224
Appendix C: The Dirac δ Function and the Normalization of Continuum States
b
a
δ(y(x)) f (x) dx =
i
f (x i )
dy
dx
−1
x=x i
(C.7)
where the index i runs over all the zeros x i of the function y(x) with x i ∈ (a, b).
The zeros must be nondegenerate, i.e., (dy/dx) x=x i = 0. If a zero falls at one of the
integration limits (x i = a or x i = b), we can apply Eq. (C.7) by counting only half
of the contribution relative to x i .
x
−∞
δ(x
) dx
= H (x)
(C.8)
where H is the Heaviside function
H (x) = 0 ∀x < 0
H (x) = 1 ∀x > 0
(C.9)
For a more rigorous treatment of distributions and of the δ function, see textbooks
of mathematics for physicists such as Dennery and Krzywicki [1] or Arfken, Weber
and Harris [2].
In the previous section we introduced Dirac’s notation for the elements of a vector
space and in particular for the quantum states. The eigenstates of an operator ˆ
O are
associated with the respective eigenvalues, although in case of degeneracy the same
eigenvalue is associated with more than one eigenstate. The degeneracy ambiguity
is solved by identifying a complete set of commuting observables. Another complication is that, in many cases, we have to deal with sets of eigenvalues that are
continuous, i.e., can be any real number in a certain interval. For instance, the eigenvalues of the Hamiltonian for a molecular system are in part made of discrete energy
levels (those corresponding to bound states) and in part by two kinds of continua: the
dissociative continuum, made of the vibrational states above the dissociation threshold reached when two or more groups of atoms get far apart from each other, and the
electronic continuum, corresponding to ionization.
When the eigenvalue spectrum is continuous, we see that we cannot number the
eigenstates as in Eq. (B.19). Leaving aside for simplicity the degenerate case, we can
identify each eigenstate by its eigenvalue:
ˆ
O |ψ λ = λ |ψ λ
(C.10)
Now, if we want to generalize the resolution of identity, Eq. (B.14), we can write
ˆ
E =
n
i=1
|ψ i ψ i | +
λ max
λ min
|ψ λ ψ λ | dλ
(C.11)
Here the index i runs over the states of the discrete spectrum, if any, while the integration range covers the whole continuum spectrum (usually λ max = ∞). If |ψ i and
Appendix C: The Dirac δ Function and the Normalization of Continuum States
b
a
δ(y(x)) f (x) dx =
i
f (x i )
dy
dx
−1
x=x i
(C.7)
where the index i runs over all the zeros x i of the function y(x) with x i ∈ (a, b).
The zeros must be nondegenerate, i.e., (dy/dx) x=x i = 0. If a zero falls at one of the
integration limits (x i = a or x i = b), we can apply Eq. (C.7) by counting only half
of the contribution relative to x i .
x
−∞
δ(x
) dx
= H (x)
(C.8)
where H is the Heaviside function
H (x) = 0 ∀x < 0
H (x) = 1 ∀x > 0
(C.9)
For a more rigorous treatment of distributions and of the δ function, see textbooks
of mathematics for physicists such as Dennery and Krzywicki [1] or Arfken, Weber
and Harris [2].
In the previous section we introduced Dirac’s notation for the elements of a vector
space and in particular for the quantum states. The eigenstates of an operator ˆ
O are
associated with the respective eigenvalues, although in case of degeneracy the same
eigenvalue is associated with more than one eigenstate. The degeneracy ambiguity
is solved by identifying a complete set of commuting observables. Another complication is that, in many cases, we have to deal with sets of eigenvalues that are
continuous, i.e., can be any real number in a certain interval. For instance, the eigenvalues of the Hamiltonian for a molecular system are in part made of discrete energy
levels (those corresponding to bound states) and in part by two kinds of continua: the
dissociative continuum, made of the vibrational states above the dissociation threshold reached when two or more groups of atoms get far apart from each other, and the
electronic continuum, corresponding to ionization.
When the eigenvalue spectrum is continuous, we see that we cannot number the
eigenstates as in Eq. (B.19). Leaving aside for simplicity the degenerate case, we can
identify each eigenstate by its eigenvalue:
ˆ
O |ψ λ = λ |ψ λ
(C.10)
Now, if we want to generalize the resolution of identity, Eq. (B.14), we can write
ˆ
E =
n
i=1
|ψ i ψ i | +
λ max
λ min
|ψ λ ψ λ | dλ
(C.11)
Here the index i runs over the states of the discrete spectrum, if any, while the integration range covers the whole continuum spectrum (usually λ max = ∞). If |ψ i and