The_Cambridge_Handbook_of_Physics_Formulas

100 Quantum physics
Angular momentum addition a
Total angular
momentum
Mutually
commuting
sets
J = L+S (4.128)
Jˆ
z = Lˆ
z + Sˆ
z (4.129)
Jˆ
2 = L ˆ2
+ S ˆ2
+2̂L·S (4.130)
Jˆ
z ψ j,mj = m j¯hψ j,mj (4.131)
Jˆ
2 ψ j,mj = j(j +1)¯h 2 ψ j,mj (4.132)
j-multiplicity = (2l +1)(2s+1) (4.133)
{L 2 ,S 2 ,J 2 ,J z ,L·S} (4.134)
{L 2 ,S 2 ,L z ,S z ,J z } (4.135)
J ,J total angular momentum
L,L orbital angular
momentum
S,S spin angular momentum
ψ eigenfunctions
m j magnetic quantum
number |m j |≤j
j (l +s) ≥ j ≥|l −s|
{} set of mutually
commuting observables
Clebsch– |j,m j 〉 = ∑ 〈j,m j |l,m l ;s,m s 〉|l,m l 〉|s,m s 〉
Gordan
m l ,m s
coefficients b m s +m l =m j
(4.136)
a Summing spin and orbital angular momenta as examples, eigenstates |s,m s 〉 and |l,m l 〉.
b Or “Wigner coefficients.” Assuming no L–S interaction.
Magnetic moments
Bohr magneton
Gyromagnetic
ratio a
µ B = e¯h
2m e
(4.137)
γ =
orbital magnetic moment
orbital angular momentum
|·〉 eigenstates
〈·|·〉 Clebsch–Gordan
coefficients
µ B Bohr magneton
−e electronic charge
¯h (Planck constant)/(2π)
m e electron mass
(4.138) γ gyromagnetic ratio
Electron orbital
gyromagnetic
ratio
γ e = −µ B
¯h
(4.139)
= −e
2m e
(4.140)
γ e
electron gyromagnetic ratio
µ e,z = −g e µ B m s (4.141)
Spin magnetic
¯h
moment of an = ±g e γ e (4.142)
2
electron b = ± g ee¯h
(4.143)
4m e
µ e,z z component of spin
magnetic moment
g e electron g-factor (≃ 2.002)
m s spin quantum number (±1/2)
√ µ
µ J = g J J(J +1)µB (4.144) J total magnetic moment
µ J,z z component of µ J
µ
Landé g-factor c J,z = −g J µ B m J (4.145) m J magnetic quantum number
J(J +1)+S(S +1)−L(L+1) J,L,S total, orbital, and spin
g J =1+
2J(J +1)
quantum numbers
(4.146) g J Landé g-factor
a Or “magnetogyric ratio.”
b The electron g-factor equals exactly 2 in Dirac theory. The modification g e =2+α/π + ..., whereα is the fine
structure constant, comes from quantum electrodynamics.
c Relating the spin + orbital angular momenta of an electron to its total magnetic moment, assuming g e =2.