This page collects the main formulas used across the Symmetry, Angular Momentum, and Spin volume. It is a quick reference, not a replacement for the explanatory pages.
A quantum symmetry preserves transition probabilities:
∣ ⟨ ϕ ∣ ψ ⟩ ∣ 2 ↦ ∣ ⟨ ϕ ∣ ψ ⟩ ∣ 2 . \left|\langle\phi|\psi\rangle\right|^2
\mapsto
\left|\langle\phi|\psi\rangle\right|^2. ∣ ⟨ ϕ ∣ ψ ⟩ ∣ 2 ↦ ∣ ⟨ ϕ ∣ ψ ⟩ ∣ 2 . Unitary operators are linear:
U ( a ∣ ψ ⟩ + b ∣ ϕ ⟩ ) = a U ∣ ψ ⟩ + b U ∣ ϕ ⟩ . U(a\lvert\psi\rangle+b\lvert\phi\rangle)
=
aU\lvert\psi\rangle+bU\lvert\phi\rangle. U ( a ∣ ψ ⟩ + b ∣ ϕ ⟩) = a U ∣ ψ ⟩ + b U ∣ ϕ ⟩ . Antiunitary operators are antilinear:
T ( a ∣ ψ ⟩ + b ∣ ϕ ⟩ ) = a ∗ T ∣ ψ ⟩ + b ∗ T ∣ ϕ ⟩ . T(a\lvert\psi\rangle+b\lvert\phi\rangle)
=
a^*T\lvert\psi\rangle+b^*T\lvert\phi\rangle. T ( a ∣ ψ ⟩ + b ∣ ϕ ⟩) = a ∗ T ∣ ψ ⟩ + b ∗ T ∣ ϕ ⟩ . A Hamiltonian symmetry satisfies
U H U − 1 = H UHU^{-1}=H U H U − 1 = H for a unitary or antiunitary symmetry operator, with antiunitarity handled carefully when complex scalars appear.
For a one-parameter unitary family,
U ( ϵ ) = exp ( − i ℏ ϵ G ) . U(\epsilon)
=
\exp\left(-\frac{i}{\hbar}\epsilon G\right). U ( ϵ ) = exp ( − ℏ i ϵ G ) . If G G G
[ G , H ] = 0 , [G,H]=0, [ G , H ] = 0 , then G G G
The translation operator by displacement a a a
T ( a ) = exp ( − i ℏ a P ) . T(a)
=
\exp\left(-\frac{i}{\hbar}aP\right). T ( a ) = exp ( − ℏ i a P ) . With the active convention used here,
( T ( a ) ψ ) ( x ) = ψ ( x − a ) . (T(a)\psi)(x)=\psi(x-a). ( T ( a ) ψ ) ( x ) = ψ ( x − a ) . The canonical commutator is
[ X , P ] = i ℏ . [X,P]=i\hbar. [ X , P ] = i ℏ. A rotation by angle θ \theta θ n ^ \hat{\mathbf n} n ^
U ( n ^ , θ ) = exp ( − i ℏ θ n ^ ⋅ J ) . U(\hat{\mathbf n},\theta)
=
\exp\left(
-\frac{i}{\hbar}\theta\,\hat{\mathbf n}\cdot\mathbf J
\right). U ( n ^ , θ ) = exp ( − ℏ i θ n ^ ⋅ J ) . The angular momentum algebra is
[ J i , J j ] = i ℏ ∑ k ϵ i j k J k . [J_i,J_j]
=
i\hbar\sum_k\epsilon_{ijk}J_k. [ J i , J j ] = i ℏ k ∑ ϵ ij k J k . The simultaneous eigenvalue equations are
J 2 ∣ j , m ⟩ = ℏ 2 j ( j + 1 ) ∣ j , m ⟩ , J^2\lvert j,m\rangle
=
\hbar^2j(j+1)\lvert j,m\rangle, J 2 ∣ j , m ⟩ = ℏ 2 j ( j + 1 ) ∣ j , m ⟩ , and
J z ∣ j , m ⟩ = ℏ m ∣ j , m ⟩ . J_z\lvert j,m\rangle
=
\hbar m\lvert j,m\rangle. J z ∣ j , m ⟩ = ℏ m ∣ j , m ⟩ . The ladder operators are
J ± = J x ± i J y . J_\pm=J_x\pm iJ_y. J ± = J x ± i J y . Their action is
J ± ∣ j , m ⟩ = ℏ j ( j + 1 ) − m ( m ± 1 ) ∣ j , m ± 1 ⟩ . J_\pm\lvert j,m\rangle
=
\hbar
\sqrt{j(j+1)-m(m\pm1)}
\lvert j,m\pm1\rangle. J ± ∣ j , m ⟩ = ℏ j ( j + 1 ) − m ( m ± 1 ) ∣ j , m ± 1 ⟩ . Orbital angular momentum is
L = R × P . \mathbf L=\mathbf R\times\mathbf P. L = R × P . For spherical harmonics,
