The ion exchange mechanism of the sodium/calcium exchanger (NCX) crystallized by Liao et al. in 2012 is studied using the Poisson-Fermi theory developed by Liu and Eisenberg in 2014. A cycle of binding and unbinding is proposed to account for the Na(+)/Ca(2+) exchange function of the NCX molecule. Outputs of the theory include electric and steric fields of ions with different sizes, correlations of ions of different charges, and polarization of water, along with number densities of ions, water molecules, and interstitial voids. We calculate the electrostatic and steric potentials of the four binding sites in NCX, i.e., three Na(+) binding sites and one Ca(2+) binding site, with protein charges provided by the software PDB2PQR. The energy profiles of Na(+) and Ca(2+) ions along their respective Na(+) and Ca(2+) pathways in experimental conditions enable us to explain the fundamental mechanism of NCX that extrudes intracellular Ca(2+) across the cell membrane against its chemical gradient by using the downhill gradient of Na(+). Atomic and numerical details of the binding sites are given to illustrate the 3 Na(+):1 Ca(2+) stoichiometry of NCX. The protein NCX is a catalyst. It does not provide (free) energy for transport. All energy for transport in our model comes from the ions in surrounding baths.