E&M.pdf


Preview of PDF document e-m.pdf

Page 1 23415

Text preview


• Orthogonality:
a

Z


n0 π
x sin
xdx =
a
a

sin
0


Z

Z

a

Z

cos
0

0

Yl,m
(θ, φ)Yl,m (θ, φ)d(cos θ)dφ = δl,l0 δm,m0

π

0
−1

Z


n0 π
a
x cos
xdx = δn,n0
a
a
2

Pl (cos θ)Pl0 (cos θ)d cos θ =
Z1 ∞

2
δl,l0
2l + 1

1
Jm (kρ)Jm (kρ)ρdρ = δ(k − k 0 )
k
0
 


Z α
xmn0
a2
xmn
ρ Jm
ρ ρdρ = [Jm+1 (xmn )]2 δn,n0
Jm
a
a
2
0
• Scalar 1/r Expansions:
- 2-D Box:
1
1
=
0
|~x − ~x |
π

Z

~

d2 k

eik·~x
1
=
2
~
π
|k|

0

Z
dαdβ

0

eiα(x−x )+iβ(y−y )
=
α2 + β 2

0

Z


0

eiα(x−x ) e−α(y−y )
α

- 2-D Polar:
ln

 m
1
1
1 ρ<

=
ln
+
Σ
cos m(φ − φ0 )
m=1
|~x − ~x0 |
ρ<
m ρ>

- 3-D Box:
1
1
=
|~x − ~x0 |
2π 2

Z

~

eik·~x
1
d k
=
2
~

|k|
3

Z

√ 2 2
0
0
0
eiα(x−x ) eiβ(y−y ) e− α +β (z−z )
dαdβ
α2 + β 2

- Spherical Harmonic:
 l 
r<
1



l
Ylm
= Σl=0 Σm=−l
(θ0 , φ0 )Ylm (θ, φ)
l+1
0
|~x − ~x |
2l + 1 r>
- Legendre Function:
 l 
r<
1

=
Σ
Pl (cos θ0 )Pl (cos θ)
l=0
l+1
|~x − ~x0 |
r>
- Ordinary Bessel Function:

Z ∞ 
1
1
0

0
0
dk
=
J0 (kρ)J0 (kρ ) + Σm=1 Jm (kρ)Jm (kρ ) cos m(φ − φ ) e−k(z> −z< )
|~x − ~x0 |
2
0

Z ∞ 

0 im(φ−φ0 )
= Σm=−∞
dk Jm (kρ)Jm (kρ )e
e−k(z> −z< )
0

- Modified Bessel Function:


Z
1
4 ∞
1

0
=
dk
I0 (kρ< )K0 (kρ> ) + Σm=1 Im (kρ< )Km (kρ> ) cos m(φ − φ ) cos k(z − z 0 )
|~x − ~x0 |
π 0
2

Z ∞ 
4 ∞
im(φ−φ0 )
= Σm=−∞
dk Im (kρ< )Km (kρ> )e
cos k(z − z 0 )
π
0