Letf (x)be a continuous function in the interval[a, b]. Dirichlet’s equation [28,29]
κlim→∞
b
a
dxsin[κ(x−x)]
π(x−x) f (x)=
0, x /∈ [a, b],
f (x), x∈(a, b), (2.1) motivates the formal definition
δ(x)= lim
κ→∞
sin(κx) π x = lim
κ→∞
1 2π
κ
−κ
dkexp(ikx)
= 1 2π
∞
−∞dkexp(ikx), (2.2)
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020
R. Dick,Advanced Quantum Mechanics, Graduate Texts in Physics, https://doi.org/10.1007/978-3-030-57870-1_2
25
such that Eq. (2.1) can (in)formally be written as b
a
dxδ(x−x)f (x)=
0, x /∈ [a, b],
f (x), x∈(a, b). (2.3) A justification for Dirichlet’s equation is given below in the derivation of Eq. (2.19).
The generalization to three dimensions follows immediately from Dirichlet’s for- mula in a three-dimensional cube, and exhaustion of an arbitrary three-dimensional volumeV by increasingly finer cubes. This yields
δ(x)= 3 i=1
κilim→∞
sin(κixi) π xi = 1
(2π )3
d3kexp(ikãx), (2.4)
V
d3xδ(x−x)f (x)=
0, x∈/V ,
f (x),x insideV . (2.5) The casex ∈ ∂V (x on the boundary ofV) must be analyzed on a case-by-case basis.
Equation (2.4) implies ψ(x, t )=
d3xδ(x−x)ψ(x, t )
= 1 (2π )3
d3kexp(ikãx)
d3x exp(−ikãx)ψ(x, t ). (2.6) This can be used to introduce Fourier transforms by splitting the previous equation into two equations,
ψ(x, t )= 1
√2π3
d3kexp(ikãx)ψ(k, t ), (2.7)
with
ψ(k, t )= 1
√2π3
d3x exp(−ikãx)ψ(x, t ). (2.8) Use ofψ(x, t )corresponds to the x-representation of quantum mechanics. Use of ψ(k, t ) corresponds to the k-representation or momentum-representation of quantum mechanics.
The notation above for Fourier transforms is a little sloppy, but convenient and common in quantum mechanics. From a mathematical perspective, the Fourier transformed function ψ(k, t ) should actually be denoted by ψ(k, t )˜ to make it clear that it isnotthe same function asψ(x, t )with different symbols for the first three variables. The physics notation is motivated by the observation thatψ(x, t )
2.1 TheδFunction and Fourier Transforms 27 and ψ(k, t ) are just different representations of the same quantum mechanical stateψ.
Another often used convention for Fourier transforms is to split the factor(2π )−3 asymmetrically, or equivalently replace it with a factor 2πin the exponents,
ψ(x, t )= 1 (2π )3
d3kexp(ikãx)ψ(k, t ), (2.9)
ψ(k, t )=
d3x exp(−ikãx)ψ(x, t ), (2.10) or equivalently
ψ(x, t )=
d3ν˜ exp(2πi˜νãx)ψ(ν, t ),˜ (2.11)
ψ(ν, t )˜ =
d3x exp(−2πiν˜ãx)ψ(x, t ), (2.12) with the vector of wave numbers
˜ ν= k
2π. (2.13)
The conventions (2.7) and (2.8) are used throughout this book.
The following is an argument for Eq. (2.1) and its generalizations to other representations of the δ function. The idea is to first construct a limit for the Heaviside step function orfunction
(x)=
1, x >0,
0, x <0, (2.14)
and go from there. The value of (0) is often chosen to suite the needs of the problem at hands. The choice(0) = 1/2 seems intuitive and is also mathemat- ically natural in the sense that any decomposition of a discontinuous functions in a complete set of functions (e.g. Fourier decomposition) will approximate the mean value between the left and right limit for a finite discontinuity, but in many applications other values of(0)are preferred.
The function helps us to explain Dirichlet’s equation (2.1) through the following construction. Supposed(x)is a normalized function,
∞
−∞dx d(x)=1. (2.15)
The integral
D(x)= x
−∞dξ d(ξ ) (2.16)
satisfies
κlim→∞D(κãx)=(x), (2.17)
where we apparently defined(0)as(0)= 0
−∞dξ d(ξ ), but this plays no role for the following reasoning.
