Chemistry Reference
In-Depth Information
(iii) Spheroidal coordinates
ðm; n; wÞ
:
r
A
þ
r
B
R
ð
1
m ¥Þ;
r
A
r
B
R
ð
1
n
1
Þ; wð
0
m ¼
n ¼
;
2
pÞ
ð
1
:
56
Þ
q
ðm
q
ðm
R
2
R
2
2
2
2
2
x
¼
1
Þð
1
n
Þ
cos
w;
y
¼
1
Þð
1
n
Þ
sin
w;
R
2
ðmn þ
1
Þ
z
¼
ð
1
:
57
Þ
3
ðm
R
2
2
2
dr
¼
n
Þ
d
m
d
n
d
w
ð
1
:
58
Þ
s
m
s
1
n
"
#
2
2
2
R
e
m
1
@
@m
þ
e
n
@
@n
þ
e
w
1
ðm
@
@w
p
r¼
m
2
n
2
m
2
n
2
2
1
Þð
1
n
2
Þ
ð
1
:
59
Þ
4
2
r
¼
R
2
ðm
2
n
2
Þ
2
2
2
@
@m
1
Þ
@
@m
þ
@
@n
Þ
@
@n
m
n
@
2
2
ðm
ð
1
n
þ
ðm
2
1
Þð
1
n
2
Þ
@w
2
ð
1
:
60
Þ
Equations (1.44)-(1.55) are used in atomic (one-centre) calculations,
whereas Equations (1.56)-(1.60) are used in molecular (at least two-
centre) calculations.
1.3 BASIC POSTULATES
We now formulate in an axiomaticway the basis of quantummechanics in
the form of three postulates.
1.3.1 Correspondence between Physical Obervables
and Hermitian Operators
In coordinate space, we have the basic correspondences
r
¼
ix
þ
jy
þ
kz
r
¼
r
p
¼
ip
x
þ
jp
y
þ
kp
z
) ^
) ^
ð
1
:
61
Þ
p
¼
i
h
r
where i is the imaginary unit (i
2
is the reduced
Planck constant. More complex observables can be treated by repeated
¼
1) and
h
¼
h
=
2
p
