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Uncertainty Relation

We have seen that if and are two noncommuting observables then a determination of the value of leaves the value of uncertain, and vice versa. It is possible to quantify this uncertainty. For a general observable , we can define an Hermitian operator

 (1.71)

where the expectation value is taken over the particular physical state under consideration. It is obvious that the expectation value of is zero. The expectation value of is termed the variance of , and is, in general, non-zero. In fact, it is easily demonstrated that

 (1.72)

The variance of is a measure of the uncertainty in the value of for the particular state in question (i.e., it is a measure of the width of the distribution of likely values of about the expectation value). If the variance is zero then there is no uncertainty, and a measurement of is bound to give the expectation value, .

Consider the Schwarz inequality,

 (1.73)

which is analogous to

 (1.74)

in Euclidian space. This inequality can be proved by noting that

 (1.75)

where is any complex number. If takes the special value then the previous inequality reduces to

 (1.76)

which is equivalent to the Schwarz inequality.

Let us substitute

 (1.77) (1.78)

into the Schwarz inequality, where the blank ket stands for any general ket. We find

 (1.79)

where use has been made of the fact that and are Hermitian operators. Note that

 (1.80)

where the commutator, , and the anti-commutator, , are defined

 (1.81) (1.82)

respectively. The commutator is clearly anti-Hermitian,

 (1.83)

whereas the anti-commutator is obviously Hermitian. Now, it is easily demonstrated that the expectation value of an Hermitian operator is a real number, whereas the expectation value of an anti-Hermitian operator is an imaginary number. (See Exercise 11.) It follows that the right-hand side of

 (1.84)

consists of the sum of an imaginary and a real number. Taking the modulus squared of both sides gives

 (1.85)

where use has been made of , et cetera. The final term on the right-hand side of the previous expression is positive definite, so we can write

 (1.86)

where use has been made of Equation (1.79). The previous expression is termed the uncertainty relation. According to this relation, an exact knowledge of the value of (i.e., ) implies no knowledge whatsoever of the value of (i.e., ), and vice versa. The one exception to this rule is when and commute, in which case exact knowledge of does not necessarily imply no knowledge of .

Next: Continuous Spectra Up: Fundamental Concepts Previous: Compatible Observables
Richard Fitzpatrick 2016-01-22