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jk7297
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- Trying to solve problem 15.9 in R.D. Mattuck's book "A Guide to Feynman Diagrams in the Many-Body Problem."
This question is for those familiar with the BCS theory of superconductivity or familiar with R.D. Mattuck's book “A Guide to Feynman Diagrams in the Many-Body Problem.” I am working my way through the book, and I am stumped by some of the problems at the end of chapter 15 (superconductivity).
In Problem 15.9(b) the reader is asked to carry out the variation of the energy as expressed in part (a) of the problem. (I note that there is a typo in the book in part (a) in that the upper case V’s should be lower case.) $$ E = 2 \sum_k \epsilon_k v_k^2 - \sum_{k \neq k'} V_{kk'} u_k u_{k'} v_k v_{k'} .$$ When I attempt the variation I get the following expression:
$$ \delta E = \sum_k \left[ 4 \epsilon_k v_k -\Delta_k \left( u_k - \frac {v_k^2} {u_k} \right) \right] \delta v_k ,$$
where ## \Delta_k ## is defined by Mattuck's equation (15.31'). The quantity ##v_k## is the probability amplitude that the particular pair state is occupied. (I note that there seems to be an unstated assumption that ##v_k## and ##u_k## are real.) I then set the expression in square brackets to zero and solve a quadratic equation for the ratio $$\frac {v_k} {u_k} .$$ But this does not lead to the desired result, namely Mattuck's equations (15.30) and (15.31).
I note that if the factor of 4 in the bracketed expression is changed to a 2, I do get the right result. But the factor of 4 arises from the factor of 2 in the first term of the energy and from another factor of 2 from the variation of ##v_k^2##: $$\delta \left(v_k^2 \right) = 2 v_k \,\delta v_k .$$
What error am I making? Am I approaching the problem incorrectly?
In Problem 15.9(b) the reader is asked to carry out the variation of the energy as expressed in part (a) of the problem. (I note that there is a typo in the book in part (a) in that the upper case V’s should be lower case.) $$ E = 2 \sum_k \epsilon_k v_k^2 - \sum_{k \neq k'} V_{kk'} u_k u_{k'} v_k v_{k'} .$$ When I attempt the variation I get the following expression:
$$ \delta E = \sum_k \left[ 4 \epsilon_k v_k -\Delta_k \left( u_k - \frac {v_k^2} {u_k} \right) \right] \delta v_k ,$$
where ## \Delta_k ## is defined by Mattuck's equation (15.31'). The quantity ##v_k## is the probability amplitude that the particular pair state is occupied. (I note that there seems to be an unstated assumption that ##v_k## and ##u_k## are real.) I then set the expression in square brackets to zero and solve a quadratic equation for the ratio $$\frac {v_k} {u_k} .$$ But this does not lead to the desired result, namely Mattuck's equations (15.30) and (15.31).
I note that if the factor of 4 in the bracketed expression is changed to a 2, I do get the right result. But the factor of 4 arises from the factor of 2 in the first term of the energy and from another factor of 2 from the variation of ##v_k^2##: $$\delta \left(v_k^2 \right) = 2 v_k \,\delta v_k .$$
What error am I making? Am I approaching the problem incorrectly?