We show how the approach to equilibrium in Kirman's ants model can be fully characterized in terms of the spectrum of a Schr\"odinger equation with a P\"oschl-Teller ($\tan^2$) potential. Among other interesting properties, we have found that in the bimodal phase where ants visit mostly one food site at a time, the switch time between the two sources only depends on the ``spontaneous conversion" rate and not on the recruitment rate. More complicated correlation functions can be computed exactly, and involve higher and higher eigenvalues and eigenfunctions of the Schr\"odinger operator, which can be expressed in terms of hypergeometric functions.