H Solutions to Problems
247
The normalized cyclic waves are thus given by
|±k=
1
√
2π
e
±ikφ
and these waves are orthogonal: −k|k=0.
2.3 The combination of transposition and complex conjugation is called the adjoint
operation, indicated by a dagger. A Hermitian matrix is thus self-adjoint. An
eigenfunction of this matrix, operating in a function space, may be expressed as
a linear combination
|ψ m =
k
c k |f k
We may arrange the expansion coefficients as a column vector c. This is called
the eigenvector. Its adjoint, c † , is then the complex-conjugate row vector. The
corresponding eigenvalue is denoted as E m . Now start by writing the eigenvalue
equation and multiply left and right with the adjoint eigenvector:
Hc = E m c
c
† Hc = E m c
† c
Now take the adjoint and use the self-adjoint property of H:
c
† H
† c = ¯
E m c
† c
c
† Hc = ¯
E m c
† c
A comparison of both results shows that the eigenvalue must be equal to its
complex conjugate and hence be real. If H is skew-symmetric, a similar argument shows that the eigenvalue must be imaginary.
3.1 The table is a valid multiplication table of a group that is isomorphic to D 2 .The
element C is the unit element. There are six ways to assign the three twofold
axes to the letters A, B, D.
3.2 Any nonlinear triatomic molecule with three different atoms has only C s symmetry, e.g., a water molecule with one hydrogen replaced by deuterium. C 2
symmetry requires a nonplanar tetra-atomic molecule, such as H 2 O 2 .I nt h e
free state the dihedral angle of this molecule is almost a right angle (see the
figure). To realize C i symmetry, one needs at least six atoms. Since three atoms
are always coplanar, the smallest molecule with no symmetry at all has at least
four atoms.
247
The normalized cyclic waves are thus given by
|±k=
1
√
2π
e
±ikφ
and these waves are orthogonal: −k|k=0.
2.3 The combination of transposition and complex conjugation is called the adjoint
operation, indicated by a dagger. A Hermitian matrix is thus self-adjoint. An
eigenfunction of this matrix, operating in a function space, may be expressed as
a linear combination
|ψ m =
k
c k |f k
We may arrange the expansion coefficients as a column vector c. This is called
the eigenvector. Its adjoint, c † , is then the complex-conjugate row vector. The
corresponding eigenvalue is denoted as E m . Now start by writing the eigenvalue
equation and multiply left and right with the adjoint eigenvector:
Hc = E m c
c
† Hc = E m c
† c
Now take the adjoint and use the self-adjoint property of H:
c
† H
† c = ¯
E m c
† c
c
† Hc = ¯
E m c
† c
A comparison of both results shows that the eigenvalue must be equal to its
complex conjugate and hence be real. If H is skew-symmetric, a similar argument shows that the eigenvalue must be imaginary.
3.1 The table is a valid multiplication table of a group that is isomorphic to D 2 .The
element C is the unit element. There are six ways to assign the three twofold
axes to the letters A, B, D.
3.2 Any nonlinear triatomic molecule with three different atoms has only C s symmetry, e.g., a water molecule with one hydrogen replaced by deuterium. C 2
symmetry requires a nonplanar tetra-atomic molecule, such as H 2 O 2 .I nt h e
free state the dihedral angle of this molecule is almost a right angle (see the
figure). To realize C i symmetry, one needs at least six atoms. Since three atoms
are always coplanar, the smallest molecule with no symmetry at all has at least
four atoms.