  We use an atomic fountain clock to measure quantum scattering phase shifts
precisely through a series of narrow, low-field Feshbach resonances at average
collision energies below $1\,\mu$K. Our low spread in collision energy yields
phase variations of order $\pm \pi/2$ for target atoms in several $F,m_F$
states. We compare them to a theoretical model and establish the accuracy of
the measurements and the theoretical uncertainties from the fitted potential.
We find overall excellent agreement, with small statistically significant
differences that remain unexplained.
