Chemistry Reference
In-Depth Information
This result may be recast in a matrix multiplication as
A
ki
A
il
=
A
×A
kl
=
T
δ
kl
(2.21)
i
or
A
T
. In order to prove the unitary property, we also have to prove that the
matrix order in this multiplication may be switched. This is achieved
1
as follows:
×A=I
A
T
×A=I
A× A
T
×A=A
(2.22)
A× A
T
=A×A
−
1
A× A
T
=I
Left or right multiplication by
A
T
into the unit matrix, and
the inverse of a matrix is unique; it thus follows that the inverse of
thus turns the matrix
A
is obtained
by taking the complex-conjugate transposed form, which means that the matrix is
unitary. Note that the result in Eq. (
2.22
) is valid only if the matrix
A
A
is nonsingular.
However, this will certainly be the case since
det
A
det
(
)
T
det
(
2
A
)
=
A
=
1
(2.23)
Spatial symmetry operations are linear transformations of a coordinate function
space. When choosing the space in orthonormal form, symmetry operations will
conserve orthonormality, and hence all transformations will be carried out by unitary
matrices. This will be the case for all spatial representation matrices in this topic.
When all elements of a unitary matrix are real, it is called an
orthogonal
matrix. As
unitary matrices, orthogonal matrices have the same properties except that complex
conjugation leaves them unchanged. The determinant of an orthogonal matrix will
thus be equal to
±
1. The rotation matrices in Chap.
1
are all orthogonal and have
determinant
+
1.
2.4 Time Reversal as an Anti-linear Operator
The fact that an operator cannot change a scalar constant in front of the function on
which it operates seems to be evident. However, in quantum mechanics there is one
important operator that does affect a scalar constant and turns it into its complex
conjugate. This is the operator of
time reversal
, i.e., the operator which inverts time,
t
→−
t
, and sends the system back to its own past. If we are looking at a stationary
1
Adapted from: [
2
, Problem 8, p. 59].
