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
∑ (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.