13 bosonic commutation relations are preserved when 2

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Unformatted text preview: counted only once, n0 = N0 /V and µ = n0 U0 . We can diagonalize (3.12) using Bogoliubov transformation bp = † up αp − vp α−p b−p = † up α−p − vp αp (3.13) Bosonic commutation relations are preserved when 2 u 2 − vp = 1 p (3.14) The mean-field Hamiltonian becomes HMF = − 2 N0 U0 2V † † (αp αp + α−p α−p ) [( p 2 + 2n0 U0 )(u2 + vp ) + n0 U0 (−2up vp )] p †† (αp α−p + αp α−p ) [( + p 2 + 2n0 U0 )(−2up vp ) + n0 U0 (u2 + vp )] p p=0 + p=0 (3.15) Cancellation of the non-diagonal terms requires ( p 2 + 2n0 U0 )(−2up vp ) + n0 U0 (u2 + vp ) = 0 p (3.16) To satisfy equation (3.14) one can take up = cosh θp vp = sinh θp (3.17) 6CHAPTER 3. BOSE-EINSTEIN CONDENSATION OF WEAKLY INTERACTING ATOMIC GASES Solution of these equations is cosh 2θp sinh 2θp Ep = ( p = = + n0 U0 Ep n0 U0 Ep p + n0 U0 )2 − (n0 U0 )2 (3.18) The diagonal form of the mean-field Hamiltonian HMF = − 2 N0 U 0 2V † † Ep (αp αp + α−p α−p ) + (3.19) p=0 Dispersion of collect...
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This document was uploaded on 02/27/2014 for the course PHYS 284 at Harvard.

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