The_Cambridge_Handbook_of_Physics_Formulas

26 Mathematics
Commutators
Commutator
definition
[A,B]=AB−BA = −[B,A] (2.89) [·,·] commutator
Adjoint [A,B] † =[B † ,A † ] (2.90) † adjoint
Distribution [A+B,C]=[A,C]+[B,C] (2.91)
Association [AB,C]=A[B,C]+[A,C]B (2.92)
Jacobi identity [A,[B,C]] = [B,[A,C]]−[C,[A,B]] (2.93)
Pauli matrices
Pauli matrices
Anticommutation
Cyclic
permutation
( )
0 1
σ 1 =
1 0
( )
1 0
σ 3 =
0 −1
( )
0 −i
σ 2 =
i 0
( )
1 0
1 =
0 1
(2.94)
σ i Pauli spin matrices
1 2×2 unit matrix
i i 2 = −1
σ i σ j +σ j σ i =2δ ij 1 (2.95) δ ij Kronecker delta
σ i σ j = iσ k (2.96)
(σ i ) 2 = 1 (2.97)
Rotation matrices a
⎛
⎞
Rotation
1 0 0
R i (θ) matrix for rotation
about x
R 1 (θ)= ⎝0 cosθ sinθ⎠ (2.98) about the ith axis
1
0 −sinθ cosθ
θ rotation angle
⎛
⎞
Rotation
cosθ 0 −sinθ
about x
R 2 (θ)= ⎝ 0 1 0 ⎠
2
sinθ 0 cosθ
(2.99)
⎛
⎞
Rotation
cosθ sinθ 0
α rotation about x 3
about x
R 3 (θ)= ⎝−sinθ cosθ 0⎠ (2.100) β rotation about x ′ 2
3
0 0 1
γ rotation about x ′′
3
Euler angles R rotation matrix
⎛
⎞
cosγcosβ cosα−sinγsinα cosγcosβ sinα+sinγcosα −cosγsinβ
R(α,β,γ)= ⎝−sinγcosβ cosα−cosγsinα −sinγcosβ sinα+cosγcosα sinγsinβ ⎠
sinβ cosα sinβ sinα cosβ
(2.101)
a Angles are in the right-handed sense for rotation of axes, or the left-handed sense for rotation of vectors. i.e., a
vector v is given a right-handed rotation of θ about the x 3 -axis using R 3 (−θ)v ↦→ v ′ . Conventionally, x 1 ≡ x, x 2 ≡ y,
and x 3 ≡ z.