Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / Section B / 9 / f

The density is symmetric about $L/2$: $p(L-x)=p(x)$. Denote the expectation by $\mu ={\int }_0^{L}xp(x)dx$. Substitution $u=L-x$ givesThus $⟨x⟩=L/2$. An expectation need not coincide with a high-density position: the centre is a node of this particular state, but the two equally weighted lobes balance about it.