Spin and Symmetry Formula Sheet

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.

Quantum Symmetries

For normalized vectors, a bijective ray symmetry preserves transition probabilities:

$$\left|\langle\phi|\psi\rangle\right|^2 \mapsto \left|\langle\phi|\psi\rangle\right|^2.$$

Unitary operators are linear:

$$U(a\lvert\psi\rangle+b\lvert\phi\rangle) = aU\lvert\psi\rangle+bU\lvert\phi\rangle.$$

Antiunitary operators are antilinear:

$$T(a\lvert\psi\rangle+b\lvert\phi\rangle) = a^*T\lvert\psi\rangle+b^*T\lvert\phi\rangle.$$

A Hamiltonian symmetry satisfies

$$UHU^{-1}=H$$

for a unitary or antiunitary symmetry operator, with antiunitarity handled carefully when complex scalars appear.

Generators and Conservation

For a strongly continuous one-parameter unitary group,

$$U(\epsilon) = \exp\left(-\frac{i}{\hbar}\epsilon G\right).$$

If $G$ has no explicit time dependence and

$$[G,H]=0,$$

then $G$ is conserved where the commutator and Heisenberg derivative are defined. For unbounded operators, invariance under the full unitary group and preservation of domains are the appropriate stronger statements.

Translations

The translation operator by displacement $a$ is

$$T(a) = \exp\left(-\frac{i}{\hbar}aP\right).$$

With the active convention used here,

$$(T(a)\psi)(x)=\psi(x-a).$$

The canonical commutator is

$$[X,P]=i\hbar.$$

Rotations and Angular Momentum

A rotation by angle $\theta$ about unit vector $\hat{\mathbf n}$ is

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{\hbar}\theta\,\hat{\mathbf n}\cdot\mathbf J \right).$$

The angular momentum algebra is

$$[J_i,J_j] = i\hbar\sum_k\epsilon_{ijk}J_k.$$

The simultaneous eigenvalue equations are

$$J^2\lvert j,m\rangle = \hbar^2j(j+1)\lvert j,m\rangle,$$

and

$$J_z\lvert j,m\rangle = \hbar m\lvert j,m\rangle.$$

The ladder operators are

$$J_\pm=J_x\pm iJ_y.$$

Their action is

$$J_\pm\lvert j,m\rangle = \hbar \sqrt{j(j+1)-m(m\pm1)} \lvert j,m\pm1\rangle.$$

Orbital Angular Momentum

Orbital angular momentum is

$$\mathbf L=\mathbf R\times\mathbf P.$$

For spherical harmonics,

$$L^2Y_\ell^m = \hbar^2\ell(\ell+1)Y_\ell^m,$$

and

$$L_zY_\ell^m = \hbar mY_\ell^m.$$

Spin-One-Half

The spin-$1/2$ operators are

$$S_i=\frac{\hbar}{2}\sigma_i.$$

The Pauli matrices obey

$$\sigma_i\sigma_j = \delta_{ij}I + i\sum_k\epsilon_{ijk}\sigma_k.$$

For commuting scalar-component vectors $\mathbf a$ and $\mathbf b$ (with a bilinear dot product if complex),

$$(\mathbf a\cdot\boldsymbol\sigma) (\mathbf b\cdot\boldsymbol\sigma) = (\mathbf a\cdot\mathbf b)I + i(\mathbf a\times\mathbf b)\cdot\boldsymbol\sigma.$$

A spinor rotation is

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{2}\theta\,\hat{\mathbf n}\cdot\boldsymbol\sigma \right).$$

Equivalently,

$$U(\hat{\mathbf n},\theta) = \cos\frac{\theta}{2}\,I -i\sin\frac{\theta}{2}\, \hat{\mathbf n}\cdot\boldsymbol\sigma.$$

The Bloch representation of a spin-$1/2$ density matrix is

$$\rho = \frac12 \left( I+\mathbf r\cdot\boldsymbol\sigma \right).$$

Here $\mathbf r$ is real and $|\mathbf r|\le1$; equality holds for pure states.

