The Dirac Delta function ( PDFDrive )

The Dirac Delta function

Ernesto Estévez Rams

estevez@imre.oc.uh.cu

Instituto de Ciencia y Tecnología de Materiales (IMRE)-Facultad de Física

Universidad de la Habana

IUCr International School on Crystallography, Brazil, 2012.

III Latinoamerican series

Introduction δ as a limit Properties Orthonormal Higher dimen. Recap Exercises Ref.

Outline

1 Introduction

Defining the Dirac Delta function

2 Dirac delta function as the limit of a family of functions

3 Properties of the Dirac delta function

4 Dirac delta function obtained from a complete set of

orthonormal functions

Dirac comb

5 Dirac delta in higher dimensional space

6 Recapitulation

7 Exercises

8 References


2 / 45


The Kronecker Delta

Suppose we have a sequence

of values {a 1, a 2, . . .} and we

wish to select algebraically a

particular value labeled by its

index i


3 / 45


The Kronecker Delta

Definition (Kronecker

delta)

K δ ij =

{ 1 i = j

0 i ≠ j

Picking one member of a set algebraically

∑

j=1

a K j δ ij = a i


4 / 45


The Kronecker Delta

Definition (Kronecker

delta)

K δ ij =

{ 1 i = j

0 i ≠ j

Picking one member of a set algebraically

∑

j=1

a K j δ ij = a i


4 / 45


Properties of the Kronecker Delta

∑

j K δ ij = 1: Normalization condition.

K δ ij

= K δ ji : Symmetry property.


5 / 45


Properties of the Kronecker Delta

∑

j K δ ij = 1: Normalization condition.

K δ ij

= K δ ji : Symmetry property.


5 / 45


Spotting a point in the mountain profile

We want to pick up

just a narrow window

of the whole view


6 / 45


How to deal with continuous functions ?

We want to do the

same with a

continuous function.


7 / 45



Consider a function f(x) continuous in the interval (a, b) and suppose we

want to pick up algebraically the value of f(x) at a particular point labeled

by x 0.

In analogy with the Kronecker delta let us define a selector function D δ(x)

with the following two properties:

∫ b

a f(x) D δ(x − x 0 )dx = f(x 0 ): Selector or sifting property

∫ b

a D δ(x − x 0 )dx = 1: Normalization condition


8 / 45



Consider a function f(x) continuous in the interval (a, b) and suppose we

want to pick up algebraically the value of f(x) at a particular point labeled

by x 0.

In analogy with the Kronecker delta let us define a selector function D δ(x)

with the following two properties:

∫ b

a f(x) D δ(x − x 0 )dx = f(x 0 ): Selector or sifting property

∫ b

a D δ(x − x 0 )dx = 1: Normalization condition


8 / 45



To be more general consider f(x) to be continuous in the interval (a, b)

except in a finite number of points where finite discontinuities occurs, then

the Dirac Delta can be defined as

Definition (Dirac delta function)

⎧

1

∫ 2 b

⎪⎨

[f(x− 0 ) + f(x+ 0 ] x0 ∈ (a, b)

1

f(x)δ(x − x 0)dx =

2 f(x+ 0 ) x0 = a

1

a

⎪⎩

2 f(x− 0 ) x0 = b

0 x 0 /∈ (a, b)


9 / 45


A bit of history

Siméon Denis Poisson (1781-1840)

In 1815 Poisson already for sees the δ(x − x 0) as a selector

function using for this purpose Lorentzian functions. Cauchy

(1823) also made use of selector function in much the same

way as Poisson and Fourier gave a series representation on

the delta function(More on this later).

Paul Adrien Maurice Dirac (1902-1984)

Dirac ”rediscovered” the delta function that now bears his

name in analogy for the continuous case with the Kronecker

delta in his seminal works on quantum mechanics.


10 / 45


What does it look like ?

