Elements of Quantum Theory
73
It is natural to interpret this equation as implying that φ n describes a state with a
well-defined value E n for the observable corresponding to A. The eigenvalues
and the eigenstates of hermitian operators satisfy the following properties:
1. The eigenvalues of a hermitian operator are real. This follows from
Eq. (3.35), If φ and ψ are taken to be the same eigenstates.
2. Eigenstates with different eigenvalues are orthogonal in the following sense.
In Eq. (3.35), if φ and ψ are eigenstates φ n and φ m of A with eigenvalues
E n and E m respectively, then
(E n – E m ) ∫φ n * φ m dτ = 0
(3.37)
or
∫ φ n * φ m dτ = 0 for E n ≠ E m
(3.38)
The states φ n and φ m are said to be orthogonal to each other. It is also
possible in the case of discrete eigenvalues to normalize the eigenstates
such that
∫φ n * φ n = 1
(3.39)
The states which satisfy the relations (3.38) and (3.39) are said to be
orthonormal. It may happen that there are more than one states which
have the same eigenvalue. These states are said to be degenerate and
the number of degenerate states is known as the degree of degeneracy
of the eigenvalue. It is possible to normalize these states and to choose a
suitable, orthonormal set of degenerate states.
3. The eigenstates of a hermitian operator are complete, and form a complete
basis. This means that any state ψ can be expressed as a linear combination
of the eigenstates φ n of a hermitian operator,
ψ =
n n
n
a φ
∑
(3.40)
The summation may include an integration over a set of states with
continuum eigenvalues.
4. Two operators A and B are said to commute if
[A, B] ≡ AB – BA = 0
(3.41)
where [A, B] is called the commutator of A and B. It is possible to choose,
as a basis, states which are simultaneous eigenstates of commuting hermitian
operators. A particularly important case is obtained if one of the operators
is the Hamiltonian (i.e. energy operator),
[H, B] = 0
(3.42)
Then the states can be chosen to be simultaneous eigenstates of H and B.
Since the eigenstates of a time-independent Hamiltonian do not change
73
It is natural to interpret this equation as implying that φ n describes a state with a
well-defined value E n for the observable corresponding to A. The eigenvalues
and the eigenstates of hermitian operators satisfy the following properties:
1. The eigenvalues of a hermitian operator are real. This follows from
Eq. (3.35), If φ and ψ are taken to be the same eigenstates.
2. Eigenstates with different eigenvalues are orthogonal in the following sense.
In Eq. (3.35), if φ and ψ are eigenstates φ n and φ m of A with eigenvalues
E n and E m respectively, then
(E n – E m ) ∫φ n * φ m dτ = 0
(3.37)
or
∫ φ n * φ m dτ = 0 for E n ≠ E m
(3.38)
The states φ n and φ m are said to be orthogonal to each other. It is also
possible in the case of discrete eigenvalues to normalize the eigenstates
such that
∫φ n * φ n = 1
(3.39)
The states which satisfy the relations (3.38) and (3.39) are said to be
orthonormal. It may happen that there are more than one states which
have the same eigenvalue. These states are said to be degenerate and
the number of degenerate states is known as the degree of degeneracy
of the eigenvalue. It is possible to normalize these states and to choose a
suitable, orthonormal set of degenerate states.
3. The eigenstates of a hermitian operator are complete, and form a complete
basis. This means that any state ψ can be expressed as a linear combination
of the eigenstates φ n of a hermitian operator,
ψ =
n n
n
a φ
∑
(3.40)
The summation may include an integration over a set of states with
continuum eigenvalues.
4. Two operators A and B are said to commute if
[A, B] ≡ AB – BA = 0
(3.41)
where [A, B] is called the commutator of A and B. It is possible to choose,
as a basis, states which are simultaneous eigenstates of commuting hermitian
operators. A particularly important case is obtained if one of the operators
is the Hamiltonian (i.e. energy operator),
[H, B] = 0
(3.42)
Then the states can be chosen to be simultaneous eigenstates of H and B.
Since the eigenstates of a time-independent Hamiltonian do not change