lecture3

# For a system without an external poten tial we take 0

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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. BOSE-EINSTEIN 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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