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Journal of Modern Optics

Uncertainty relations from fisher information

J. [Rcirc]eháĉek, Z. Hradil

Published Apr 1, 2004• 12 citations
Publisher

Abstract

Heisenberg uncertainty relations yield sometimes rather trivial statements. For instance, the uncertainty of the transversal momentum of a particle behind a rectangular slit is infinite, which yields ∞.Δχh/2. Motivated by the estimation theory we will show that a “good” uncertainty relation can be recovered provided one gives the uncertainties a slightly different meaning. This elementary example will also hint at the importance of the information theory for the quantum physics.

DOI
10.1080/09500340408233612

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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.)

•Dec 1, 1991•0 citations
Heisenberg uncertainty relations yield sometimes rather trivial statements. For instance, the uncertainty of the transversal momentum of a particle behind a rectangular slit is infinite, which yields ∞.Δχh/2. Motivated by the estimation theory we will show that a “good” uncertainty relation can be recovered provided one gives the uncertainties a slightly different meaning. This elementary example will also hint at the importance of the information theory for the quantum physics.
Since the publication of Heisenberg’s paper1 on the “anschaulichen Inhalt” of quantum mechanics, discussions of the fundamental limitations on the accuracy of physical measurements have been much in the foreground. According to Heisenberg, the quantum mechanics implies that it is impossible to measure simultaneous values of a coordinate and its conjugate momentum with unlimited precision. Instead, if Δp be the estimated error or uncertainty in a momentum and Δq that in the associated coordinate one must have the inequality, ΔpΔq>h2π \Delta p\Delta q>\frac{h}{{2\pi}} ΔpΔq>2πh​ (1) This inequality has come to be known quite generally as Heisenberg’s uncertainty relation.
Analyzing general uncertainty relations one can find that there can exist such pairs of non-commuting observables A and B and such vectors that the lower bound for the product of standard deviations ΔA and ΔB calculated for these vectors is zero: ΔA·ΔB≥0. Here we discuss examples of such cases and some other inconsistencies which can be found performing a rigorous analysis of the uncertainty relations in some special cases. As an illustration of such cases matrices (2×2) and (3×3) and the position–momentum uncertainty relation for a quantum particle in the box are considered. The status of the uncertainty relation in PT–symmetric quantum theory and the problems associated with it are also studied.
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.)