240b_ps4 - states (use the BCS model) is W N ! W S = N(0)...

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Physics 240B M. L. Cohen Spring 2008 Problem Set #4 Due: 04/15/08 (20) 1. For the Cooper pair problem: (a) Calculate the mean square radius of a Cooper pair. (b) How does the energy of the bound pair vary with center of mass momentum? What does this have to do with the coherence length? Is the situation different for isolated pairs than for overlapping pairs? Why? (c) Does the single-pair model give a continuous spectrum above the ground state or one with an energy gap? (d) Which spin state (singlet or triplet) of a bound pair has lower energy for an isotropic potential? Why? Give examples of potentials in which the singlet state has lower energy and explain why. Do the same for the triplet state. (20) 2. (a) Find <H BCS > in terms of the u's and v's for the wavefunction ! = k " u k + v k b k + # $ % 0 , where ! 0 is the vacuum state. (b) Use part (a) to derive the BCS gap equation. Show that the energy difference between the superconducting and normal
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Unformatted text preview: states (use the BCS model) is W N ! W S = N(0) &quot; 2 2 (20) 3. Find the transition temperature T c for a three square-well model with wells K p , K e , and K c having cutoff energies ! 1 , ! 2 , and ! 3 , respectively. Assume K e and K p attractive and K c repulsive with K c &lt; K p &lt; K e . Show that your result reduces to the usual two square-well model solution when K e = 0. (20) 4. (a) Calculate the Bardeen-Pines interaction (following the discussion given in class) including finite temperatures; that is, put in the phonon occupation numbers. (b) Calculate the electron-phonon self-enrgy (with phonon occupation numbers). (c) Estimate N(0) V (Coulomb) and N(0) V (Bardeen-Pines) for a typical metal. (20) 5. For the pair creation and destruction operators, calculate b k ,b &quot; k + [ ] # , b k ,b &quot; k [ ] + , b k ,b &quot; k + [ ] + , b k 2 , and b k + 2 ....
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