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arXiv:quant-ph/0307007

On the uncertainty relations for angular observables and beyond them

Spiridon Dumitru

Published Jul 1, 2003• 0 citations
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Abstract

For angular observables pairs (angular momentum-angle and number-phase) the adequate reference element of normality is not the Robertson-Schrödinger uncertainty relation but a Schwarz formula regarding the quantum fluctuations. Beyond such a fact the traditional interpretation of the uncertainty relations appears as an unjustified doctrine.

DOI
10.48550/arxiv.quant-ph/0307007

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For angular observables pairs (angular momentum-angle and number-phase) the adequate reference element of normality is not the Robertson-Schrödinger uncertainty relation but a Schwarz formula regarding the quantum fluctuations. Beyond such a fact the traditional interpretation of the uncertainty relations appears as an unjustified doctrine.
The disputed question of uncertainty relations (UR) on a circle is regarded as a particular element of a more general problem which refers to the quantum description of angular observables LzL_zLz​ and ϕϕϕ. The improvised Lz−ϕL_z-ϕLz​−ϕ UR are found to be affected by unsourmontable shortcomings. Also in contradiction with a largely accepted belief it is proved that the usual procedures of quantum mechanics are accurately applicable for the Lz−ϕL_z-ϕLz​−ϕ pair. The applicability regards both the known circular motions and the less known non-circular rotational motions. The presented facts contribute as arguments to the following indubitable conclusions: (i) the traditional interpretation of UR must be denied as an incorrect doctrine, (ii) for a natural physical consideration of the the Lz−ϕL_z-ϕLz​−ϕ pair the results from the usual quantum mechanics are sufficient while the improvised Lz−ϕL_z-ϕLz​−ϕ UR must be rejected as senseless formulas and (iii) the descriptions of quantum measurements have to be done in a framework which is distinct and additional in respect with the usual quantum mechanics.
Quantum-mechanical uncertainty relations for position and momentum are expressed in the form of inequalities involving the R\'enyi entropies. The proof of these inequalities requires the use of the exact expression for the (p,q)(p,q)(p,q)-norm of the Fourier transformation derived by Babenko and Beckner. Analogous uncertainty relations are derived for angle and angular momentum and also for a pair of complementary observables in NNN-level systems. All these uncertainty relations become more attractive when expressed in terms of the symmetrized R\'enyi entropies.
The uncertainty principle places fundamental limits on the accuracy with which we are able to measure the values of different physical quantities (Heisenberg 1949 The Physical Principles of the Quantum Theory (New York: Dover); Robertson 1929 Phys. Rev. 34 127). This has profound effects not only on the microscopic but also on the macroscopic level of physical systems. The most familiar form of the uncertainty principle relates the uncertainties in position and linear momentum. Other manifestations include those relating uncertainty in energy to uncertainty in time duration, phase of an electromagnetic field to photon number and angular position to angular momentum (Vaccaro and Pegg 1990 J. Mod. Opt. 37 17; Barnett and Pegg 1990 Phys. Rev. A 41 3427). In this paper, we report the first observation of the last of these uncertainty relations and derive the associated states that satisfy the equality in the uncertainty relation. We confirm the form of these states by detailed measurement of the angular momentum of a light beam after passage through an appropriate angular aperture. The angular uncertainty principle applies to all physical systems and is particularly important for systems with cylindrical symmetry.
Unsolved controversies about uncertainty relations and quantum measurements still persists nowadays. They originate around the shortcomings regarding the conventional interpretation of uncertainty relations. Here we show that the respective shortcomings disclose veridic and unavoidable facts which require the abandonment of the mentioned interpretation. So the primitive uncertainty relations appear as being either thought fictions or fluctuations formulae. Subsequently we reveal that the conventional approaches of quantum measurements are grounded on incorrect premises. We propose a new approach in which : (i) the quantum observables are considered as generalized stochastic variables, (ii) the view is focused only on the pre-existent state of the measured system, without any interest for the collapse of the respective state, (iii) a measurement is described as an input-output transformation which modify the probability density and current but preserve the expressions of the operators. The measuring uncertainties are evaluated as changes in the probabilistic estimators of observables. Related to different observables we do not find reasons of principle neither for uncertainties-connections nor for measuring compatibility/incompatibility.