Root-mean-square positional uncertainty

NOTE 2. Correlation between the measurements influences metrological compatibility of measurement results. If the measurements are completely uncorrelated, the standard measurement uncertainty of their difference is equal to the root mean square sum of their standard measurement uncertainties, while it is lower for positive covariance or higher for negative covariance. 

With some algebra, it follows from Eq. (2.65) that the product of the uncertainties (root-mean-square deviations) in the expectation values for position and momentum must be > Hjl [4]. This is a statement of Heisenberg s uncertainty principle. 

Strictly, the uncertainty in momentum is the root mean square (r.m.s.) deviation of the momentum from its mean value, Ap = ((p ) — pyY, where the angle brackets denote mean values. Likewise, the uncertainty in position is the r.m.s. deviation in the mean value of position,... 

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