Formulas

A-1
Useful Formulae and Integrals
∞
∫0
∞
∫0
∞
∫0
∞
∫0
∞
∫0
xn e-x dx = n!
(n = positive integer)
1
e-βx2 dx = 2 (π/β) 1/2
x e-βx2 dx =
1
2β
1
x2 e-βx2 dx = 4 (π/β3) 1/2
x3 e-βx2 dx =
1
2β2
Stirling's approximation:
loge (n!) ≃ n logen – n
n! =
(n >> 1)
1
2π n nn exp -n + 12n + O

c
c!
(a+b)c = ∑ ar b(c-r) r!(c-r)!
r=0
 1 
n2
 
A-2
Spherical Harmonics Ym
Legendre Polynomials PA (z)
0
Y0 = 1
4π
Po(z) = 1
ℓ
Y1(θ , φ) = 1
0
Y1(θ , φ) =
3 sinθ e iφ
8π
3 cosθ
4π
P1(z) = z
-1
3 sinθ e -iφ
8π
2
15 sin 2 θ e 2iφ
32π
Y1 (θ , φ) =
Y2(θ,φ) =
1
15 sinθ cosθ e iφ
8π
0
5 (3 cos 2 θ - 1)
16π
Y2(θ,φ) = Y2(θ,φ) =
-1
15 sinθ cosθ e -i φ
8π
-2
15 sin 2 θ e -2i φ
32π
Y2 (θ,φ) =
Y2 (θ,φ) =
P2(z) =
1
2
2 (3z
- 1)
1
P3(z) = 2 (5z3 - 3z)
o
YA (θ,φ) =
2A + 1
PA (cos θ)
4π
Jn(x)
The Bessel and Neumann functions Zn(x) = Y (x) are solutions of Bessel's equation
 n
d2Zn(x) 1 dZn(x)
n2
+
+
(
1
–
) Zn(x) = 0
x dx
dx2
x2
e = 4.80 ×10-10 esu = 1.60 × 10-19 Coulomb
No = 6.02 × 1023 particles/mole
c = 3.00 × 1010 cm/sec = 3.00 × 108 m/sec
kB = 1.38 × 10-23 J K-1 =1.38 × 10-16 ergK-1
–h = 1.05 × 10-27 erg sec = 1.05 × 10-34 J sec
ao = 0.529 × 10-8 cm
me = 9.11 × 10-28 g = 9.11 × 10-31 kg
εo ≅
mp = 1.67 × 10-24 g = 1.67 × 10-27 kg
Tesla.m
µo = 4π × 10-7 Ampere
1
Farad
m
9
4π×9×10
A-3
VECTOR OPERATIONS IN CYLINDRICAL AND SPHERICAL COORDINATES
CYLINDRICAL COORDINATES
Coordinat es(r, ϕ, z)
Gradient
Curl
Divergence
∂f
∂f i ∂f
+
∇f = i1 r + i2 1r
∂
∂ϕ 3 ∂z
∂A z ∂A ϕ
∂A r ∂A z
∂
∂Ar
+ i2
+ i 3 1r
(rAϕ) - 1r
∇ x A = i 1 1r
z
z
r
r
∂ϕ ∂
∂
∂
∂
∂ϕ
∇ • A = 1r
2
Laplacian
Unit vectors (i 1, i 2, i 3)
∂A ϕ ∂A z
∂
(r A r) + 1r
+
∂r
∂ϕ
∂z
2
2
∂ ∂f
∂ f ∂ f
+ 12
r
+
∇ f = 1r
∂r ∂r r ∂ 2 ∂z 2
ϕ
z
r
z
y
ϕ
x
A-4
VECTOR OPERATIONS IN CYLINDRICAL AND SPHERICAL COORDINATES
SPHERICAL COORDINATES
Coordinates (r, θ, ϕ)
Unit vectors (i 1, i 2, i 3)
∂ f i 1 ∂f
f
+ r
i3 1 ∂
+
∂r 2 ∂θ
r sin θ ∂ϕ
Gradient
∇f = i 1
Curl
∇ x A = i1
∂
∂A
1
1 ∂Ar - 1 ∂ (r A )
(sin θ Aϕ) - θ + i
ϕ
r ∂r
2
∂ϕ
r sin θ ∂θ
r sin θ ∂ϕ
∂
∂Ar
+ i 1r
(r Aθ) - 1r
3
∂r
∂θ
Divergence
Laplacian
∂A ϕ
∂ 2
∂
r Ar + 1
sin θ Aθ + 1
∇ • A = 12
r ∂r
r sin θ ∂θ
r sin θ ∂ϕ
2
2
∂ 2 ∂f
∂
∂f
∂ f
1
1
+
r
sin θ
+
∇ f = 12
2
∂r r 2 sin θ ∂θ
r ∂r
∂θ
r 2 sin θ ∂ϕ 2
2
2
∂
∂
∂f
∂ f
1
1
rf +
sin θ
+
= 1r
2
2
2
r sin θ ∂θ
∂r
∂θ
r 2 sin θ ∂ϕ 2
z
r
θ
z
y
ϕ
x