Generalizing the Heisenberg uncertainty relation

Abstract

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.Comment: 12 pages, no figures. Contains corrections to errors in the published editio

Quantization of the canonically conjugate pair angle and orbital angular momentum

“…The correlation (72), and therefore γ, vanish for the wave function (112) with A = Q, B = P . In our case the parameter γ here describes the correlations (98). A combination of γ and s, namely |σ| 2 = γ 2 + s 2 determines the squeezing properties of the distribution [see Eq.…”

“…Finally, there seems to be no chance to derive the existence of quasi-OAM δ by starting from finite dimensional vector spaces! Other authors [98][99][100] discussed related problems associated with uncertainty relations for wave functions on the circle.…”

Uncertainty relations in curved spaces

“…whereφ is a mean value of the observable. Using the Cauchy inequality | Ψ 1 |Ψ 2 | |Ψ 1 | · |Ψ 2 |, one gets [4,11]:…”

“…In the case of two dimensions one can write: are written down in terms of old variables, ϑ and ϕ. Commutational relations η ,p η = i , ξ ,p ξ = i , p η ,p ξ = 0 can be verified directly but, as in section 2, one has to remember that the lhs of these equations are not defined for the majority of states (it is again a question of periodicity). One may derive uncertainty relations in two different ways: either using the δ-function approach and a relation analogous to (4) or applying the inequality (3). In any case the result is ∆η · ∆p η 0, ∆ξ · ∆p ξ 0.…”

“…The foundations of quantum mechanics (see, for example, [3]) and, in particular, uncertainty relations [4,5] have been in the focus of unprecedented activity. Some subtle mathematical points are taken into account, such as self-adjointness of operators, their domains and so on [4,6], which are usually disregarded in physical papers.…”