Chemistry Reference
In-Depth Information
; ...;
n. Then, if A is a
n-dimensional set of basis functions
fw
k
ð
x
Þg
k
¼
1
Hermitian operator:
A
w
i
ð
x
Þ¼
X
k
w
k
ð
x
Þ
A
ki
¼
X
jw
k
ihw
k
j
A
w
i
i
ð
1
:
25
Þ
k
where the expansion coefficients now have two indices and are the
elements of the square matrix A (order n):
ð
dx
0
A
ki
¼hw
k
j
A
w
i
i¼
k
ð
x
0
Þð
A
w
i
ð
x
0
ÞÞ
w
ð
1
:
26
Þ
0
1
A
11
A
12
A
1n
@
A
¼
w
A
w
A
21
A
22
A
2n
f
A
ki
g
Y
A
¼
ð
1
:
27
Þ
A
n1
A
n2
A
nn
which is called the matrix representative of the operator A in the basis
fw
k
g
, and we use matrix multiplication rules (Chapter 2). In this way,
the eigenvalue equations of quantum mechanics transform into eigen-
value equations for the corresponding representative matrices. We
must recall, however, that a complete set implies matrices of infinite
order.
Under a unitary transformation U of
the basis functions
w
¼
ðw
1
w
2
...w
n
Þ
:
0
¼
w
U
ð
1
:
28
Þ
w
the representative A of the operator A is changed into
A
w
A
0
¼
w
0
0
¼
U
AU
ð
1
:
29
Þ
1.2.14 Properties of the Operator
r
We have seen that in Cartesian coordinates the vector operator
r
(the
gradient, a vector whose components are operators) is defined as
(Rutherford, 1962)
r¼
i
@
@
x
þ
j
@
@
y
þ
k
@
ð
1
:
30
Þ
@
z
