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CRC Press eBooks

- Heisenberg Uncertainty Principle

S. Rajasekar, R. Velusamy

Published Dec 11, 2014• 1 citation
Publisher

Abstract

One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. Let a measurement of an observable A yields the value a and is the eigenvalue of the operator A belonging to the eigenfunction φa. The individual measurements will deviate from the average value (〈a〉) computed over large number (N) of measurements. Denote the deviation of a from 〈a〉 as a˜. What would be the value of average deviation? We obtain 〈a˜〉 = 1 N ∑ (a− 〈a〉) = 0 . (8.1) This is because some of the values of a are below 〈a〉 while the others are above 〈a〉. Therefore, 〈a˜〉 is not useful for describing how much an individual value of an observable deviates from the expected or mean value. The mean-square deviation or variance (σ2), that is, the average of the squares of the deviations would be nonzero. The variance is given by σ2 = (∆a)2 = 1 N ∑ (a− 〈a〉)2 = 〈a2〉 − 〈a〉2 . (8.2) The square-root of mean-square deviation is called standard deviation or dispersion. This quantity could be taken as a measure of the spread in the measured values. When most of the values a are near 〈a〉 then the distribution of a peaks about 〈a〉 and in this case σ is small. If most of the values of a are found over a wide range about 〈a〉 then the distribution spreads over a wide range of a with a large value of σ. This spread, denoted as ∆A, in the measured values of A, is called the uncertainty in the measurement of A.

