HW8 soln unchecked - Solutions of Problem set 8 PHYS...

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Solutions of Problem set # 8 PHYS 361 (Solid State Physics), Autumn 2009 1. Localizing a Fermi-surface electron (a) ψ ( r ) = Z k F k 2 dk Z 2 π 0 Z π 0 sin 2 cos( kr cos θ ) e - ( k - k F ) / = - 4 π Z - kr kr du cos u Z k F dk k 2 kr e - ( k - k F ) / , · · · used θ u = kr cos θ = 8 π Z k F k 2 dk sin kr kr e - ( k - k F ) / (b) C ( ) Z d 3 r | ψ ( r ) | 2 = 64 π 2 Z d 3 r Z k 2 1 dk 1 sin k 1 r k 1 r e - ( k 1 - k F ) / Z k 2 2 dk 2 sin k 2 r k 2 r e - ( k 2 - k F ) / (c) Z d 3 r sin k 1 r sin k 2 r k 1 rk 2 r = 4 π k 1 k 2 1 2 Z -∞ dr e ik 1 r - e - ik 1 r 2 i · e ik 2 r - e - ik 2 r 2 i = 2 π 2 k 1 k 2 δ ( k 1 - k 2 ) C ( ) = 128 π 4 Z Z k 1 k 2 dk 1 dk 2 e - ( k 1 - k F ) / e - ( k 2 - k F ) / δ ( k 1 - k 2 ) = 128 π 4 Z k F k 2 dke - 2( k - k F ) / = 128 π 4 Z 0 2 dke - k ( k F + k/ 2) 2 , · · · usedk 2 k - k F = 64 π 4 ( k 2 F + k F ) + O ( 3 ) (d) | ψ (0) | = Z k F d 3 k 2 e - ( k - k F ) / = 8 π k 2 F + O ( 2 ) | ψ (0) | 2 C ( ) = k 2 F π 2
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(e) < U > = 1 C ( ) Z d 3 * ( ~ r ) ( ~ r ) , where U = - V, 0 < r < a = - 4 πV C Z a 0 drψ * ( ~ r ) ψ ( ~ r ) = - (8 π ) 2 4 π 2 C Z k F Z k F dk 1 dk 2 · k 1 k 2 e - ( k 1 - k F ) / e - ( k 2 - k F ) / Z a 0 dr sin( k 1 r ) sin( k 2 r ) = - V (8 π ) 3 4 C Z k F Z
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