382
E. Dirac δ-Function
we can interpret it to mean that the wavefunctions e
ikx /(2π)
1/2 and e
ik
' x /(2π)
1/2
are orthogonal on the real axis −∞ ≤ x ≤ ∞ for k / k
′ (since the left-hand
=
side is zero), while for k = k
′ their overlap is infinite, in such a way that the
integral of this overlap is unity. This is the continuum analogue of orthonormality for wavefunctions labelled by a discrete index, as in (E.12). We say that
the plane waves in (E.26) are ‘normalized to a δ-function’. There is, however,
a problem with this: plane waves are not square integrable and thus do not
strictly belong to a Hilbert space. Mathematical physicists concerned with
such matters have managed to deal with this by introducing ‘rigged’ Hilbert
spaces in which such a normalization is legitimate. Although we often, in the
text, appear to be using ‘box normalization’ (i.e. restricting space to a finite
volume V ), in practice when we evaluate integrals over plane waves the limits
will be extended to infinity, and results like (E.26) will be used repeatedly.
Important three- and four-dimensional generalizations of (E.25) are:
∫
e
ik·x d
3
k = (2π)
3 δ(x)
(E.27)
and
∫
e
ik·x d
4 k = (2π)
4 δ(x)
(E.28)
where k · x = k
0 x
0
− k · x (see appendix D) and δ(x) = δ(x
0 )δ(x).
Properties of the δ-function
The basic properties of the δ-function are exemplified by the equations (see
(E.5) and (E.6))
∫ ∞
δ(x − a) dx = 1, δ(x − a) = 0 for x = / a,
(E.29)
−∞
where a is any real number; and
∫ ∞
f (x) δ(x − a) dx = f (a),
(E.30)
−∞
where f (x) is any continuous function of x. Other useful properties follow:
1
(i)
δ(ax) =
δ(x).
(E.31)
|a|
Proof
For a > 0,
∫
∫
∞
∞
dy
1
δ(ax) dx =
δ(y)
= ;
(E.32)
a
a
for a < 0,
−∞
−∞
∫
∫
∫
∞
−∞
∞
dy
dy
1
δ(ax) dx =
δ(y)
=
δ(y)
=
.
(E.33)
a
|a|
|a|
−∞
∞
−∞
E. Dirac δ-Function
we can interpret it to mean that the wavefunctions e
ikx /(2π)
1/2 and e
ik
' x /(2π)
1/2
are orthogonal on the real axis −∞ ≤ x ≤ ∞ for k / k
′ (since the left-hand
=
side is zero), while for k = k
′ their overlap is infinite, in such a way that the
integral of this overlap is unity. This is the continuum analogue of orthonormality for wavefunctions labelled by a discrete index, as in (E.12). We say that
the plane waves in (E.26) are ‘normalized to a δ-function’. There is, however,
a problem with this: plane waves are not square integrable and thus do not
strictly belong to a Hilbert space. Mathematical physicists concerned with
such matters have managed to deal with this by introducing ‘rigged’ Hilbert
spaces in which such a normalization is legitimate. Although we often, in the
text, appear to be using ‘box normalization’ (i.e. restricting space to a finite
volume V ), in practice when we evaluate integrals over plane waves the limits
will be extended to infinity, and results like (E.26) will be used repeatedly.
Important three- and four-dimensional generalizations of (E.25) are:
∫
e
ik·x d
3
k = (2π)
3 δ(x)
(E.27)
and
∫
e
ik·x d
4 k = (2π)
4 δ(x)
(E.28)
where k · x = k
0 x
0
− k · x (see appendix D) and δ(x) = δ(x
0 )δ(x).
Properties of the δ-function
The basic properties of the δ-function are exemplified by the equations (see
(E.5) and (E.6))
∫ ∞
δ(x − a) dx = 1, δ(x − a) = 0 for x = / a,
(E.29)
−∞
where a is any real number; and
∫ ∞
f (x) δ(x − a) dx = f (a),
(E.30)
−∞
where f (x) is any continuous function of x. Other useful properties follow:
1
(i)
δ(ax) =
δ(x).
(E.31)
|a|
Proof
For a > 0,
∫
∫
∞
∞
dy
1
δ(ax) dx =
δ(y)
= ;
(E.32)
a
a
for a < 0,
−∞
−∞
∫
∫
∫
∞
−∞
∞
dy
dy
1
δ(ax) dx =
δ(y)
=
δ(y)
=
.
(E.33)
a
|a|
|a|
−∞
∞
−∞