A particle of mass m is confined inside a one-dimensional region of length L bordered by perfectly reflecting walls. A researcher measures the separation in energy between two adjacent quantum states labeled by consecutive integer quantum numbers n and n+1.
(a) Starting from the condition that only standing waves that fit an integer number of half-wavelengths in the region are permitted, derive an expression for the allowed energy of the nth state in terms of n, m, and L.
(b) From your result in part (a), obtain a formula for the energy spacing ΔE between adjacent states as a function of n, m, and L.
(c) For a fixed value of n, determine quantitatively how the spacing ΔE changes if the length L of the confining region is tripled. Express your answer as a factor by which ΔE increases or decreases.
(d) Discuss qualitatively what your result implies about the spacing of spectral lines emitted by semiconductor quantum dots as their physical size is varied during fabrication.