2.2 Linear Operators and Transformation Matrices
13
i.e.,
ˆ
R|f i =
n
j =1
D ji (R)|f j
(2.7)
When multiplying this equation left and right with a given bra-function in an orthonormal basis, one obtains
f k | ˆ
R|f i =
n
j =1
D ji (R)f k |f j =
n
j =1
D ji (R)δ kj = D ki (R)
(2.8)
where the summation index j has been restricted to k by the Kronecker delta. Hence,
the elements of the transformation matrix are recognized as matrix elements of the
symmetry operators. The transformation of bra-functions runs entirely parallel with
the transformation of ket-functions, except that the complex conjugate of the transformation matrix has to be taken, and hence,
ˆ
Rf i |=
n
j =1
¯
D ji (R)f j |
(2.9)
For convenience, we sometimes abbreviate the row vector of the function space
as |f, so that the transformation is written as
R|f=|fD(R)
(2.10)
When the bra-functions are also ordered in a row vector, we likewise have:
ˆ
Rf|==f| ¯
D(R)
(2.11)
A product of two operators is executed consecutively, and hence the one closest to
the ket acts first. In detail,
ˆ
R ˆ
S|f i = ˆ
R
j
D ji (S)|f j
=
k,j
D kj (R)D ji (S)|f k
=
k
D(R) × D(S)
ki
|f k
(2.12)
Here, the symbol × refers to the product of two matrices.
ˆ
R ˆ
S|f=|fD(R) × D(S)
(2.13)
This is an important result. It shows that the consecutive action of two operators can
be expressed by the product of the corresponding matrices. The matrices are said to
13
i.e.,
ˆ
R|f i =
n
j =1
D ji (R)|f j
(2.7)
When multiplying this equation left and right with a given bra-function in an orthonormal basis, one obtains
f k | ˆ
R|f i =
n
j =1
D ji (R)f k |f j =
n
j =1
D ji (R)δ kj = D ki (R)
(2.8)
where the summation index j has been restricted to k by the Kronecker delta. Hence,
the elements of the transformation matrix are recognized as matrix elements of the
symmetry operators. The transformation of bra-functions runs entirely parallel with
the transformation of ket-functions, except that the complex conjugate of the transformation matrix has to be taken, and hence,
ˆ
Rf i |=
n
j =1
¯
D ji (R)f j |
(2.9)
For convenience, we sometimes abbreviate the row vector of the function space
as |f, so that the transformation is written as
R|f=|fD(R)
(2.10)
When the bra-functions are also ordered in a row vector, we likewise have:
ˆ
Rf|==f| ¯
D(R)
(2.11)
A product of two operators is executed consecutively, and hence the one closest to
the ket acts first. In detail,
ˆ
R ˆ
S|f i = ˆ
R
j
D ji (S)|f j
=
k,j
D kj (R)D ji (S)|f k
=
k
D(R) × D(S)
ki
|f k
(2.12)
Here, the symbol × refers to the product of two matrices.
ˆ
R ˆ
S|f=|fD(R) × D(S)
(2.13)
This is an important result. It shows that the consecutive action of two operators can
be expressed by the product of the corresponding matrices. The matrices are said to