Past exam of the theoretical physics I course of the University of Cambridge / 2019 / 3 / d / Solution

For independent $\chi ,{\chi }^{\ast }$, the derivatives needed in the ${\chi }^{\ast }$ Euler-Lagrange field equation areConsequently the two conjugate field equations areWrite $\chi ={\chi }_1+i{\chi }_2$ with real components, and put $A={\hslash }^2/(2M)$. Separating real and imaginary parts givesFor each real field take the real part of a complex plane wave amplitude, ${\chi }_j=\mathrm{Re}[q_j{e}^{i(\mathbf{k}\cdot \mathbf{r}-\omega t)}]$. With ${ϵ}_k={\hslash }^2|\mathbf{k}{|}^2/(2M)$ the two equations becomeA nonzero normal mode requires their determinant to vanish, soThe positive-frequency Bogoliubov quasiparticle dispersion is thereforeThere is also the negative-frequency conjugate solution, as required for real component fields. Since $\mu >0$, ${\omega }^2\ge 0$, consistent with the stability of the chosen uniform ground state.