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arXiv:quant-ph/9901041 · Physics Letters A

Average local values and local variances in quantum mechanics

J.G. Muga, J.P. Palao, R. Sala

Published Feb 1, 1998• 29 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(97)00902-x

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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.

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On local observables in quantum mechanics

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Physics Letters A•Dec 1, 1984•6 citations
Halls exact variance decomposition [Phys. Rev. A 64, 052103 (2001)] splits the quantum variance of an observable into the ensemble variance of an optimal position based estimate and a residual nonclassical inaccuracy. We evaluate this decomposition in Bohmian mechanics. For momentum, the optimal estimate coincides with the Bohmian guidance field, and the inaccuracy is proportional to the ensemble average of the quantum potential. This gives a variance level identity separating momentum fluctuations into classical statistical dispersion and a quantum contribution from amplitude variations. The real and imaginary parts of the weak value map directly onto the two decomposition terms. By contrast, the inaccuracy vanishes for spin. This distinction is traced to the kinematic status of velocity in the primitive ontology, showing how the decomposition distinguishes observables dynamically coupled to local beables from merely contextual ones.
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.