Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 344 / 2 / d / Solution

For candidate (ii), $|\mathbf{p}|=p_0$ everywhere and each component has wavevector magnitude $q_0$. The quadratic density is therefore $\hat{a}{p}_0^2/2$, while the quartic density is $b{p}_0^4/4$ without trigonometric averaging:For $\hat{a}<0$,Since $(-1/4)/(-1/6)=3/2$, the helical structure's free energy is $50\%$ more negative than that of candidate (i).