  In quantum mechanics, the Heisenberg uncertainty relation presents an
ultimate limit to the precision by which one can predict the outcome of
position and momentum measurements on a particle. Heisenberg explicitly stated
this relation for the prediction of "hypothetical future measurements", and it
does not describe the situation where knowledge is available about the system
both earlier and later than the time of the measurement. We study what happens
under such circumstances with an atomic ensemble containing $10^{11}$
$^{87}\text{Rb}$ atoms, initiated nearly in the ground state in presence of a
magnetic field. The collective spin observables of the atoms are then well
described by canonical position and momentum observables, $\hat{x}_A$ and
$\hat{p}_A$ that satisfy $[\hat{x}_A,\hat{p}_A]=i\hbar$. Quantum non-demolition
measurements of $\hat{p}_A$ before and of $\hat{x}_A$ after time $t$ allow
precise estimates of both observables at time $t$. The capability of assigning
precise values to multiple observables and to observe their variation during
physical processes may have implications in quantum state estimation and
sensing.