δ p(x − x 0) =

{ p x ∈ (x0 − 1/2p, x 0 + 1/2p)

0 x /∈ (x 0 − 1/2p, x 0 + 1/2p)


11 / 45



p ∫ x 0 +1/2p

f(x)dx x 0 −1/2p


12 / 45



∫ ∞

−∞ δp(x) dx = ∫ 1/2p

−1/2p p dx = 1


{ ∞ x = x0

δ(x − x 0)dx =

0 x ≠ x 0

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{ ∞ x = x0

δ(x − x 0) =

0 x ≠ x 0

The above expression is just ”formal”, the δ(x) must be always understood in

the context of its selector property i.e. within the integral

δ(x) is defined more rigorously in terms of a distribution or a functional

(generalized function)


14 / 45


Dirac delta function as the limit of a family of functions

The Dirac delta function can be pictured as the limit in a sequence of

functions δ p which must comply with two conditions:

∫

lím ∞

p→∞ −∞ δp(x)dx = 1: Normalization condition

lím p→∞

δ p(x≠0)

lím x→0 δ = 0 Singularity condition. p(x)


15 / 45


Dirac delta function as the limit of a family of functions

The Dirac delta function can be pictured as the limit in a sequence of

functions δ p which must comply with two conditions:

∫

lím ∞

p→∞ −∞ δp(x)dx = 1: Normalization condition

lím p→∞

δ p(x≠0)

lím x→0 δ = 0 Singularity condition. p(x)


15 / 45


... as the limit of Gaussian functions

∆ px

δ p(x) Gaussian

family

pΠ

δ p(x) =

√

p

π exp (−px2 )

1 p

Π

1 p

1 p

x


16 / 45



Normalization condition

∫ ∞

−∞

√

p

δ p(x)dx =

π

∫ ∞

−∞

√ ∫

exp (−px 2 1 ∞

)dx = exp (−px 2 )d( √ px) =

π −∞

√ ∫ 1 ∞

√ ∫

exp (−t 2 1 ∞

)dt = 2 exp (−t 2 )dt

π

π

−∞

0

I = ∫ ∞

0

e −t2 dt

I 2 = ∫ ∞

0

e −y2 dy ∫ ∞

0

e −z2 dz = ∫ ∫ ∞

0

exp (y 2 + z 2 )dydz


17 / 45



⎧

⎨

⎩

r 2 = y 2 + z 2

y = r cos φ

z = r sin φ

⎫

⎬

⎭

I 2 = ∫ π/2

dφ ∫ ∞

∫

e −r2 rdr = π ∞

e −s ds = π 0 0 4 0 4

I = √ π

2

∫ ∞ δp(x)dx = 1 −∞


18 / 45



Singularity condition

lím

p→∞

δ p(x ≠ 0)

lím x→0 δ = p(x)

lím

p→∞

√ p

π e−px2

√ p

π

= 0


19 / 45


... as the limit of Lorentzian functions

∆ px

δ p(x) Lorentzian

family

δ p(x) = 1 p

p

p

Π

p

2Π

1 + p 2 x 2 x

1p

1p


20 / 45




∫ ∞

−∞

δ p(x)dx = 1 π

∫ ∞

−∞

pdx

1 + p 2 x = 1 ∫ ∞

2 π −∞

dt

1 + t 2 =

1

π lím arctan t|t=k t=−k = 2

k→∞ π lím arctan k = 1

k→∞

Π2

y

ArcTanx

x


21 / 45




lím

p→∞

δ p(x ≠ 0)

lím x→0 δ p(x) =

1

π lím

p→∞

1

π lím

p→∞

p

1+p 2 x 2

p

1

=

1 + p 2 x = 0 2 The Dirac Delta function

22 / 45


... as the limit of Sinc functions

∆ px

p

Π

δ p(x) Sinc family

δ p(x) = p π

sin px

px

Π

p

Π

2

p

Π

3

p

x


23 / 45




∫ ∞

−∞

δ p(x)dx = 1 π

∫

2 ∞

π 0

∫ ∞

−∞

∫ ∞

1

π

sin (px)dx

=

x

−∞

sin z

z dz =

sin z

z

dz = 2 π

π 2 = 1 The Dirac Delta function

24 / 45




1

π

lím

p→∞

sin (px)

x

p

π

lím

p→∞

δ p(x ≠ 0)

lím x→0 δ p(x) =

sin (px)

= lím = 0

p→∞ px

δ p(x) alternative definition of the

Sinc family

∫

δ p(x) = 1 p

2π −p e±itx dt

∫

δ p(x) = 1 p cos (tx)dt π 0


25 / 45


Properties of the Dirac delta function

Let us denote by x n the roots of the equation f(x) = 0 and suppose that

f ′ (x n) ≠ 0 then

Composition of functions

δ(f(x)) = ∑ n

δ(x−x n)

|f ′ (x n|


26 / 45


Important consequences of the composition property are

δ(−x) = δ(x) (symmetry property).

