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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
u 2 − vp = 1
p (3.14) The mean-ﬁeld Hamiltonian becomes
HMF = − 2
(αp αp + α−p α−p ) [( p 2
+ 2n0 U0 )(u2 + vp ) + n0 U0 (−2up vp )]
(αp α−p + αp α−p ) [( + p 2
+ 2n0 U0 )(−2up vp ) + n0 U0 (u2 + vp )]
p p=0 +
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
Ep = ( p =
= + n0 U0
p + n0 U0 )2 − (n0 U0 )2 (3.18) The diagonal form of the mean-ﬁeld Hamiltonian
HMF = − 2
N0 U 0
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.
- Fall '10