PhD thesis in English

4. Mean-field description of an interacting BECaction principle [19] based on the Lagrangian∫L GP = d⃗r L GP (⃗r) , (4.16)where the Lagrangian density is given byL GP (⃗r) = i 2) (ψ ∂ψ∗∂t − ψ∗∂ψ + 2∂t 2M |∇ψ|2 + V (⃗r)|ψ| 2 + g 2 |ψ|4 . (4.17)By performing the extremization of the action with respect to ψ(⃗r, t),δ ∫ t 2t 1dtL GPδψ ∗ (⃗r, t)= 0, (4.18)we again arrive at the GP Eq. (4.15). This is an important observation, since itrepresents the basis of the time-dependent variational analysis. We exploit thisapproach later in Chapter 5.4.2 Finite-temperature properties of a BECAfter briefly exploring the properties of a weakly interacting BEC in the zerotemperaturelimit, we now continue the study of its finite-temperature propertiesin the HF approximation, Eqs. (4.9)-(4.11). To facilitate an almost analytical approach,we abolish Eq. (4.11) and treat excited states within a semiclassical approximation,while keeping the generalized GP equation (4.9) for the condensate phase.To apply the semiclassical approximation, similarly to derivation of Eq. (1.13), weuse the classical Hamiltonian H(⃗r, ⃗p) = ⃗p 2 /2M + 2g(n 0 (⃗r) + n th (⃗r)), in accordancewith Eq. (4.11), and replace a summation over the discrete eigenstates in Eq. (4.10)by an integration over momentum ⃗p. We assume a spherically symmetric trap,i.e. V (⃗r) = V (r), and search for the equilibrium configuration (∂ψ(⃗r, τ)/∂τ = 0),when ψ(⃗r, τ) ≡ ψ(r). Eqs. (4.10) and (4.11) are now reduced to a single equation,and, together with the equation for a total number of particles, we have the following90