vector space can be written as
ðtÞ¼XN
n¼ 1cn (^) n, ½ 2 : 18
in which thecnare complex numbers and may depend upon time.
The eigenvalue equation [2.11] leads to the following interpretation
of the relationship between an operator and its associated observable:
the result of making a measurement ofAupon a system is one of the
eigenvalues of A. This statement illustrates the discrete nature of
quantum mechanics: only a limited set of outcomes is possible for
the measurement. In practice, however, the expectation value ofAis
measured experimentally. The expectation value is defined as the average
magnitude of a particular property obtained following a large number
of measurements of that property carried out over an ensemble of
identically prepared systems. The expectation value of some property,
hiA, is calculated mathematically as the scalar product of(t) andA(t),
hiA ¼
Z
ðtÞAðtÞd : ½ 2 : 19
If the wavefunction for the system is an eigenfunction of the operator,
(t)¼ (^) n, then
hiA ¼
Z
ðtÞAðtÞd ¼
Z
nA (^) nd ¼n
Z
n (^) nd ¼n: ½ 2 : 20
This result shows that if(t) is an eigenfunction of the operatorA,
then measuringAfor each member of the ensemble yields the identical
resultn. In general, the wavefunction for the system will not be an
eigenfunction ofA, and [2.18] is used to express [2.19] in terms of the
eigenfunctions ofA. The derivation ofhiA proceeds as follows:
hiA ¼
Z
ðtÞAðtÞd
¼
Z XN
i¼ 1
ci (^) i
"#
A
XN
j¼ 1
cj (^) j
"#
d ¼
Z XN
i¼ 1
ci i
"#
A
XN
j¼ 1
cj (^) j
"#
d
¼
XN
i¼ 1
XN
j¼ 1
cicj
Z
iA (^) jd ¼
XN
i¼ 1
XN
j¼ 1
cicjj
Z
i (^) jd
¼
XN
j¼ 1
cjcjj: ½ 2 : 21
2.1 POSTULATES OFQUANTUMMECHANICS 33