Addition of Angular Momentum

For independent angular momenta satisfying $[J_{1i},J_{2j}]=0$,

$$\mathbf J=\mathbf J_1+\mathbf J_2.$$

Allowed total angular momenta are

$$J = \lvert j_1-j_2\rvert, \lvert j_1-j_2\rvert+1, \ldots, j_1+j_2.$$

The coupled basis expansion is

$$\lvert j_1,j_2;J,M\rangle = \sum_{m_1,m_2} \langle j_1,m_1;j_2,m_2|J,M\rangle \lvert j_1,m_1\rangle \lvert j_2,m_2\rangle.$$

The magnetic quantum number rule is

$$M=m_1+m_2.$$

For two spin-$1/2$ particles,

$$\frac12\otimes\frac12 = 1\oplus0.$$

The states are

$$\begin{aligned} \lvert 1,1\rangle &= \lvert\uparrow\uparrow\rangle,\\ \lvert 1,0\rangle &= \frac{1}{\sqrt2} \left( \lvert\uparrow\downarrow\rangle + \lvert\downarrow\uparrow\rangle \right),\\ \lvert 1,-1\rangle &= \lvert\downarrow\downarrow\rangle,\\ \lvert 0,0\rangle &= \frac{1}{\sqrt2} \left( \lvert\uparrow\downarrow\rangle - \lvert\downarrow\uparrow\rangle \right). \end{aligned}$$

Parity

Parity acts as spatial inversion:

$$\Pi\,\mathbf X\,\Pi^{-1}=-\mathbf X, \qquad \Pi\,\mathbf P\,\Pi^{-1}=-\mathbf P.$$

For ordinary scalar wavefunctions,

$$\Pi^2=I.$$

Orbital angular momentum is parity even:

$$\Pi\,\mathbf L\,\Pi^{-1} = \mathbf L.$$

For an operator $O$ with parity $\pi_O$,

$$\Pi O\Pi^{-1} = \pi_O O.$$

A matrix element between parity eigenstates can be nonzero only if

$$\pi_a\pi_O\pi_b=1.$$

Time Reversal

Time reversal is antiunitary and satisfies

$$TiT^{-1}=-i.$$

Its basic action is

$$T\,\mathbf X\,T^{-1}=\mathbf X, \qquad T\,\mathbf P\,T^{-1}=-\mathbf P.$$

Angular momentum and spin are time-reversal odd:

$$T\,\mathbf L\,T^{-1}=-\mathbf L, \qquad T\,\mathbf S\,T^{-1}=-\mathbf S.$$

For spinless particles in the position representation,

$$T=K.$$

For spin-$1/2$, a common convention is

$$T=-i\sigma_yK,$$

which gives

$$T^2=-I.$$

Berry Phase

For a parameter-dependent Hamiltonian,

$$H(R)\lvert n(R)\rangle = E_n(R)\lvert n(R)\rangle.$$

The Berry connection convention is

$$\mathbf A_n(R) = i\langle n(R)|\nabla_R n(R)\rangle.$$

For a closed loop $C$ of an isolated nondegenerate eigenspace, choose a periodic normalized eigenvector along the loop. Then

$$\gamma_n[C] = \oint_C\mathbf A_n(R)\cdot dR.$$

Under a gauge change

$$\lvert n(R)\rangle \mapsto e^{i\chi(R)} \lvert n(R)\rangle,$$

the connection transforms as

$$\mathbf A_n \mapsto \mathbf A_n-\nabla_R\chi.$$

In a nonperiodic gauge along a closed ray path, add $\arg\langle n(R(0))|n(R(T))\rangle$ to the line integral. The phase is defined modulo $2\pi$. For a three-dimensional parameter space the curvature can be written as

$$\mathbf B_n = \nabla_R\times\mathbf A_n.$$

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• 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.