Advanced Quantum Mechanics Schwabl · Problem 3.4

Correlation Functions, Scattering, and Response

Physics — Quantum Mechanics · bogoliubov-transformation-diagonalization

Problem

By means of the Bogoliubov transformation, the Hamiltonian of the weakly interacting Bose gas can be brought into diagonal form. One thereby finds the condition (3.2.17): (k22m+nVk)ukvk+n2Vk(uk2+vk2)=0.\left(\frac{k^2}{2m} + nV_\mathbf{k}\right)u_\mathbf{k}v_\mathbf{k} + \frac{n}{2}V_\mathbf{k}\left(u_\mathbf{k}^2 + v_\mathbf{k}^2\right) = 0. Confirm the results (3.2.18) and (3.2.19).

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