Cold Collision Frequency Shift of the $1\mathit{S}$- $2\mathit{S}$ Transition in Hydrogen

We have observed the cold collision frequency shift of the $1S$- $2S$ transition in trapped spin-polarized atomic hydrogen. We find $\Delta {\nu }_{1S-2S}=-3.8\mathrm{}\pm 0.8\times {10}^{-10\mathrm{}}n\mathrm{Hz}{\mathrm{cm}}^3$, where $n$ is the sample density. From this we derive the $1S$- $2S$ $s$-wave triplet scattering length, $a_{1S-2S}=-1.4\mathrm{}\pm 0.3\mathrm{nm}$, which is in fair agreement with a recent calculation. The shift provides a valuable probe of the distribution of densities in a trapped sample.