Unformatted text preview: wavefunction, and project dynamics under Hamiltonian (3.31)
into this state . This procedure gives equation (3.34)(we will discuss mathematical formalism for ﬁnding such projected equations of motion in section ??. For
now we will assume that this statement sounds reasonable).
In the simplest case of Vext = 0 we observe that equation (3.34) has a static
√
solution Ψcl = n0 eiφ , provided that equation (3.5) is satisﬁed. Phase φ can
be arbitrary.
Equation (3.33) can be used to obtain an alternative derivation of the spectrum of Bogoliubov quasiparticles. For a system without an external poten√
tial we take Ψ0 = n0 and then consider small ﬂuctuations around this state
Ψ(r, t) = Ψ0 + δ Ψ(r, t). Linearized equations of motion are
˙
i δΨ
˙
−i δ Ψ∗ 1
2m
1
=−
2m
=− 2 δ Ψ + U0 n0 (δ Ψ + δ Ψ∗ ) 2 δ Ψ∗ + U0 n0 (δ Ψ + δ Ψ∗ ) (3.36) 10CHAPTER 3. BOSEEINSTEIN CONDENSATION OF WEAKLY INTERACTING ATOMIC GASE
Keeping in mind representation of the instantaneous wavefunction Ψ(r, t) =
n0 + δn(r, t) eiδφ(r,t) , it...
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This document was uploaded on 02/27/2014 for the course PHYS 284 at Harvard.
 Fall '10
 EugeneDemler
 Physics

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