DOI
10.1201/b17899-12

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One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. Let a measurement of an observable A yields the value a and is the eigenvalue of the operator A belonging to the eigenfunction φa. The individual measurements will deviate from the average value (〈a〉) computed over large number (N) of measurements. Denote the deviation of a from 〈a〉 as a˜. What would be the value of average deviation? We obtain 〈a˜〉 = 1 N ∑ (a− 〈a〉) = 0 . (8.1) This is because some of the values of a are below 〈a〉 while the others are above 〈a〉. Therefore, 〈a˜〉 is not useful for describing how much an individual value of an observable deviates from the expected or mean value. The mean-square deviation or variance (σ2), that is, the average of the squares of the deviations would be nonzero. The variance is given by σ2 = (∆a)2 = 1 N ∑ (a− 〈a〉)2 = 〈a2〉 − 〈a〉2 . (8.2) The square-root of mean-square deviation is called standard deviation or dispersion. This quantity could be taken as a measure of the spread in the measured values. When most of the values a are near 〈a〉 then the distribution of a peaks about 〈a〉 and in this case σ is small. If most of the values of a are found over a wide range about 〈a〉 then the distribution spreads over a wide range of a with a large value of σ. This spread, denoted as ∆A, in the measured values of A, is called the uncertainty in the measurement of A.
One of the most striking deviations of quantum mechanics from classical mechanics is that any measurement of a microscopic system involves the interaction with the measuring instruments. The interaction cannot be neglected and in fact leads to uncontrollable disturbance on the system. In 1927, Werner Karl Heisenberg in his celebrated paper established the quantitative expression for unavoidable momentum disturbance caused by any position measurement. The crucial word here is simultaneously. The uncertainty relation gives the lower bound on the product of the uncertainties of two pairs of variables. But it says nothing about the accuracy with which individual observables can be measured. The physical meaning of energy-time uncertainty is different from that of x-px uncertainty. Position and momentum are observables to which Hermitian operators are assigned and which are measured at a particular time. An important implication of the uncertainty principle is that quantum systems do follow classical trajectories which conflict with our everyday experience.
Textbooks often emphasize the philosophical aspects of Heisenberg's Principle calling it “The Uncertainty Principle”. However, as a physical principle there is nothing uncertain about it. In its “exact” form it is a useful relationship between physical parameters. It is an expression of the fact that particles of matter behave simultaneously in two modes: one that is particle like, and the other that is wave like. These two modes are intimately connected, and it is this connection that is expressed in the Heisenberg Principle, and more generally in quantum mechanics including Schrödinger's equation (Atkins and Friedman, 1997). From this viewpoint, the principle can be used to deduce some of the properties of atoms and molecules in a simple way. This method of analysis is quantitative, but it is limited to cases that involve only a single coordinate that is given in advance (Born, 1989). The method has been extended somewhat by Simons and Bloch (1973) to cover some cases involving the angular momentum quantum number. It is a rudimentary form of density functional theory. In a somewhat different form than the one used here, it has been applied to molecules by Parr and his collaborators (Borkman and Parr, 1968). Physical properties can never be measured exactly. Errors, however small, are always present in a measurement procedure. Therefore, measured values of variables are average values derived from a distribution of individual measurements.
We revisit the definitions of error and disturbance recently used in error-disturbance inequalities derived by Ozawa and others by expressing them in the reduced system space. The interpretation of the definitions as mean-squared deviations relies on an implicit assumption that is generally incompatible with the Bell-Kochen-Specker-Spekkens contextuality theorems, and which results in averaging the deviations over a non-positive-semidefinite joint quasiprobability distribution. For unbiased measurements, the error admits a concrete interpretation as the dispersion in the estimation of the mean induced by the measurement ambiguity. We demonstrate how to directly measure not only this dispersion but also every observable moment with the same experimental data, and thus demonstrate that perfect distributional estimations can have nonzero error according to this measure. We conclude that the inequalities using these definitions do not capture the spirit of Heisenberg's eponymous inequality, but do indicate a qualitatively different relationship between dispersion and disturbance that is appropriate for ensembles being probed by all outcomes of an apparatus. To reconnect with the discussion of Heisenberg, we suggest alternative definitions of error and disturbance that are intrinsic to a single apparatus outcome. These definitions naturally involve the retrodictive and interdictive states for that outcome, and produce complementarity and error-disturbance inequalities that have the same form as the traditional Heisenberg relation.
The uncertainty principle, formulated by Heisenberg in 1927, is a fundamental principle in quantum mechanics. It forces us to review key concepts of classical physics, and from this analysis are born the concepts of quantum physics. Consequences of the Uncertainty principle also introduces the concept of incompatible sizes, a completely new concept compared to classical physics. A state of the system, which we call a quantum state, will be determined from the knowledge of a set maximum of certain quantities that are independent and compatible with each other. A measurable physical quantity is called an observable. If at a given moment a measurement is performed by one or more compatible observables it is said they are performing an observation of the system. The basic principles of quantum mechanics are expressed by a number of postulates. Their numbers and can vary from one text to another.
Does the Heisenberg uncertainty principle refer to errors of measurement, or to the spread of values of the physical variables intrinsic to a particle’s state, or to some combination of these? The most commonly quoted form of the uncertainty principle relates the spread of an ensemble of separate measurements of some variable to the analogous spread of its conjugate variable. In contrast, Heisenberg’s original argument for the uncertainty principle involved the perturbation to a particle’s state by a measurement of one variable, which affects one’s ability to predict the outcome of a subsequent measurement of the conjugate variable. The relation between these two views of the uncertainty principle is discussed in this paper. A familiar example is considered: an ensemble of identically prepared particles passing through a slit, and after further propagation being detected. From this arrangement it is possible to infer joint (although necessarily imprecise) information on both transverse position and conjugate momentum for each member of the ensemble. It is shown that in this case of joint measurement the product of standard deviations for the measurement outcomes is at least twice as large as the lower bound implied by the usual uncertainty principle. The discussion is meant to help clarify the different roles played in the various statements of the uncertainty principle by the initial state and by measurement error.
Recent years have witnessed a controversy over Heisenberg's famous error-disturbance relation. Here the conflict is resolved by way of an analysis of the possible conceptualizations of measurement error and disturbance in quantum mechanics. Two approaches to adapting the classic notion of root-mean-square error to quantum measurements are discussed. One is based on the concept of a noise operator; its natural operational content is that of a mean deviation of the values of two observables measured jointly, and thus its applicability is limited to cases where such joint measurements are available. The second error measure quantifies the differences between two probability distributions obtained in separate runs of measurements and is of unrestricted applicability. We show that there are no nontrivial unconditional joint-measurement bounds for state-dependent errors in the conceptual framework discussed here, while Heisenberg-type measurement uncertainty relations for state-independent errors have been proven.