HW8 soln unchecked - Solutions of Problem set # 8 PHYS 361...

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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
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This note was uploaded on 03/02/2010 for the course PHYSICS 336 taught by Professor Pucker during the Spring '10 term at King's College London.

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HW8 soln unchecked - Solutions of Problem set # 8 PHYS 361...

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