In the classical limit, a simple relation has been shown to exist between the thermal expansion coefficient \ensuremath{\alpha} and the cumulants of the vibrational amplitude that are measured in x-ray-absorption fine structure (XAFS), i.e., \ensuremath{\alpha}rT${\mathrm{\ensuremath{\sigma}}}^{2}$/${\mathrm{\ensuremath{\sigma}}}^{(3)}$=1/2, where ${\mathrm{\ensuremath{\sigma}}}^{2}$ is the mean-square vibrational amplitude, ${\mathrm{\ensuremath{\sigma}}}^{(3)}$ the third cumulant, T the absolute temperature, and r the equilibrium bond length. We generalize this relation to the quantum case using a correlated Einstein model and thermodynamic perturbation theory, and find \ensuremath{\alpha}rT${\mathrm{\ensuremath{\sigma}}}^{2}$/${\mathrm{\ensuremath{\sigma}}}^{(3)}$=[3z(1+z)ln(1/z)]/[(1-z)(1+10z+${\mathit{z}}^{2}$)], where z=exp(-${\mathrm{\ensuremath{\Theta}}}_{\mathit{E}}$/T), and ${\mathrm{\ensuremath{\Theta}}}_{\mathit{E}}$ is the Einstein temperature. This result is found to be in agreement with the measured thermal expansion coefficient and XAFS cumulants in RbBr at 30 K and 125 K.