[PDF] [PDF] Dirac Delta Function

[PDF] Dirac Delta Function

Dirac Delta Function 1 Definition Dirac's delta function is defined by the following property δ(t) = { 0 t = 0 ∞ t = 0 (1) with ∫ t2 t1 dtδ(t) = 1 (2) if 0 ∈ [t1,t2 ] (and 

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of the Kronecker delta δmn, and thus to permit unified discussion of discrete that the “delta function”—which he presumes to satisfy the conditions ∫ +∞ − ∞

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Appendix A

Dirac Delta Function

In 1880 the self-taught electrical scientist Oliver Heaviside introduced the following function ?(x)=?1forx>0

0forx<0(A.1)

which is now called Heaviside step function. This is a discontinous function, with a discontinuity of first kind (jump) atx=0, which is often used in the context of the analysis of electric signals. Moreover, it is important to stress that the Haviside step function appears also in the context of quantum statistical physics. In fact, the Fermi-Dirac function (or Fermi-Dirac distribution) Fβ (x)=1 e x+1,(A.2) proposed in 1926 by Enrico Fermi and Paul Dirac to describe the quantum statistical distribution of electrons in metals, whereβ=1/(k B

T)is the inverse of the absolute

temperatureT(withkB the Boltzmann constant) andx=?-μis the energy?of the electron with respect to the chemical potentialμ, becomes the function?(-x in the limit of very small temperatureT, namely limβ→+∞ F (x)=?(-x)=?0forx> 0

1forx<0.(A.3)

Inspired by the work of Heaviside, with the purpose of describing an extremely localized charge density, in 1930 Paul Dirac investigated the following "function"

δ(x)=?

+∞forx=0

0forx?=0(A.4)L. Salasnich,Quantum Physics of Light and Matter, UNITEXT for Physics, 179

DOI: 10.1007/978-3-319-05179-6, © Springer International Publishing Switzerland 2014

180 Appendix A: Dirac Delta Function

imposing that?

δ(x)dx=1.(A.5)

Unfortunately, this property ofδ(x)is not compatible with the definition (A.4). In fact, from Eq. (A.4) it follows that the integral must be equal to zero. In other words, it does not exist a functionδ(x)which satisfies both Eqs. (A.4) and (A.5). Dirac suggested that a way to circumvent this problem is to interpret the integral of Eq. (A.5)as

δ(x)dx=lim

?→0 (x)dx,(A.6) whereδ (x)is a generic function of bothxand?such that lim ?→0 (x)=?+∞forx=0

0forx?=0,(A.7)

(x)dx=1.(A.8) not a function) which satisfy Eqs. (A.4) and (A.5) with the caveat that the integral in Eq. (A.5) must be interpreted according to Eq. (A.6) where the functionsδ (x) satisfy Eqs. (A.7) and (A.8).

Thereareinfinitefunctionsδ

there is, for instance, the following Gaussian (x)=1 ?⎷πe -x 2 2 ,(A.9) which clearly satisfies Eq. (A.7) and whose integral is equal to 1 for any value of?.

Another example is the function

(x)=? 1

0for|x|>?/2,(A.10)

which again satisfies Eq. (A.7) and whose integral is equal to 1 for any value of?. In the following we shall use Eq. (A.10) to study the properties of the Dirac delta function. as the limit of the corresponding integral involvingδ (x), namely

δ(x)f(x)dx=lim

?→0 (x)f(x)dx,(A.11)

Appendix A: Dirac Delta Function 181

for any functionf(x). It is then easy to prove that

δ(x)f(x)dx=f(0).(A.12)

by using Eq. (A.10) and the mean value theorem. Similarly one finds

δ(x-c)f(x)dx=f(c).(A.13)

In particular

δ(-x)=δ(x),(A.14)

δ(ax)=1

|a|δ(x)witha?=0,(A.15)

δ(f(x))=?

i 1 |f (x i )|δ(x-x i )withf(x i )=0.(A.16) x. It is not difficult to define a Dirac delta functionδ (D) (r)in the case of a D- dimensional domainR D , wherer=(x 1 ,x 2 ,...,x D )?R D is a D-dimensional vector: (D) (r)=?+∞forr=0

0forr?=0(A.17)

and R D (D) (r)d D r=1.(A.18)

Notice that sometimesδ

(D) (r)is written using the simpler notationδ(r). Clearly, also in this case one must interpret the integral of Eq. (A.18)as R D (D) (r)d D r=lim ?→0 R D (D)? (r)d D r,(A.19) whereδ (D)? (r)is a generic function of bothrand?such that lim ?→0 (D)? (r)=?+∞forr=0

