL 2 π ~ 2 . Problem 2, Kittel 7.2: Energy of a Relativistic Fermi Gas (a) For a relativistic gas in 3D the orbitals are still of the form ψ ~n ( x,y,z ) = A sin( n x πx/L ) sin( n y πy/L ) sin( n z πz/L ) but now the energy of the orbitals takes the form ± n = | ~ p | c = π ~ c L n (3) 1
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where n = q n 2 x + n 2 y + n 2 z . We can still, however, use Kittel’s result that in n -space the occupied modes fill up a sphere of radius n F = ± 3 N π ² 1 / 3 (4) since this result is independent of the dispersion relation (3). The Fermi energy is then just ± F ± n F = π ~ c L n F = ~ πc ± 3 n π ² 1 / 3 . (b) Now using (3) and (4) the ground state energy is easily found: U 0 = 2 × 1 8 Z n F 0 4 πn 2 ± n dn = π 2 ~ c L Z n F 0 n 3 dn = π 2 ~ c 4 L ± 3 N π ² 4 / 3 = 3 4 F . Problem 3, Kittel 7.5: Liquid 3 He as a Fermi Gas (a) The Fermi velocity v F is defined in terms of the Fermi energy: ± F 1 2 mv 2 F . To find the Fermi energy, we need to know the number density; we can find this from the given mass density of 3 He
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PS8Sol_f10 - Physics 112 Fall 2010 Professor William...

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