ps3 - C es = T dS es dT , (5) where S es is the entropy of...

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Physics 880.06: Problem Set 3 Note: please turn these problems 11:59 P. M. on Tuesday, April 26, 2011. Each problem is worth 10 points. 1. The “Bogoliubon” operators were introduced in class by the transfor- mation c k = u k γ k , 0 + v k γ k , 1 (1) c k = v k γ k , 0 + u k γ k , 1 , (2) where the coeFcients u k and v k satisfy the normalization condition | u k | 2 + | v k | 2 = 1 . (3) Show that the operators γ , k , 0 , γ k , 1 and their Hermitean conjugates sat- isfy the standard ±ermi anticommutation relations [ γ k ,i , γ k ,j ] + = δ ij , (4) where i and j can take on the values 1 and 2, and [ .., . .. ] + denotes an anticommutator. 2. The operator a k = c k c k creates a pair of electrons with opposite wave vector and opposite spin. Show that if k n = k , the commutator (not the anticommutator) [ a k , a k ] = 0, as expected for Bose operators, but that if k = k , this commuta- tor does not equal unity, as one would expect for Bose operators, and calculate [ a k , a k ]. 3. The electronic speci²c heat of the Bogoliubons is, as stated in class,
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Unformatted text preview: C es = T dS es dT , (5) where S es is the entropy of the Bogoliubons and is given by the standard result for a collection of ±ermions: S = − 2 k B s k [(1 − f k ) ln (1 − f k ) + f k lnf k ] . (6) 1 Here f k = 1 e βE k + 1 (7) and E k = r Δ 2 + ξ 2 k , (8) Δ being the energy gap, which we assume is real, and ξ k = ǫ k − E F . Show that the specifc heat can be written as C es = 2 βk B s k p − ∂f k ∂E k Pp E 2 k + 1 2 β d Δ 2 dβ P , (9) where β = 1 / ( k B T ). Show explicitly that, iF Δ is temperature-independent, the specifc heat varies at low temperatures as exp( − Δ /k B T ) multiplied by a Function which varies more slowly with temperature. (This is also true even iF Δ is dependent on temperature, provided that it remains fnite at T = 0.) 2...
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This note was uploaded on 07/28/2011 for the course PHYSICS 880.06 taught by Professor Stroud during the Fall '10 term at Ohio State.

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ps3 - C es = T dS es dT , (5) where S es is the entropy of...

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