L 2 Y ℓ m = ℏ 2 ℓ ( ℓ + 1 ) Y ℓ m , L^2Y_\ell^m
=
\hbar^2\ell(\ell+1)Y_\ell^m, L 2 Y ℓ m = ℏ 2 ℓ ( ℓ + 1 ) Y ℓ m , and
L z Y ℓ m = ℏ m Y ℓ m . L_zY_\ell^m
=
\hbar mY_\ell^m. L z Y ℓ m = ℏ m Y ℓ m . The spin-1 / 2 1/2 1/2
S i = ℏ 2 σ i . S_i=\frac{\hbar}{2}\sigma_i. S i = 2 ℏ σ i . The Pauli matrices obey
σ i σ j = δ i j I + i ∑ k ϵ i j k σ k . \sigma_i\sigma_j
=
\delta_{ij}I
+
i\sum_k\epsilon_{ijk}\sigma_k. σ i σ j = δ ij I + i k ∑ ϵ ij k σ k . For vectors a \mathbf a a b \mathbf b b
( a ⋅ σ ) ( b ⋅ σ ) = ( a ⋅ b ) I + i ( a × b ) ⋅ σ . (\mathbf a\cdot\boldsymbol\sigma)
(\mathbf b\cdot\boldsymbol\sigma)
=
(\mathbf a\cdot\mathbf b)I
+
i(\mathbf a\times\mathbf b)\cdot\boldsymbol\sigma. ( a ⋅ σ ) ( b ⋅ σ ) = ( a ⋅ b ) I + i ( a × b ) ⋅ σ . A spinor rotation is
U ( n ^ , θ ) = exp ( − i 2 θ n ^ ⋅ σ ) . U(\hat{\mathbf n},\theta)
=
\exp\left(
-\frac{i}{2}\theta\,\hat{\mathbf n}\cdot\boldsymbol\sigma
\right). U ( n ^ , θ ) = exp ( − 2 i θ n ^ ⋅ σ ) . Equivalently,
U ( n ^ , θ ) = cos θ 2 I − i sin θ 2 n ^ ⋅ σ . U(\hat{\mathbf n},\theta)
=
\cos\frac{\theta}{2}\,I
-i\sin\frac{\theta}{2}\,
\hat{\mathbf n}\cdot\boldsymbol\sigma. U ( n ^ , θ ) = cos 2 θ I − i sin 2 θ n ^ ⋅ σ . The Bloch representation of a spin-1 / 2 1/2 1/2
ρ = 1 2 ( I + r ⋅ σ ) . \rho
=
\frac12
\left(
I+\mathbf r\cdot\boldsymbol\sigma
\right). ρ = 2 1 ( I + r ⋅ σ ) . For two angular momenta,
J = J 1 + J 2 . \mathbf J=\mathbf J_1+\mathbf J_2. J = J 1 + J 2 . Allowed total angular momenta are
J = ∣ j 1 − j 2 ∣ , ∣ j 1 − j 2 ∣ + 1 , … , j 1 + j 2 . J
=
\lvert j_1-j_2\rvert,
\lvert j_1-j_2\rvert+1,
\ldots,
j_1+j_2. J = ∣ j 1 − j 2 ∣ , ∣ j 1 − j 2 ∣ + 1 , … , j 1 + j 2 . The coupled basis expansion is
∣ j 1 , j 2 ; J , M ⟩ = ∑ m 1 , m 2 ⟨ j 1 , m 1 ; j 2 , m 2 ∣ J , M ⟩ ∣ j 1 , m 1 ⟩ ∣ j 2 , m 2 ⟩ . \lvert j_1,j_2;J,M\rangle
=
\sum_{m_1,m_2}
\langle j_1,m_1;j_2,m_2|J,M\rangle
\lvert j_1,m_1\rangle
\lvert j_2,m_2\rangle. ∣ j 1 , j 2 ; J , M ⟩ = m 1 , m 2 ∑ ⟨ j 1 , m 1 ; j 2 , m 2 ∣ J , M ⟩ ∣ j 1 , m 1 ⟩ ∣ j 2 , m 2 ⟩ . The magnetic quantum number rule is
M = m 1 + m 2 . M=m_1+m_2. M = m 1 + m 2 . For two spin-1 / 2 1/2 1/2
1 2 ⊗ 1 2 = 1 ⊕ 0. \frac12\otimes\frac12
=
1\oplus0. 2 1 ⊗ 2 1 = 1 ⊕ 0. The states are
∣ 1 , 1 ⟩ = ∣ ↑ ↑ ⟩ , ∣ 1 , 0 ⟩ = 1 2 ( ∣ ↑ ↓ ⟩ + ∣ ↓ ↑ ⟩ ) , ∣ 1 , − 1 ⟩ = ∣ ↓ ↓ ⟩ , ∣ 0 , 0 ⟩ = 1 2 ( ∣ ↑ ↓ ⟩ − ∣ ↓ ↑ ⟩ ) . \begin{aligned}
\lvert 1,1\rangle
&=
\lvert\uparrow\uparrow\rangle,\\
\lvert 1,0\rangle
&=
\frac{1}{\sqrt2}
\left(
\lvert\uparrow\downarrow\rangle
+
\lvert\downarrow\uparrow\rangle
\right),\\
\lvert 1,-1\rangle
&=
\lvert\downarrow\downarrow\rangle,\\
\lvert 0,0\rangle
&=
\frac{1}{\sqrt2}
\left(
\lvert\uparrow\downarrow\rangle
-
\lvert\downarrow\uparrow\rangle
\right).
