The_Cambridge_Handbook_of_Physics_Formulas

104 Quantum physics
Nuclear collisions
σ(E)= π Breit–Wigner
k 2 g Γ ab Γ c
(E −E 0 ) 2 +Γ 2 /4
formula a 2J +1
g =
(2s a +1)(2s b +1)
(4.174)
(4.175)
Total width Γ=Γ ab +Γ c (4.176)
Resonance
lifetime
Born scattering
formula b
τ = ¯h Γ
∣ ∫
dσ ∣∣∣
dΩ = 2µ ∞
¯h 2
0
sinKr
Kr
(4.177)
2
V (r)r 2 dr
∣
(4.178)
Mott scattering formula c
dσ
( α
) 2
[csc
dΩ = 4 χ χ 4E 2 +sec4 2 + Acos( α
¯hv lntan2 χ )]
2
sin 2 χ 2 cos χ 2
(4.179)
dσ
( α
) 2 4−3sin 2
dΩ ≃ χ
2E sin 4 (A = −1,α≪ v¯h) (4.180)
χ
σ(E) cross-section for a+b → c
k incoming wavenumber
g spin factor
E total energy (PE + KE)
E 0 resonant energy
Γ width of resonant state R
Γ ab partial width into a+b
Γ c partial width into c
τ resonance lifetime
J total angular momentum
quantum number of R
s a,b spins of a and b
dσ
dΩ differential collision
cross-section
µ reduced mass
K = |k in −k out | (see footnote)
r radial distance
V (r) potential energy of interaction
¯h
α/r
χ
v
A
(Planck constant)/2π
scattering potential energy
scattering angle
closing velocity
= 2 for spin-zero particles, = −1
for spin-half particles
a For the reaction a+b ↔ R → c in the centre of mass frame.
b For a central field. The Born approximation holds when the potential energy of scattering, V , is much less than
the total kinetic energy. K is the magnitude of the change in the particle’s wavevector due to scattering.
c For identical particles undergoing Coulomb scattering in the centre of mass frame. Nonidentical particles obey the
Rutherford scattering formula (page 72).
Relativistic wave equations a
Klein–Gordon
equation
(massive, spin
zero particles)
Weyl equations
(massless, spin
1/2 particles)
Dirac equation
(massive, spin
1/2 particles)
a Written in natural units, with c =¯h =1.
(∇ 2 −m 2 )ψ = ∂2 ψ
∂t 2 (4.181)
(
)
∂ψ
∂t = ± ∂ψ
σ x
∂x +σ ∂ψ
y
∂y +σ ∂ψ
z
∂z
(4.182)
(iγ µ ∂µ−m)ψ = 0 (4.183)
( )
∂
where ∂µ=
∂t , ∂
∂x , ∂
∂y , ∂
(4.184)
∂z
(γ 0 ) 2 = 1 4 ; (γ 1 ) 2 =(γ 2 ) 2 =(γ 3 ) 2 = −1 4 (4.185)
ψ wavefunction
m particle mass
t time
ψ spinor wavefunction
σ i Pauli spin matrices
(see page 26)
i i 2 = −1
γ µ Dirac( matrices: )
γ 0 12 0
=
0 −1
( ) 2
γ i 0 σi
=
−σ i 0
1 n n×n unit matrix