Common diagonal dipole phase in an infinite well ($⟨n|x|n⟩=w/2$)

Every stationary state in $0<x<w$ has mean position $w/2$. Splitting a uniform field coupling into $-qE(t)(x-w/2)-qE(t)w/2$ separates its transition-driving odd operator from an identity term. The latter gives the common phase $qw\int E(t)dt/(2\hslash )$ and cannot change populations. Removing it by an origin shift is equivalent to a time-dependent scalar phase; relative transition amplitudes are unchanged.