Comment on ” On the uncertainty relations and squeezed states for the quantum mechanics on a circle ”

It is shown by examples that the position uncertainty on a circle, proposed recently by Kowalski and Rembieli´nski [J. Phys. A 35 (2002) 1405] is not consistent with the state localization. We argue that the relevant uncertainties and uncertainty relations (UR's) on a circle are that based on the Gram-Robertson matrix. Several of these generalized UR's are displayed and related criterions for squeezed states are discussed. 1. In the recent paper [1] the problem of relevant uncertainty relation (UR) for the angular momentum and the angle variables of a particle on a circle was discussed and a new UR was proposed. Noting a contradiction in the previously obtained UR [2] the authors defined new quantities˜∆ 2 (ˆ ϕ) and˜∆ 2 (ˆ J) (ˆ ϕ = ϕ, ˆ J = −id/dϕ) as measures for the uncertainty of the angle ϕ and the angular momentum J and suggest the inequality ˜ ∆ 2 (ˆ ϕ) + ˜ ∆ 2 (ˆ J) ≥ 1. (1) The quantities˜∆ 2 (ˆ ϕ), ˜ ∆ 2 (ˆ J) are defined as [1] (note a change in notation: ∆ 2 → ˜ ∆ 2) ˜ ∆ 2 (ˆ ϕ) = − 1 4 ln U 2 2 , ˜ ∆ 2 (ˆ J) = 1 4 ln e −2ˆJ e 2 ˆ J , (2) where U = exp(i ˆ ϕ). The authors of [1] find that for the eigenstates |z of the operator Z = exp(− ˆ J + 1/2)U , Z|z = z|z (the " genuine coherent states (CS's) for a quantum particle on a circle " [3]) both quantities (2) equal 1/2, and suggest that˜∆ 2 (ˆ ϕ) and˜∆ 2 (ˆ J) obey the inequality (1) in any state. Henceforth, the quantities˜∆ 2 (ˆ ϕ) and˜∆ 2 (ˆ J) should be referred to as Kowalski-Rembieli´nski uncertainties ((K-R) uncertainties), the UR (1) – as K-R UR and |z – as Kowalski-Rembieli´nski-Papaloucas CS's (K-R-P CS's). 2. Next we shall demonstrate that the K-R uncertainty˜ ∆ 2 (ˆ ϕ) is not consistent with the state localization on a circle. For this purpose we compare the ϕ–probability distributions p ψ (ϕ) (defined as p ψ (ϕ) = |ψ(ϕ)| 2 = |ϕ|ψ| 2) in K-R-P CS's with ϕ–distributions in certain states with squeezed˜∆ 2 (ˆ ϕ). The quantity˜∆ 2 (ˆ ϕ) is called [1] squeezed if it is less that than 1/2. The authors of [1] constructed a family …