  The Heisenberg inequality \Delta X \Delta P \geq \hbar/2 can be replaced by
an exact equality, for suitably chosen measures of position and momentum
uncertainty, which is valid for all wavefunctions. The significance of this
"exact" uncertainty relation is discussed, and results are generalised to
angular momentum and phase, photon number and phase, time and frequency, and to
states described by density operators. Connections to optimal estimation of an
observable from the measurement of a second observable, Wigner functions,
energy bounds, and entanglement are also given.
