IV-30
The Dirac Delta Function, į(x-xo)
Dirac Delta Function
In one dimension, į(x-xo) is defined to be such that:
ma to b f(x) į(x-xo)dx
/
+
*0
if xo is not in [a,b].
*½f(xo) if xo = a or b;
*f(xo)
if xo İ (a,b).
.
Properties of į(x-xo): (you should know those marked with *)
*1. į(x-xo) = 0
if x … xo
*2. m-4 to +4 į(x)dx
= 1
3. į(ax) = į(x)/|a|
*4. į(-x) = į(x)
5. į(x²-a²) = [į(x-a) + į(x+a)]/(2a); a $ 0
6. m-4 to +4 į(x-a)į(x-b)dx = į(a-b)
+------------------------------------------------------------------------------------------------------------------------------------------**7. į(g(x)) = 3i į(x-xoi)/|dg/dx|x=xoi
where g(xoi) = 0 and dg/dx exists at and in a region around xoi.
.------------------------------------------------------------------------------------------------------------------------------------------
*8. f(x)į(x-a) = f(a)į(x-a)
9. į(x) is a "symbolic" function which provides convenient notation for many mathematical expressions. Often
one "uses" į(x) in expressions which are not integrated over. However, it is understood that eventually
these expressions will be integrated over so that the definition of į (box above) applies.
10. No ordinary function having exactly the properties of į(x) exists. However, one can approximate į(x) by the
limit of a sequence of (non-unique) functions, įn(x). Some examples of įn(x) which work are given below.
In all these cases,
m-4 to +4 įn(x)dx = 1 œ n and limitn --> 4 m-4 to +4 įn(x-xo)f(x)dx = f(xo). œ n.
(a) įn(x) /
+
*0
*n
*0
.
for x < -1/(2n)
for -1/(2n) # x # 1/(2n)
for x > 1/(2n)
(b) įn(x) / n//ʌ exp[-n²x²]
(c) įn(x) / (n//ʌ)· 1/(1 + n²x²)
=
(d) įn(x) / sin(nx)/ʌx
[1/(2ʌ)]m-n to n exp(ixt)dt
IV-31
+ (-1)r drf/dxr|xo if xo İ (a,b)
11. ma to b f(x) d /dx į(x-xo) dx =
r
r
r r
*½[(-1) d f/dxr|xo if xo = a or b
.0 otherwise
f(x) is arbitrary, continuous function at x = xo
12. ma to b xr dr/dxr į(x-xo)dx = ma to b (-1)rr! į(x-xo) dx where xo İ (a,b).
important expressions involving į(x-xo)
13. į(x-xo) = [1/2ʌ] m-4 to +4 eik(x-xo)dk
14. į(r-ro) / į(x-xo)į(y-yo)į(z-zo)= [1/2ʌ]3mmmall k space exp[ik·(r-ro)]dkxdkydkz
15. į(g(x)) = 3n į(x-ȕn)/|dg/dx|x = ȕn where g(ȕn) = 0.
16. į(r-ro) = į(q1-q1o)į(q2-q2o)į(q3-q3o)//g in general system.
Dirac Delta Function in 3 Dimensions: į (r - ro) / į(x-x )į(y-y )į(z-z )
o
17.
į(k - ko) = [1/2ʌ]3 m-4 to +4m-4 to +4m-4 to +4 exp[ir·(k - ko)] dxdydz
18.
į(r - ro) = į(q1-q1o)į(q2-q2o)į(q3-q3o)//g
o
o
derivation: mmmį(r - ro)f(r) d3x = f(ro) = mmmį(q1-q1o)į(q2-q2o)į(q3-q3o)//g· f(r(qi)) /gdq1dq2dq3
19.
į(r - ro)spherical coordinates = į(r-ro)į(ij-ijo)į(ș-șo)/(r²sinș)
20.
½[į(x-a)+į(x+a)] = [1/ʌ] m0 to 4 cos(ka) cos(kx) dk
Exercises: Evaluate the following integrals.
a) m-1 to 5 į(2x-ʌ) exp[ sin3(x-ʌ) ] dx
b) mmmall spaceį(r·r-a²)į(cosș-1//2)į(sinij-½)exp[ik·r]d3x
IV-32
Helmholtz Theorem
If
(a) L·F(r)
= ȡ(r) everywhere for finite r;
(b) L x F(r) = J(r) "
(c) limitr-->4ȡ(r) = 0;
(d) limitr-->4|J(r)| = 0;
then
F(r) = -Lĭ(r) + L x A(r)
where ĭ and A are determined from ȡ and J as shown below.
proof:
1. Define ĭ and A as follows:
ĭ(r)
=
[1/4ʌ]mmmV= all space ȡ(r')/|r - r'| d3x' + ĭo(r),
where L²ĭo(r) = 0;
A(r)
=
[1/4ʌ]mmmV= all space J(r')/|r - r'| d3x' + Ao(r),
where Lx(LxAo(r)) = 0;
2. Let F(r) = -Lĭ(r) + LxA(r); we shall show that this F satisfies the conditions (a) and (b) if (c) and (d) hold:
L·F
LxF
3. L·F
=
=
= -L²ĭ(r)
4. But L²[1/|r - r'|]
L·[-Lĭ(r) + LxA(r)]
Lx[-Lĭ(r) + LxA(r)]
=
=
-L²ĭ(r)
Lx[LxA(r)].
= -[1/4ʌ]mmmV= all space ȡ(r')L²[1/|r - r'|] d3x' + 0
= L·L[1/|r - r'|] = L·[-(r - r')/|r - r'|3]
=-[3/|r - r'|3 + [-3/|r - r'|4](r - r')·(r - r')/|r - r'| ]
=-[3/|r - r'|3 + [-3/|r - r'|3] ]
=0
if r … r'
5 What happens if r = r'? We shall see that the expression --> 4, but with a crucial additional property!
CLAIM: L²
1
' &4ʌ į(r&r')
|r&r'|
Derivation:
(a) Consider the following integral, mmmV L·L[1/r] d3x = mmmV L²[1/r] d3x
where V = all space and V' = all space except a sphere of radius į centered on the origin and
a "funnel" extending from r = 0 to r = 4. See figure below.
We shall show that this integral = -4ʌ if r = 0 is in V.
Note: V contains r = 0 and V' does not.