A particle is confined to a one-dimensional box with perfectly rigid walls. Which category best describes the relationship between the energy of the particle and the uncertainty in its position?

In a one‑dimensional box the particle’s wavefunction is spread over the whole box, so its position uncertainty Δx is roughly the same (about half the box length) for all energy levels. As the particle’s energy rises, the momentum uncertainty Δp grows because the wavefunction contains higher‑frequency components. The product Δx Δp remains on the order of ħ/2, so Δx stays essentially fixed while Δp increases. Thus, the energy of the particle is not directly related to its position uncertainty; it is the momentum uncertainty that changes with energy. For example, the ground state has a low Δp, while the second excited state has a larger Δp, yet both have a similar Δx across the box.