Page 99 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

90 General principles of quantum theory

jØiˆ c(ë)jëi dë

where

c(ë) ˆhëjØi
and the expectation value of A takes the form

2
hAiˆ jc(ë)j ë dë (3:53)
If dP ë is the probability of obtaining a value of A between ë and ë ‡ dë, then
equation (3.47) is replaced by

hAiˆ ë dP ë

and we see that
2
2
dP ë ˆjc(ë)j dë ˆjhëjØij dë
The probability dP ë is often written in the form
dP ë ˆ r(ë)dë
where r(ë)isthe probability density of obtaining the result ë and is given by
2 2
r(ë) ˆjc(ë)j ˆjhëjØij
In terms of the probability density, equation (3.53) becomes

hAiˆ ër(ë)dë (3:54)

^
In some applications to physical systems, the eigenkets of A possess a
partially discrete and a partially continuous spectrum, in which case equations
(3.51) and (3.53) must be combined.
The scalar product hØjØi may be evaluated from equations (3.48) and
(3.50) as

g j g i g i
X X X X X X
2
hØjØiˆ c c iá hjâjiáiˆ jc iá j
jâ
j âˆ1 i áˆ1 i áˆ1
g i
X X X
2
ˆ jhiájØij ˆ P i
i áˆ1 i
Since the state vector Ø is normalized, this expression gives
X
P i ˆ 1
i
Thus, the sum of the probabilities P i equals unity as it must from the de®nition
of probability. For a continuous set of eigenkets, this relationship is replaced by