Advanced Quantum Mechanics Schwabl · Problem 3.5

Correlation Functions, Scattering, and Response

Physics — Quantum Mechanics · bose-condensate-temperature-dependence

Problem

Determine the temperature dependence of the number of particles in the condensate, N0(T)N_0(T), for a contact potential Vk=λV_\mathbf{k} = \lambda. a) Proceed by first calculating the thermodynamic expectation value of the particle number operator N^=∑kak†ak\hat{N} = \sum_\mathbf{k} a^\dagger_\mathbf{k}a_\mathbf{k} by rewriting it in terms of the quasiparticle operators αk\alpha_\mathbf{k}. One finds (in the continuum limit: 1V∑k→∫d3k(2π)3\frac{1}{V}\sum_\mathbf{k} \to \int \frac{d^3k}{(2\pi)^3}) N=N0(T)+2(mnλ)3/2π2(16+U1(γ)),N = N_0(T) + 2\frac{(mn\lambda)^{3/2}}{\pi^2}\left(\frac{1}{6} + U_1(\gamma)\right), where γ=βk022m\gamma = \beta\frac{k_0^2}{2m}, k02=4mnλk_0^2 = 4mn\lambda, β=1/kT\beta = 1/kT, kk the Boltzmann constant, and Un(γ)=∫0∞dy xneγy−1,with y=xx2+1.U_n(\gamma) = \int_0^\infty dy\,\frac{x^n}{e^{\gamma y}-1}, \quad \text{with } y = x\sqrt{x^2+1}. b) Show that, for low temperatures, the depletion of the condensate increases quadratically with temperature N0(T)V=N0(0)V−m12c(kT)2,\frac{N_0(T)}{V} = \frac{N_0(0)}{V} - \frac{m}{12c}(kT)^2, where c=nλmc = \sqrt{\frac{n\lambda}{m}}. Also, discuss the limiting case of high temperatures and compare the result obtained with the results from the theory of the Bose–Einstein condensation of noninteracting bosons below the transition temperature. *Lit.:* R.A. Ferrell, N. Menyhárd, H. Schmidt, F. Schwabl and P. Szépfalusy, Ann. Phys. (N.Y.) 47, 565 (1968); Phys. Rev. Lett. 18, 891 (1967); Phys. Letters 24A, 493 (1967); K. Kehr, Z. Phys. 221, 291 (1969).

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