δ(ax) = δ(x)

|a|

(scaling property).

δ(ax − x 0 ) = δ(x− x 0

a

)

(a more general formulation of the scaling property).

|a|


27 / 45




δ(ax) = δ(x)

|a|

(scaling property).

δ(ax − x 0 ) = δ(x− x 0

a

)


|a|

δ(x 2 − a 2 ) = δ(x−a)+δ(x+a)

2|a| .


27 / 45




δ(ax) = δ(x)

|a|

(scaling property).

δ(ax − x 0 ) = δ(x− x 0

a

)


|a|

δ(x 2 − a 2 ) = δ(x−a)+δ(x+a)

2|a| .

∫ ∞

−∞ g(x)δ(f(x))dx = ∑ n

g(x n)

|f ′ (x n)|


27 / 45




δ(ax) = δ(x)

|a|

(scaling property).

δ(ax − x 0 ) = δ(x− x 0

a

)


|a|

δ(x 2 − a 2 ) = δ(x−a)+δ(x+a)

2|a| .

∫ ∞

−∞ g(x)δ(f(x))dx = ∑ n

g(x n)

|f ′ (x n)|


27 / 45


Convolution

Convolution

f(x)⊗δ(x+x 0) = ∫ ∞

f(ς)δ(ς−(x+x0))dς = f(x+x0)

−∞

The effect of convolving with the position-shifted Dirac delta is to shift f(t)

by the same amount.

∫ ∞

−∞ δ(ζ − x)δ(x − η)dx = δ(ζ − η)) The Dirac Delta function

28 / 45


Convolution

Convolution

f(x)⊗δ(x+x 0) = ∫ ∞

f(ς)δ(ς−(x+x0))dς = f(x+x0)

−∞

The effect of convolving with the position-shifted Dirac delta is to shift f(t)

by the same amount.

∫ ∞

−∞ δ(ζ − x)δ(x − η)dx = δ(ζ − η)) The Dirac Delta function

28 / 45


Heaviside

Definition (Step function)

Θ(x) = 1 2 (1 + x

|x| )


{ 1 x ≥ 0

Θ(x) =

0 x < 0

dΘ

dx = 1 d x

2 dx |x| = 1 2 lím d 2

arctan (px) =

p→∞ dx π

1

π lím p

p→∞ 1 + p 2 x = δ(x)

2

Heaviside

δ(x) = dΘ(x)

dx


29 / 45


Heaviside


Θ(x) = 1 2 (1 + x

|x| )


{ 1 x ≥ 0

Θ(x) =

0 x < 0

dΘ

dx = 1 d x

2 dx |x| = 1 2 lím d 2

arctan (px) =

p→∞ dx π

1

π lím p

p→∞ 1 + p 2 x = δ(x)

2

Heaviside

δ(x) = dΘ(x)

dx


29 / 45


Heaviside

y

y

p⩵0.5

x

p⩵2

x

y

y

p⩵1

x

p⩵5

x


30 / 45


Fourier transform of the Dirac delta function

Fourier transform

Γ [δ(x)] = ̂δ(x ∗ ) ≡ ∫ ∞

−∞ δ(x) exp (−2πix∗ x)dx = 1

This property allow us to state yet another definition of the Dirac delta as

the inverse Fourier transform of f(x) = 1


δ(x) =

∫ ∞

−∞

exp (2πix ∗ x)dx ∗ The Dirac Delta function

31 / 45


Fourier transform of the Dirac delta function

Fourier transform

Γ [δ(x)] = ̂δ(x ∗ ) ≡ ∫ ∞

−∞ δ(x) exp (−2πix∗ x)dx = 1

This property allow us to state yet another definition of the Dirac delta as

the inverse Fourier transform of f(x) = 1


δ(x) =

∫ ∞

−∞

exp (2πix ∗ x)dx ∗ The Dirac Delta function

31 / 45


Dirac delta function obtained from a complete set of


Let the set of functions {ψ n} be a complete system of orthonormal functions

in the interval (a, b) and let x and x 0 be inner points of that interval. Then

Theorem (Orthonormal functions)

