2.5 Problems
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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
C =
|a|e iα |b|e iβ
|c|e iγ |d|e iδ
Impose now the requirement that this matrix is unitary. This will introduce relationships 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, 2π[.
Normalize these waves over the interval. Are they mutually orthogonal?
2.3 A matrix H which is equal to its complex-conjugate transpose, H = ¯
H T ,i s
called Hermitian. It follows that the diagonal elements of such a matrix are
real, while corresponding off-diagonal elements form complex-conjugate pairs:
H Hermitian → H ii ∈ R; H ij = ¯
H ji
Prove that the eigenvalues of a Hermitian matrix are real. If the matrix is skewHermitian, 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)