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 a Hermitian operator
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
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
which is analogous to
in Euclidian space. This inequality can be proved by noting that
where is any complex number. If takes the special value then the above inequality reduces to
which is the same as the Schwarz inequality.
Let us substitute
into the Schwarz inequality, where the blank ket stands for any general ket. We find
where use has been made of the fact that and are Hermitian operators. Note that
where the commutator, , and the anti-commutator, , are defined
The commutator is clearly anti-Hermitian,
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. It follows that the right-hand side of
consists of the sum of an imaginary and a real number. Taking the modulus squared of both sides gives
where use has been made of , etc. The final term on the right-hand side of the above expression is positive definite, so we can write
where use has been made of Equation (76). The above expression is termed the uncertainty relation. According to this relation, an exact knowledge of the value of implies no knowledge whatsoever of the value of , 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 .
- Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)