0forr?=0,(A.20)

lim ?→0 (D)? (r)d D r=1.(A.21)

182 Appendix A: Dirac Delta Function

Several properties ofδ(x)remain valid also forδ (D) (r). Nevertheless, some properties ofδ (D) (r)depend on the space dimensionD. For instance, one can prove the remarkable formula (D) (r)=? 1 2π 2 (ln|r|)forD=2 1

D(D-2)V

D 2 1 |r| D-2 forD≥3,(A.22) where? 2 2 ∂x 21
2 ∂x 22
2 ∂x 2D andV D D/2 /?(1+D/2)is the volume of a D-dimensional ipersphere of unitary radius, with?(x)the Euler Gamma function.

In the caseD=3 the previous formula becomes

(3) (r)=-1

4π?

2 ?1 |r|? ,(A.23) which can be used to transform the Gauss law of electromagnetism from its integral form to its differential form.

Appendix B

Fourier Transform

It was known from the times of Archimedes that, in some cases, the infinite sum of decreasing numbers can produce a finite result. But it was only in 1593 that the written as the infinite sum of power functions. This function is nothing else than the geometric series, given by 1 1-x= n=0 x n ,for|x|<1.(B.1) In 1714 Brook Taylor suggested that any real functionf(x)which is infinitely differentiable inx 0 and sufficiently regular can be written as a series of powers, i.e. f(x)= n=0 c n (x-x 0 n ,(B.2) where the coefficientsc n are given by c n =1 n!f (n) (x 0 ),(B.3) withf (n) (x)the n-th derivative of the functionf(x). The series (B.2) is now called Taylor series and becomes the so-called Maclaurin series ifx 0 =0. Clearly, the geometric series (B.1) is nothing else than the Maclaurin series, wherec n =1. We observe that it is quite easy to prove the Taylor series: it is sufficient to suppose that Eq. (B.2) is valid and then to derive the coefficientsc n by calculating the derivatives off(x)atx=x 0 ; in this way one gets Eq. (B.3). In 1807 Jean Baptiste Joseph Fourier, who was interested on wave propagation and periodic phenomena, found that any sufficiently regular real function function f(x)which is periodic, i.e. such that L. Salasnich,Quantum Physics of Light and Matter, UNITEXT for Physics, 183 DOI: 10.1007/978-3-319-05179-6, © Springer International Publishing Switzerland 2014

184 Appendix B: Fourier Transform

f(x+L)=f(x),(B.4) whereLis the periodicity, can be written as the infinite sum of sinusoidal functions, namely f(x)=a 0 2+ n=1 a n cos? n2π Lx? +b n sin? n2π Lx?? ,(B.5) where a n =2 L? L/2 -L/2 f(y)cos? n2π Ly? dy,(B.6) b n =2 L? L/2 -L/2 f(y)sin? n2π Ly? dy.(B.7) it is sufficient to suppose that Eq. (B.5) is valid and then to derive the coefficients a n andb n by multiplying both side of Eq. (B.5) by cos?n 2π L x?and cos?n 2π L x? respectively and integrating over one periodL; in this way one gets Eqs. (B.6) and (B.7). It is important to stress that, in general, the real variablexof the functionf(x) can represent a space coordinate but also a time coordinate. In the former caseL gives the spatial periodicity and 2π/Lis the wavenumber, while in the latter caseL is the time periodicity and 2π/Lthe angular frequency.

Taking into account the Euler formula

e in 2π L x =cos? n2π Lx? +isin? n2πLx? (B.8) withi=⎷ Š1 the imaginary unit, Fourier observed that his series (B.5) can be re-written in the very elegant form f(x)= n=-∞ f n e in 2π L x ,(B.9) where f n =1 L? L/2 -L/2 f(y)e -in 2π L y dy(B.10) are complex coefficients, withf 0 =a 0 /2,f n =(a n -ib n )/2ifn>0 andf n (a -n +ib -n )/2ifn<0, thusf ?n =f -n The complex representation (B.9) suggests that the functionf(x)can be periodic but complex, i.e. such thatf:R→C. Moreover, one can consider the limit L→+∞of infinite periodicity, i.e. a function which is not periodic. In this limit

Appendix B: Fourier Transform 185

Eq. (B.9) becomes the so-called Fourier integral (or Fourier anti-transform) f(x)=1

2π?

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