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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
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
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