  Many-body fermion systems are important in many branches of physics,
including condensed matter, nuclear, and now cold atom physics. In many cases,
the interactions between fermions can be approximated by a contact interaction.
A recent theoretical advance in the study of these systems is the derivation of
a number of exact universal relations that are predicted to be valid for all
interaction strengths, temperatures, and spin compositions. These equations,
referred to as the Tan relations, relate a microscopic quantity, namely, the
amplitude of the high-momentum tail of the fermion momentum distribution, to
the thermodynamics of the many-body system. In this work, we provide
experimental verification of the Tan relations in a strongly interacting gas of
fermionic atoms. Specifically, we measure the fermion momentum distribution
using two different techniques, as well as the rf excitation spectrum and
determine the effect of interactions on these microscopic probes. We then
measure the potential energy and release energy of the trapped gas and test the
predicted universal relations.
