Chemistry Reference
In-Depth Information
Fig. 2.1
Matrix
representation of a group: the
operators (
left
) are mapped
onto the transformations
(
right
) of a function space.
The consecutive action of two
operators on
the left
(symbolized by
•
) is replaced
by the multiplication of two
matrices on
the right
(symbolized by
×
)
represent
the action of the corresponding operators. The relationship between both
is a mapping. In this mapping the operators are replaced by their respective matrices,
and the product of the operators is mapped onto the product of the corresponding
matrices. In this mapping the order of the elements is kept.
D
(RS)
=D
(R)
×D
(S)
(2.14)
In mathematical terms, such a mapping is called a
homomorphism
(see Fig.
2.1
). In
Eq. (
2.14
) both the operators and matrices that represent them are
right-justified
; that
is, the operator (matrix) on the
right
is applied first, and then the operator (matrix)
immediately to the left of it is applied to the result of the action of the right-hand op-
erator (matrix). The conservation of the order is an important characteristic, which
in the active picture entirely relies on the convention for collecting the functions in a
row vector. In the column vector notation the order would be reversed. Further con-
sequences of the homomorphism are that the unit element is represented by the unit
matrix,
I
, and that an inverse element is represented by the corresponding inverse
matrix:
=I
D
R
−
1
=
D
D
(E)
(2.15)
(R)
−
1
2.3 Unitary Matrices
A matrix is unitary if its rows and columns are orthonormal. In this definition the
scalar product of two rows (or two columns) is obtained by adding pairwise prod-
ucts of the corresponding elements,
A
ij
A
kj
, one of which is taken to be complex
conjugate:
A
ij
A
kj
=
A
ji
A
jk
=
δ
ik
↔A
is unitary
(2.16)
j
j
A unitary matrix has several interesting properties, which can easily be checked
from the general definition:
