The emergence of ferroelectricity in nanometer-thick films of doped hafnium oxide (HfO2) makes this material a promising candidate for use in Si-compatible non-volatile memory devices. The switchable polarization of ferroelectric HfO2 controls functional properties of these devices through the electric potential distribution across the capacitor. The experimental characterization of the local electric potential at the nanoscale has not so far been realized in practice. Here, we develop a new methodology which allows us, for the first time, to experimentally quantify the polarization-dependent potential profile across few-nanometer-thick ferroelectric Hf0.5Zr0.5O2 thin films. Using a standing-wave excitation mode in synchrotron based hard X-ray photoemission spectroscopy, we depth-selectively probe TiN/Hf0.5Zr0.5O2/W prototype memory capacitors and determine the local electrostatic potential by analyzing the core-level line shifts. We find that the electric potential profile across the Hf0.5Zr0.5O2 layer is non-linear and changes with in situ polarization switching. Combined with our scanning transmission electron microscopy data and theoretical modeling, we interpret the observed non-linear potential behavior in terms of defects in Hf0.5Zr0.5O2, at both interfaces, and their charge state modulated by the ferroelectric polarization. Our results provide an important insight into the intrinsic electronic properties of HfO2 based ferroelectric capacitors and are essential for engineering memory devices.