∑

n ψ∗ n(x)ψ n(x 0) = δ(x − x 0)

To proof the theorem we shall demonstrate that the left hand side has the

sifting property of the Dirac distribution

I = ∫ b

f(x) ∑

a n ψ∗ n(x)ψ n(x 0)dx = f(x 0)

f(x) = ∑ m cmψm(x) cm = ∫ b

a f(x)ψ∗ m(x)dx


32 / 45


Dirac delta function obtained from a complete set of


I =

∫ b

a

∑

c ∑ mψ m(x) ψn(x)ψ ∗ n(x 0)dx =

m

n

∑ ∑

∫ b

c m ψ n(x 0) ψn(x)ψ ∗ m(x)dx

m n

a

∫ b

a ψ∗ n(x)ψ m(x)dx = K δ mn

I = ∑ n

∑

c m ψ n(x 0) K δ mn =

n

∑

c mψ m(x 0) = f(x 0)

n


33 / 45


Dirac comb

Definition (Dirac comb)

∑ ∞

m=−∞ δ(x − ma) The Dirac Delta function

34 / 45


Dirac comb

1

The set { √ exp (2πinx/a)} forms a complete set of orthonormal

(|a|)

functions, then

∑

δ(x) = 1 ∞

|a| n=−∞

exp (2πinx/a)

each summand in the LHS in the above expression is periodic with period

|a| therefore the whole sum is periodic with the same period and

Dirac comb

∑ ∞

∑

1

m=−∞

δ(x − ma) = ∞

|a| n=−∞ exp (2πinx/a) The Dirac Delta function

35 / 45


Fourier transform of a Dirac comb

∞∑

∫ ∞

∫ ∞

−∞ m=−∞

δ(x − ma) exp(−2πix ∗ x)dx =

∞∑

δ(x − ma) exp(−2πix ∗ x)dx =

m=−∞ −∞

m=−∞

∞∑

exp(2πix ∗ ma)

Theorem (Fourier transform of a Dirac comb)

Γ [ ∑ ∞

1

m=−∞

δ(x − ma)] =

|a|

∑ ∞

h=−∞ δ(x∗ − h/a) = |a ∗ | ∑ ∞

h=−∞ δ(x∗ − ha ∗ )

The Fourier transform of a Dirac comb is a Dirac comb


36 / 45


Dirac delta in higher dimensional space

Dirac delta in higher dimensions. Cartesian coordinates

∫

···

∫

f(⃗x)δ(⃗x − ⃗x0)d N x = f( ⃗x 0)

which comes from

∫

···

∫

f(x1, x 1, . . . x N )δ(x 1 −x 01)δ(x 2 −x 02) . . . δ(x N −x 0N )dx 1dx 2 . . . dx N =

f( x 01, x 02, ⃗ . . . x 0N )

Definition (Dirac delta in higher dimensions. Cartesian

coordinates)

δ(⃗x − ⃗x 0) = δ(x 1 − x 01)δ(x 2 − x 02) . . . δ(x N − x 0N ) = ∏ N

s=1 δ(xs − x0s)


37 / 45


General coordinates

{x i}: Cartesian coordinates

{y i}: General coordinates

x 1 = x 1(y 1, . . . y N )

x 2 = x 2(y 1, . . . y N )

. . .

x N = x N (y 1, . . . y N )

J(y 1, y 2, . . . , y N ) =

∣

∂x 1 ∂x 1

∂y 1

∂x 2 ∂x 2

∂y 1

∂x N

∂y 1

∂x 1

∂y 2

. . .

∂y N

∂y 2

. . .

∂x 2

∂y N

. . . . . . . . . . . .

∂x N

∂y 2

. . .

∂x N

∂y N

∣ ∣∣∣∣∣∣∣∣

Definition (Dirac delta in higher

dimensions. General coordinates)

δ(⃗x − ⃗x 0) =

1

δ(⃗y − ⃗y0) |J(y 1 ,y 2 ,...,y N )|


38 / 45


Oblique coordinates

x 1 = a 11y 1 + a 12y 2 . . . a 1N y N

J(y 1, y 2, . . . , y N ) =

∣

x 2 = a 11y 1 + a 22y 2 . . . a 2N y N

a 11 a 12 . . . a 1N ∣∣∣∣∣∣∣

.

a 21 a 22 . . . a 2N

x N = a N1y 1 + a N2y 2 . . . a NN y N ∣

. . . . . . . . . . . .

a N1 a N2 . . . a NN

⃗x =

⎛

⎜

⎝

a 11 a 12 . . . a 1N

⎞

a 21 a 22 . . . a 2N

. . . . . . . . . . . .

