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Optics and Spectroscopy

Some surprises of quantum measurements and states

Yu.I. Vorontsov

Published Jun 8, 2004• 0 citations
Publisher

Abstract

The problem of physical reality and measurability of observables is discussed on the example of phenomena revealed by the author. The essence of the phenomena is as follows. After measurement of the coordinate, the variance of the momentum can be smaller than the initial one and depends on the result of the coordinate measurement even if the coordinate and the momentum are uncorrelated. This refers to any pair of conjugate observables, in particular, to the number of quanta and the phase. In a state with a minimum product of uncertainties of the coordinate and the momentum, the mean value of the observable $${{\left\{ {\hat x^2 ,\hat p^2 } \right\}} \mathord{\left/ {\vphantom {{\left\{ {\hat x^2 ,\hat p^2 } \right\}} 2}} \right. \kern-\nulldelimiterspace} 2}$$ , which corresponds to the classical product of the squares of the coordinate and the momentum, is negative.

DOI
10.1134/1.1753630

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The problem of physical reality and measurability of observables is discussed on the example of phenomena revealed by the author. The essence of the phenomena is as follows. After measurement of the coordinate, the variance of the momentum can be smaller than the initial one and depends on the result of the coordinate measurement even if the coordinate and the momentum are uncorrelated. This refers to any pair of conjugate observables, in particular, to the number of quanta and the phase. In a state with a minimum product of uncertainties of the coordinate and the momentum, the mean value of the observable {{\left\{ {\hat x^2 ,\hat p^2 } \right\}} \mathord{\left/ {\vphantom {{\left\{ {\hat x^2 ,\hat p^2 } \right\}} 2}} \right. \kern-\nulldelimiterspace} 2} , which corresponds to the classical product of the squares of the coordinate and the momentum, is negative.
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.
We suggest a somewhat non-standard view on a set of curious, paradoxical from the standpoint of simple classical physics and everyday experience phenomena. There are the quantisation (discrete set of values) of the observables (e.g., energy, momentum, angular momentum); forbidden simultaneous measurements of the observables in the most cases (e.g., of a coordinate and momentum, of angular momentum projections on difference axis); counter-intuitive relations on the simultaneously measurable quantities (e.g., the famous expression for the square momentum l(l+1)l(l+1)l(l+1) with the maximal projection lll). These and other paradoxes are traditionally related to "purely quantum" phenomenon, i.e., having no analogue in the "classical world" ones. However, there are deep analogies between classical and "quantum" worlds, as soon as the quantum technique is applied to the classical phenomenon. We follow these analogies with the examples of relatively simple and well known models of classical physics, such as a simplified model of light transition through the media, a system of electric charges close to each other and far from the observer; the specific of motion in the Coulomb/Newtonian field. This text can be considered as a mini-course addressed to higher school and undergraduate students who are interested in basics of quantum mechanics, but are not yet ready for systematic study of standard courses. The text may be also useful to those who supervise such students.
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Uncertainty relations between a bounded coordinate operator and a conjugate momentum operator frequently appear in quantum mechanics. We prove that physically reasonable minimum-uncertainty solutions to such relations have quantized expectation values of the conjugate momentum. This implies, for example, that the mean angular momentum is quantized for any minimum-uncertainty state obtained from any uncertainty relation involving the angular-momentum operator and a conjugate coordinate. Experiments specifically seeking to create minimum-uncertainty states localized in angular coordinates therefore must produce packets with integer angular momentum. \textcopyright{} 1996 The American Physical Society.
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The state of a quantum-mechanical system after a simultaneous measurement of conjugate variables such as coordinate and momentum is derived. In the case of the minimum-uncertainty, or ideal, simultaneous measurement, the state of the system after the measurement is a coherent state corresponding to that representing a minimum-uncertainty wave packet. An operator formalism to describe simultaneous measurements of noncommuting observables is introduced which parallels that familiar for a single measurement, and an expression for the joint probability distribution predicting the results of an ideal simultaneous measurement is derived and applied to the example of a harmonic oscillator. It is found that the minimum uncertainty in a simultaneous measurement of noncommuting variables has two causes: (1) the unavoidable perturbations introduced by the measuring process, and (2) the unavoidable lack of precision in the state of the system itself. For position and momentum measurements, these two independent uncertainties contribute to give a net minimum uncertainty of $\ensuremath{\Delta}{q}^{\ensuremath{'}}\ensuremath{\Delta}{p}^{\ensuremath{'}}=\ensuremath{\hbar}$. It is also shown that the formalism of simultaneous measurement leads to a well-defined quantum-mechanical phase space.