\end{aligned} ∣ 1 , 1 ⟩ ∣ 1 , 0 ⟩ ∣ 1 , − 1 ⟩ ∣ 0 , 0 ⟩ = ∣ ↑↑ ⟩ , = 2 1 ( ∣ ↑↓ ⟩ + ∣ ↓↑ ⟩ ) , = ∣ ↓↓ ⟩ , = 2 1 ( ∣ ↑↓ ⟩ − ∣ ↓↑ ⟩ ) . Parity acts as spatial inversion:
Π X Π − 1 = − X , Π P Π − 1 = − P . \Pi\,\mathbf X\,\Pi^{-1}=-\mathbf X,
\qquad
\Pi\,\mathbf P\,\Pi^{-1}=-\mathbf P. Π X Π − 1 = − X , Π P Π − 1 = − P . For ordinary scalar wavefunctions,
Π 2 = I . \Pi^2=I. Π 2 = I . Orbital angular momentum is parity even:
Π L Π − 1 = L . \Pi\,\mathbf L\,\Pi^{-1}
=
\mathbf L. Π L Π − 1 = L . For an operator O O O π O \pi_O π O
Π O Π − 1 = π O O . \Pi O\Pi^{-1}
=
\pi_O O. Π O Π − 1 = π O O . A matrix element between parity eigenstates can be nonzero only if
π a π O π b = 1. \pi_a\pi_O\pi_b=1. π a π O π b = 1. Time reversal is antiunitary and satisfies
T i T − 1 = − i . TiT^{-1}=-i. T i T − 1 = − i . Its basic action is
T X T − 1 = X , T P T − 1 = − P . T\,\mathbf X\,T^{-1}=\mathbf X,
\qquad
T\,\mathbf P\,T^{-1}=-\mathbf P. T X T − 1 = X , T P T − 1 = − P . Angular momentum and spin are time-reversal odd:
T L T − 1 = − L , T S T − 1 = − S . T\,\mathbf L\,T^{-1}=-\mathbf L,
\qquad
T\,\mathbf S\,T^{-1}=-\mathbf S. T L T − 1 = − L , T S T − 1 = − S . For spinless particles in the position representation,
T = K . T=K. T = K . For spin-1 / 2 1/2 1/2
T = − i σ y K , T=-i\sigma_yK, T = − i σ y K , which gives
T 2 = − I . T^2=-I. T 2 = − I . For a parameter-dependent Hamiltonian,
H ( R ) ∣ n ( R ) ⟩ = E n ( R ) ∣ n ( R ) ⟩ . H(R)\lvert n(R)\rangle
=
E_n(R)\lvert n(R)\rangle. H ( R ) ∣ n ( R )⟩ = E n ( R ) ∣ n ( R )⟩ . The Berry connection convention is
A n ( R ) = i ⟨ n ( R ) ∣ ∇ R n ( R ) ⟩ . \mathbf A_n(R)
=
i\langle n(R)|\nabla_R n(R)\rangle. A n ( R ) = i ⟨ n ( R ) ∣ ∇ R n ( R )⟩ . For a closed loop C C C
γ n [ C ] = ∮ C A n ( R ) ⋅ d R . \gamma_n[C]
=
\oint_C\mathbf A_n(R)\cdot dR. γ n [ C ] = ∮ C A n ( R ) ⋅ d R . Under a gauge change
∣ n ( R ) ⟩ ↦ e i χ ( R ) ∣ n ( R ) ⟩ , \lvert n(R)\rangle
\mapsto
e^{i\chi(R)}
\lvert n(R)\rangle, ∣ n ( R )⟩ ↦ e i χ ( R ) ∣ n ( R )⟩ , the connection transforms as
A n ↦ A n − ∇ R χ . \mathbf A_n
\mapsto
\mathbf A_n-\nabla_R\chi. A n ↦ A n − ∇ R χ . The curvature preview is
B n = ∇ R × A n . \mathbf B_n
=
\nabla_R\times\mathbf A_n. B n = ∇ R × A n .
J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics , 3rd ed., Cambridge University Press, 2020.
A. R. Edmonds, Angular Momentum in Quantum Mechanics , Princeton University Press, 1957.
A. Shapere and F. Wilczek, eds., Geometric Phases in Physics , World Scientific, 1989.
D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum , World Scientific, 1988.