⎟

⎠ ⃗y

Definition (Dirac delta in higher

dimensions. Oblique coordinates)

δ(⃗x− ⃗x 0) = 1 δ(⃗y − ⃗y0) = √

1

δ(⃗y − ⃗y

| ||A|| | | ||G|| | 0)


39 / 45


Recap: Definitions

δ(x − x 0)dx =

{ ∞ x = x0

δ(x) = d ( 1

dx 2 (1 + x )

|x| ) 0 x ≠ x 0

√

p

δ(x) = lím

p→∞ π exp (−px2 )

p sin px

δ(x) = lím

p→∞ π px

δ(x) = lím

p→∞

δ(x) =

∫ ∞

−∞

1 p

p 1 + p 2 x 2

exp (2πix ∗ x)dx ∗

δ(x − x 0) = ∑ n

ψ ∗ n(x)ψ n(x 0)

where {ψ ∗ n(x)} is a complete set of orthornormal functions.


40 / 45


Recap: Properties

Γ [

∫ b

a

δ(f(x)) = ∑ n

δ(x 2 − a 2 ) =

∞∑

m=−∞

⎧

1

2 ⎪⎨

[f(x− 0 ) + f(x+ 0 ] x0 ∈ (a, b)

1

f(x)δ(x − x 0)dx =

2 f(x+ 0 ) x0 = a

1

⎪⎩

2 f(x− 0 ) x0 = b

0 x 0 /∈ (a, b)

δ(x − x n)

|f ′ (x n|

∫ ∞

g(x)δ(f(x))dx = ∑

−∞ n

δ(−x) = δ(x) δ(ax − x 0) = δ(x − x 0

a

)

|a|

δ(x − ma)] = |a ∗ |

δ(x − a) + δ(x + a)

2|a|

∞∑

m=−∞

∞∑

h=−∞

g(x n)

|f ′ (x n)|

f(x) ⊗ δ(x + x 0) = f(x + x 0)

δ(x ∗ − ha ∗ ) Γ [δ(x)] = ̂δ(x ∗ ) = 1

δ(x − ma) = 1

|a|

∞∑

n=−∞

exp (2πinx/a)


41 / 45


Recap: Higher Dimensions

Cartesian coordinates:

∫

∫

· · ·

f(⃗x)δ(⃗x − ⃗x 0)d N x = f( ⃗x 0)

δ(⃗x − ⃗x 0) = δ(x 1 − x 01)δ(x 2 − x 02) . . . δ(x N − x 0N ) =

General Coordinates {y i}:

δ(⃗x − ⃗x 0) =

Oblique Coordinates ⃗x = A · ⃗y:

1

δ(⃗y − ⃗y0)

|J(y 1, y 2, . . . , y N )|

N∏

δ(x s − x 0s)

s=1

δ(⃗x − ⃗x 0) = 1

| ||A|| | δ(⃗y − ⃗y0) = 1

√

| ||G|| |

δ(⃗y − ⃗y 0)


42 / 45


Exercises

Prove:

δ(f(x)) = ∑ δ(x−x n)

n |f ′ (Hint: develop f(x) in Taylor series around x n and

(x n|

prove the sifting property with δ(f(x)))

∑ ∞

n=−∞ f(na) = |x∗ | ∑ ∞

m=−∞ ̂f(mx ∗ ) (Poisson summation formula)

sin ((2p+1)π x−x 0 )

a

lím p→∞

sin (π x−x0 = |a| ∑ ∞

a ) m=−∞ δ(x − x 0 − ma)


43 / 45


Exercises

Prove that on the limit p → ∞ the sawtooth funtion tends to the Dirac

comb


44 / 45


References

Prof. RNDr. Jiri Komrska.

The Dirac distribution

http://physics.fme.vutbr.cz/∼komrska

V. Balakrishnan

All about the Dirac Delta Function(?)


45 / 45

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