Page 97 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

88 General principles of quantum theory

hxiˆhØjxjØi
* +

" @
hp x iˆ Ø Ø
i @x
hriˆhØjrjØi
* +

"

hpiˆ Ø = Ø
i

* +
2
"
2

E ˆhHiˆ Ø ÿ = ‡ V(r) Ø

2m
The expectation value hAi is not the result of a single measurement of the
property A, but rather the average of a large number (in the limit, an in®nite
number) of measurements of A on systems, each of which is in the same state
Ø. Each individual measurement yields one of the eigenvalues ë i , and hAi is
then the average of the observed array of eigenvalues. For example, if the
eigenvalue ë 1 is observed four times, the eigenvalue ë 2 three times, the
eigenvalue ë 3 once, and no other eigenvalues are observed, then the expectation
value hAi is given by

4ë 1 ‡ 3ë 2 ‡ ë 3
hAiˆ
8
In practice, many more than eight observations would be required to obtain a
reliable value for hAi.
In general, the expectation value hAi of the observable A may be written for
a discrete set of eigenfunctions as

X
hAiˆ P i ë i (3:47)
i
where P i is the probability of obtaining the value ë i . If the state function Ø for
a system happens to coincide with one of the eigenstates jii, then only the
eigenvalue ë i would be observed each time a measurement of A is made and
therefore the expectation value hAi would equal ë i
^
hAiˆhijAjiiˆhijë i jiiˆ ë i
It is important not to confuse the expectation value hAi with the time average
of A for a single system.
For an arbitrary state Ø at a ®xed time t, the ket jØi may be expanded in
^
terms of the complete set of eigenkets of A. In order to make the following
discussion clearer, we now introduce a slightly more complicated notation.
Each eigenvalue ë i will now be distinct, so that ë i 6ˆ ë j for i 6ˆ j. We let g i be