  The proof of the Heisenberg uncertainty relation is modified to produce two
improvements: (a) the resulting inequality is stronger because it includes the
covariance between the two observables, and (b) the proof lifts certain
restrictions on the state to which the relation is applied, increasing its
generality. The restrictions necessary for the standard inequality to apply are
not widely known, and they are discussed in detail. The classical analog of the
Heisenberg relation is also derived, and the two are compared. Finally, the
modified relation is used to address the apparent paradox that eigenfunctions
of the z component of angular momentum L_z do not satisfy the \phi-L_z
Heisenberg relation; the resolution is that the restrictions mentioned above
make the usual inequality inapplicable to these states. The modified relation
does apply, however, and it is shown to be consistent with explicit
calculations.
