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 $Lz$ do not satisfy the $\phi \u2013Lz$ 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.

## REFERENCES

*Quantum Theory*(Prentice–Hall, Englewood Cliffs, NJ, 1951), pp. 205–207.

*Modern Quantum Mechanics*(Addison–Wesley, New York, 1994), revised ed., pp. 34–36.

*Quantum Mechanics*(Springer-Verlag, New York, 1990), Vol. 1, pp. 201–206.

*Quantum Noise*(Springer-Verlag, New York, 1991), pp. 1–2.

*Invariants and the Evolution of Nonstationary Quantum Systems,*Vol. 183 of the Proceedings of the Lebedev Physics Institute, ed. M. A. Markov (Nova Science, Commack, NY, 1989), pp. 3–101. This article also discusses the $\phi -Lz$ uncertainty relation as well as higher-order uncertainty relations, relations among an arbitrary number of observables, and entropy-based uncertainty relations, and it has a substantial list of references.

*Theory of Linear Operators in Hilbert Space*(Ungar, New York, 1961 and 1963), 2 Vols.

*Linear Operators*(Interscience, New York, 1958, 1963, and 1971), 3 Vols.

*Functional Analysis*(Ungar, New York, 1955).

*Foundations of Quantum Mechanics*(Addison–Wesley, New York, 1968).

*Quantum Mechanics*(Wiley, New York, 1977), Vol. 1, pp. 286–287.

*American Journal of Physics*and

*The Physics Teacher*as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.