PC 251 Equation Sheet
s
f0 = f
1 ∓ u/c
1 ± u/c
x0
=
x − ut
p
1 − u2 /c2
y0
= y
Classical Physics
1
p2
mv 2 =
2
2m
K=
p~ = m~v
z
~ = ~r × p~
L
U=
= z
t − (u/c2 )x
= p
1 − u2 /c2
t0
1 |q1 ||q2 |
4π0 r2
F =
0
m~v
p~ = p
1 − v 2 /c2
1 q1 q2
4π0 r
mc2
E=p
1 − v 2 /c2
∆U = q∆V
B=
mc2
− mc2
1 − v 2 /c2
p
E = (pc)2 + (mc2 )2
K=p
µ0 i
2r
~
U = −~
µ·B
E & M Radiation
Kav =
3
kT
2
2d sin θ = nλ
2N
1
N (E) = √
E 1/2 e−E/kT
π (kT )3/2
Edof =
E = hf = hc/λ
1
kT
2
Kmax = eVs = hf − φ
λc = hc/φ
Special Relativity
∆t = p
∆t0
I = σT 4
1 − u2 /c2
p
L = L0 1 − u2 /c2
λmax T = 2.8978 × 10−3 mK
uLc2
∆t0 = p
1 − u2 /c2
v=
I(λ) =
v0 + u
1 + v 0 u/c2
2πhc2
1
λ5 ehc/λkT − 1
1
1
1
−
=
(1 − cos θ)
0
E
E
me c2
1
Wave Properties of Particles
Rutherford-Bohr Model
λ = h/p
n=
∆x∆p ∼ ~
b =
∆E∆t ∼ ~
p
∆p =
(p2av ) − (pav )2
=
N (θ)
=
d =
Schrödingers Equation
−~2 d2 ψ
+ U (x)ψ(x) = Eψ(x)
2m dx2
Ψ(x, t) = ψ(x)e−iωt
ω = E/~
P (x) = |ψ(x)|2
Z
f>θ
zZ e2
cot
2K 4π0
|ψ(x)|2 dx = 1
rn
=
a0
=
En
=
EH
=
λ
=
rn =
x2
Z
|ψ(x)|2 dx
P (x1 : x2 ) =
x1
Z
[f (x)]av =
En
=
ψ0 (x)
En
ω0
=
a0 n2
Z2
, En = −EH 2
Z
n
me mp
m=
me + mp
∞
|ψ(x)|2 f (x)dx
Hydrogen Atom
−∞
r
=
1
θ
2
a0 n2
4π0 ~2
= 0.0529 nm
m e2
EH
− 2
n
m e4
= 13.6 eV
32π 2 20 ~2
2 2 1
n1 n2
2
R∞ n1 − n22
~ =
|L|
ψn (x)
ntπb2
2 2 2
nt zZ
e
1
4
4r2 2K
4π0
sin (θ/2)
1 zZe2
4π0 K
∞
−∞
NA ρ
M
Lz
2
sin
L
L
h2 n2
for n = 1, 2, 3, ...
8mL2
mω 1/4
0
nπx e−(
√
p
l(l + 1) ~
= ml ~
∆Lz ∆φ ≥ ~
Hydrogen quantum numbers
principle: n = 1, 2, 3, ...
angular momentum: l = 0, 1, 2, ..., n − 1
magnetic: ml = 0, ±1, ±2, ..., ±l
spin magnetic: ms = ± 21
km/2~)x2
~π 1
=
n+
~ω0 for n = 0, 1, 2, ...
2
r
k
=
m
~
µ
~ L = −(e/2m)L
~
µ
~ S = −(e/2m)S
2
~ =
|S|
p
Decay processes
Alpha: A
ZXN →
3/4~ for s = 1/2
Beta:
= ±1/2~
Sz
Selection rules for photon emission:
∆l = ±1, ∆ml = 0, ±1
A−4 0
4
Z−2 X N −2 + 2 He2
A
A 0
−
Z X N → Z+1 X N −1 + e + ν̄e ,
A
A 0
+
Z X N → Z−1 X N +1 + e + νe ,
A
A 0
−
Z X N + e → Z−1 X N +1 + νe
Gamma:
A ∗
ZXN
→
A
ZXN
+γ
Spectroscopic notation:
Nuclear Reactions
s(l = 0), p(l = 1), d(l = 2), f (l = 3), ...
R = σN I0 /S = φσN
Normalization condition in spherical polar coordinates:
Z ∞ Z 2π Z π
|ψ(r, θ, φ)|2 r2 sin θ dθ dφ dr
1=
0
0
a(t) = λN = R(1 − e−λt )
Q = (mi − mf )c2
0
Particle Physics
Many-Electron Atoms
Filling order:
1s, 2s, 2p, 3s, 3d, 4s, 3d, 4p, 5s,
4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d
Adding angular momenta l1 and l2 :
Lmax
=
l1 + l2
Lmin
=
|l1 + l2 |
ML
=
ml1 + ml2
Nuclear Structure & Decay
R = R0 A1/3 , R0 = 1.2 fm
Conservation Laws
Baryon number
Lepton flavor
Charge
Quark flavor (except in Weak)
2
B = [N mn + Zm(11 H0 ) − m(A
Z XN )]c
mc2 = ~c/x
a
= λN
λ
=
Cosmology
ln 2/t1/2
N
= N0 e−λt
a
= a0 e−λt
v = H0 d
H0 = 72 km/s/Mps
3