Equation (2.17) yields for every function f (x) which is differentiable in the interval[a, b]the equation
b a
dx κ d(κãx)f (x)=D(κãx)f (x)b
a− b
a
dx D(κãx)f(x), (2.18) and therefore
κlim→∞
b
a
dx κ d(κãx)f (x)=(b)f (b)−(a)f (a)− b
a
dx (x)f(x)
=(b)f (b)−(a)f (a)−(b)[f (b)−f (0)] +(a)[f (a)−f (0)]
= [(b)−(a)]f (0), (2.19)
where we simply split the integral according to b
a
dx (x)f(x)= b
0
dx (x)f(x)− a
0
dx (x)f(x) (2.20) to arrive at the final result. Equation (2.19) confirms
κlim→∞κ d(κx)=δ(x), (2.21) or after shifting the argument,
κlim→∞κ d[κ(x−x0)] =δ(x−x0). (2.22) From a mathematical perspective, equations like (2.21) mean that the action of theδ distribution on a smooth funtion corresponds to integration with a kernel κ d(κx)and then taking the limitκ → ∞.
Equation (2.2) is an important particular realization of Eq. (2.21) with the normalized sinc function d(x) = sinc(x)/π = sin(x)/π x. Another important realization uses the functiond(x)=(π+π x2)−1,
2.1 TheδFunction and Fourier Transforms 29
δ(x)= lim
κ→∞
1 π
κ
1+κ2x2 = lim
a→0
1 π
a a2+x2
= lim
a→0
1 2π
∞
−∞dkexp(ikx−a|k|). (2.23) Note that we did not required(x)to have a maximum atx =0 to derive (2.21), and indeed we do not need this requirement. Consider the following example,
d(x)=1 2
α
πexp[−α(x−a)2] +1 2
β
π exp[−β(x−b)2]. (2.24) This function has two maxima ifαãβ =0 and ifaandbare sufficiently far apart, and it even has a minimum atx=0 ifα=β anda= −b. Yet we still have
κlim→∞κ d(κãx)= lim
κ→∞
κ 2
α
πexp[−α(κx−a)2] +κ
2 β
π exp[−β(κx−b)2]
=δ(x), (2.25)
because the scaling withκ scales the initial maxima nearaandbtoa/κ →0 and b/κ→0.
Sokhotsky–Plemelj Relations
The Sokhotsky–Plemelj relations are very useful relations involving a δ distribu- tion,1
1
x−i =P1
x +iπ δ(x), 1
x+i =P1
x −iπ δ(x). (2.26)
Indeed, for the practical evaluation of integrals involving singular denominators, we virtually never use these relations but evaluate the integrals with the left-hand sides directly using the Cauchy and residue theorems. The primary use of the Sokhotsky–Plemelj relations in physics and technology is to establish relations between different physical quantities. The relation between retarded Green’s func- tions and local densities of states is an example for this and will be derived in Sect.21.1.
1Sokhotsky [161], Plemelj [135]. The “physics” version (2.26) of the Sokhotsky–Plemelj relations is of course more recent than the original references because theδdistribution was only introduced much later.
I will give a brief justification for the Sokhotsky–Plemelj relations. The relations 1
x+i = 1 i
∞
0
dkexp[ik(x+i)] = 1 i
0
−∞dk exp[−ik(x+i)] (2.27) imply
1
x+i = −1 2
∞
−∞dkcos(kx)= −π δ(x). (2.28) On the other hand, the real part is
1
x+i = 1
2(x+i)+ 1
2(x−i) = x
x2+2. (2.29) This implies for integration with a bounded functionf (x)in[a, b]
b a
dx f (x) x+i =
b a
dx xf (x)
x2+2 −iπ[(b)−(a)]f (0). (2.30) However, the weight factor
K(x)= x
x2+2 (2.31)
esentially cuts the region −3 < x < 3 symmetrically from the integral b
adx f (x)/x (the value 3 is chosen becausexK(x) = 0.9 forx = ±3), see Fig.2.1. Therefore we can use this factor as one possible definition of a principal value integral,
P b
a
dx f (x) x =lim
→0
b
a
dx K(x)f (x). (2.32)