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Compressibility measurements, sensitive to charge rearrangements, are performed on a quantum point contact (QPC). Screening due to mobile charges in the QPC is quantitatively measured, using a second point contact to detect the screened electrical potential. These measurements are performed from pinch-off through the opening of the first few modes in the QPC. While the measured signal closely matches a Thomas-Fermi-Poisson prediction, deviations from the classical behavior, in the form of additional dips, are apparent near the openings of the different modes, with the largest dip at the opening of the first mode. Density functional calculations attribute the first dip to exchange interactions. The other dips reflect the diverging density of states at the opening of each one-dimensional mode, which affects both kinetic and exchange contributions to the energy., Comment: 6 pages, 5 figures. Published in PRL. Supplementary Information is included. Changes: Two references and journal reference added