Chemistry Reference
In-Depth Information
2.5 Problems
2.1 A complex number can be characterized by an absolute value and a phase.
A2
×
2 complex matrix thus contains eight parameters, say
|
a
|
e
iα
e
iδ
|
b
|
e
iβ
C=
e
iγ
|
c
|
|
d
|
Impose now the requirement that this matrix is unitary. This will introduce rela-
tionships between the parameters. Try to solve these by adopting a reduced set
of parameters.
2.2 The cyclic waves
e
ikφ
and
e
−
ikφ
are defined in a circular interval
φ
∈[
0
,
2
π
[
.
Normalize these waves over the interval. Are they mutually orthogonal?
2.3 A matrix
H = H
T
,is
called
Hermitian
. It follows that the diagonal elements of such a matrix are
real, while corresponding off-diagonal elements form complex-conjugate pairs:
H
which is equal to its complex-conjugate transpose,
H
ij
=
H
ji
H
→
H
ii
∈
R
;
Hermitian
Prove that the eigenvalues of a Hermitian matrix are real. If the matrix is
skew
-
Hermitian,
H=−H
T
, the eigenvalues are all imaginary.
References
1. Dirac, P.A.M.: The Principles of Quantum Mechanics. Clarendon Press, Oxford (1958)
2. Altmann, S.L.: Rotations, Quaternions, and Double Groups. Clarendon Press, Oxford (1986)
3. Wigner, E.P.: Group Theory. Academic Press, New York (1959)
