We will see in Section 9.9 that when a charged particle in a one-dimensional box (PIB) absorbs or emits a photon, the quantum number n must change by an odd integer: Δn = ±1, ±3, ±5,...... Transitions with n changing by an even integer are not allowed for the PIB. Show that the allowed transition frequencies have the form v = k(h/8ml2), where k takes on certain whole-number values. Find the smallest value of k for which two different Δn values give the same emission frequency.