Formula Sheet (SI units)

1 Physical constants
fine structure constant : α = e2 1
≈
4πɛ0c 137
Rydberg energy : Eo =
Comp Exam Formula Sheet (SI) 1
mee 4
2 2 (4πɛo) 2
= mec 2 α 2
2
Bohr magneton : µB = e
Bohr radius : ao =
2me
4πɛ0 2
mee 2
2 Vector calculus relationships
Triple products:
A × (B × C) = B(A · C) − C(A · B)
A · (B × C) = B · (C × A) = C · (A × B)
Product rules:
∇(A · B) = (A · ∇)B + (B · ∇)A
+ A × (∇ × B) + B × (∇ × A)
∇ · (φA) = φ∇ · A + A · ∇φ
∇ · (A × B) = B · (∇ × A) + A · (∇ × B)
∇ × (A × B) = A∇ · B − B∇ · A+
Second derivatives:
Green’s theorem:
V
+ (B · ∇)A − (A · ∇)B
∇ × (∇ × A) = ∇(∇ · A) − ∇ 2 A
∇ · (∇ × A) = 0
2 2
ψ∇ φ − φ∇ ψ dV =
S
(ψ∇φ − φ∇ψ) · dS
Spherical coordinates:
∇f = ∂f 1 ∂f
ˆr +
∂r r ∂θ ˆ θ + 1 ∂f
r sin θ ∂φ ˆ φ
∇ · A = 1
r2 ∂ 2 1 ∂
r Ar +
∂r r sin θ ∂θ (sin θAθ) + 1
r sin θ
∇ × A = 1
∂
r sin θ ∂θ (sin θAφ) − ∂Aθ
ˆr
∂φ
+ 1
1 ∂Ar ∂
−
r sin θ ∂φ ∂r (rAφ)
ˆθ
+ 1
∂
r ∂r (rAθ) − ∂Ar
ˆφ
∂θ
∇ 2 f = 1
r2
∂ 2 ∂f 1
r +
∂r ∂r r2
∂
sin θ
sin θ ∂θ
∂f
∂θ
+
1
r2 sin 2 ∂
θ
2f ∂φ2 Cylindrical coordinates:
∇f = ∂f 1 ∂f
ˆρ +
∂ρ ρ ∂φ ˆ φ + ∂f
∂z ˆz
∇ · A = 1 ∂
ρ ∂ρ (ρAρ) + 1 ∂Aφ ∂Az
+
ρ ∂φ ∂z
1 ∂Az ∂Aφ
∇ × A = − ˆρ
ρ ∂φ ∂z
∂Aρ ∂Az
+ − ˆφ
∂z ∂ρ
+ 1
∂
ρ ∂ρ (ρAφ) − ∂Aρ
ˆz
∂φ
∇ 2 f = 1
∂
ρ
ρ ∂ρ
∂f
+
∂ρ
1
ρ2 ∂2f ∂φ2 + ∂2f ∂z2 3 Quantum mechanics
Raising and lowering operators for ang. momentum:
J± = Jx ± iJy
J± |j, m〉 = j(j + 1) − m(m ± 1) |j, m ± 1〉
Perturbation theory for nondegenerate states:
En ≈ E o n + 〈n| V |n〉 +
m=n
|〈n| V |m〉| 2
+ · · ·
En − Em
∂Aφ
∂φ

Harmonic oscillator: [a, a † ] = 1
mω p
a = x + i√
2 2mω
a †
mω p
= x − i√
2 2mω
a † |n〉 = √ n + 1|n + 1〉
a|n〉 = √ n|n − 1〉
4 Electromagnetism
Maxwell’s equations:
∇ · D = ρ ∇ × E = − ∂B
∂t
∇ · B = 0 ∇ × H = ∂D
+ J
∂t
Magnetic dipole field:
B(r) = µ0 3ˆr(ˆr · m) − m
4π r3 Energy density: U = 1
2 (E · D + B · H)
Poynting vector: S = E × H
General solutions of Laplace’s equation
in cylindrical coordinates (independent of z):
Φ(ρ, φ) = ao log(ρ)
∞
an n
+ + bnρ (cn cos nφ + dn sin nφ)
ρn n=1
in spherical coordinates:
Φ(r, θ, φ) =
Φ(r, θ) =
∞
l
l=0 m=−l
∞
l=0
Almr l + Blm
r l+1
Comp Exam Formula Sheet (SI) 2
Ylm(θ, φ)
Alr l + Bl
rl+1
Pl(cos θ)
(with azimuthal symmetry)
5 Useful math formulas
e ikr cos θ =
∞
(2l + 1)i l jl(kr)Pl(cos θ)
l=0
∞
e ixy dy = 2πδ(x)
∞
0
−∞
x n e −x dx = n! , integer n
(1 + x) n =
n
k=1
n!
k!(n − k)! xk
log(n!) ≈ 1
log(2πn) + n log(n) − n
2
sin(x ± y) = sin x cos y ± cos x sin y
cos(x ± y) = cos x cos y ∓ sin x sin y
1
|x − x ′
=
|
lm
4π
2l + 1
r l <
r l+1
>
1
|x − r ′
=
ˆz|
l
Spherical Bessel functions:
Y ∗
lm(θ ′ , φ ′ )Ylm(θ, φ)
r l <
r l+1
>
Pl(cos θ)
sin z
j0(z) =
z
cos z
n0(z) = −
z
sin z cos z
j1(z) = −
z2 z
Legendre polynomials:
cos z sin z
n1(z) = − −
z2 z
P0(x) = 1 P2(x) = 1
P1(x) = x
2
3x − 1
2
P3(x) = 1 3
5x − 3x
2
P m
l (x) = 1 − x 2 m/2 d m Pl
dx m
Spherical harmonics:
Y00 = 1
√
4π
3
Y11 = −
Y10 =
15
Y22 =
32π sin2 θe i2φ
8π sin θeiφ
15
Y21 = − sin θ cos θeiφ
8π
3
4π cos θ Y20
5 3
=
4π 2 cos2 θ − 1
2