On the uncertainty relation. II

J. R. Croca

K. Kowalski, J Rembieli ski

In the preceding Comment (quant-ph/0209032) Trifonov disputes our recently proposed uncertainty relations for a quantum particle on a circle. He states that (i) the quantity $Δ^2(\hatϕ)$ introduced by us representing the uncertainty of the angle is not a proper measure of the position uncertainty and therefore the proposed inequalities can hardly be qualified as a relevant uncertainty relations on a circle; and that (ii) the most suitable uncertainty relations on a circle are those based on the Gram-Robertson matrix. It is shown by examples that both points are erroneous.

Otto Halpern

Kenjiro Yanagi

We give a trace inequality related to the uncertainty relation of generalized Wigner-Yanase-Dyson skew information which includes our result in JMAA, vol.365, pp.12-18, 2010.

K. Kraus

シロウ イシカワ, Shiro Ishikawa, 史郎 石川

In this note, we shall propose a new measurement-axiom in nonrelativistic quantum theory, which can be considered a kind of generalized Copenhagen interpretation. This interpretation can assert rather radical statements, which may irritate our minds familiar with Copenhagen interpretation. We shall, under this interpretation, assert the following two statements: (i) the uncertainties (i.e. Δq and Δp) in Heisenberg's uncertainty relation "Δq・Δp≥■/2" can be characterized as (average) errors in the approximates simultaneous measurement, and so, Heisenberg's uncertainty relation can be clarified as the statement in physics for the first time, (ii) the (discrete) trajectory of a particle is enough significant (though this, of course, includes errors). Furthermore, concerning (ii) we shall show the numerical results of the trajectory of a particle in two slit experiment. (Though almost all parts of this note are composed of summaries of references [11], [12], [13] and [14], we add some discussions from other angles.)