Formula Sheet (SI units)

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