Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 4 / 15C / ii

Set ${\xi }_j=\sqrt{m\omega /\hslash }{x}_j$. Separation into three one-dimensional oscillators givesThe ground state isThe number of triples of nonnegative integers summing to $N$ iswhich is the degeneracy of $E_N$.

Finally,is a pure level-$N=2$ oscillator state: the constant terms in the two degree-two Hermite contributions cancel. Part (i) gives $L_3\chi =2\hslash \chi$, while $N=2$ gives $H\chi =(7/2)\hslash \omega \chi$.