Spin and Angular-Momentum Formulas

Spin and angular momentum share one rotation algebra but appear in several distinct calculation settings: orbital wavefunctions, intrinsic spin, effective pseudospin, coupled multiplets, and tensor-operator selection rules. These cards provide a convention-aware route from the basic commutators to matrix elements and transition amplitudes.

Each card is a lookup layer. The Symmetry, Angular Momentum, and Spin volume remains the canonical home for derivations, representation theory, and physical interpretation.

Find the right card

Task	Card
Check commutators, spectra, dimensions, or rotation phases	Angular Momentum Algebra
Convert a spinor, axis, or two-level Hamiltonian into matrices and probabilities	Spin-Half Matrices
Raise, lower, repeat, or assemble a fixed-$j$ matrix	Ladder-Operator Action
Convert coupled and uncoupled bases or evaluate a scalar coupling	Addition of Angular Momentum
Factor spherical-tensor amplitudes and apply angular selection rules	Wigner–Eckart Theorem

Core formula map

Structure	Formula
Rotation algebra	$[J_i,J_j]=i\hbar\epsilon_{ijk}J_k$
Multiplet eigenvalues	$J^2\lvert j,m\rangle=\hbar^2j(j+1)\lvert j,m\rangle$ and $J_z\lvert j,m\rangle=\hbar m\lvert j,m\rangle$
Ladder action	$J_\pm\lvert j,m\rangle=\hbar\sqrt{j(j+1)-m(m\pm1)}\lvert j,m\pm1\rangle$
Spin one-half	$S_i=\hbar\sigma_i/2$
Total angular momentum	$\mathbf J=\mathbf J_1+\mathbf J_2$
Allowed coupled sectors	$J=\lvert j_1-j_2\rvert,\ldots,j_1+j_2$
Scalar coupling	$\mathbf J_1\cdot\mathbf J_2=(J^2-J_1^2-J_2^2)/2$
Tensor factorization	$\langle j'm'\rvert T_q^{(k)}\lvert jm\rangle=C_{jm,kq}^{j'm'}\langle j'\lVert T^{(k)}\rVert j\rangle/\sqrt{2j'+1}$

The same symbol $J$ can denote an operator vector, its total quantum number, or a particular component when decorated. Keep boldface, subscripts, and ket labels visible in intermediate work.

How the cards fit together

Algebra and representations

The Angular Momentum Algebra card gives the common skeleton:

$$[J_i,J_j] = i\hbar\epsilon_{ijk}J_k, \qquad [J^2,J_i]=0.$$

It distinguishes dimensionful operators $J_i$ from dimensionless generators $J_i/\hbar$, orbital integer labels from spinor half-integer labels, and full rotational symmetry from conservation of one component.

Spin one-half

The Spin-Half Matrices card specializes the algebra to two dimensions. It is organized around directional projectors,

$$P_\pm(\boldsymbol n) = \frac12 \left( I\pm\boldsymbol n\cdot\boldsymbol\sigma \right),$$

active spinor rotations, Bloch-vector probabilities, and exact exponentiation of

$$H=c_0I+\boldsymbol c\cdot\boldsymbol\sigma.$$

Use the Pauli Matrices Table when the task is a fixed identity lookup rather than a measurement or evolution workflow.

Ladders and finite multiplets

The Ladder-Operator Action card extends the one-step formula to repeated powers, normalized highest-weight construction, general matrix elements, and the differential action on spherical harmonics. It also distinguishes finite angular-momentum ladders from the unbounded ideal harmonic-oscillator ladder.

Use the Spin Matrices Table for fixed low-$j$ matrix entries. Use the formula card to construct or verify an arbitrary representation.

Coupled angular momenta

The Addition of Angular Momentum card connects the uncoupled basis

$$\lvert j_1,m_1;j_2,m_2\rangle$$

to the coupled basis

$$\lvert j_1,j_2;J,M\rangle.$$

It gives triangle and dimension checks, Clebsch–Gordan unitarity, exchange symmetry, spin-half couplings, total-$J$ projectors, and the eigenvalues of isotropic scalar interactions. Coefficient tables remain separate because their phase and factor-order conventions must be stated row by row.

Tensor operators and selection rules

The Wigner–Eckart Theorem card factors the dependence on $m,m'$, and $q$ from the reduced matrix element. It gives a fixed reduced-element convention, selection-rule workflow, line-strength checks, Hermitian reciprocity, and scalar and vector specializations.

The theorem is not a full spectroscopy model. Rotational permission must be combined with parity, exchange symmetry, state mixing, operator content, populations, and dynamics.

Convention ledger

Angular-momentum calculations are reliable only when several conventions are kept aligned.

Operator normalization

This compendium uses dimensionful generators:

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

If a source instead uses $T_i=J_i/\hbar$, its commutator has no explicit $\hbar$. Do not combine that commutator with dimensionful eigenvalues or ladder actions.

For spin one-half,

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

Pauli matrices are dimensionless and have eigenvalues $\pm1$; physical spin components have eigenvalues $\pm\hbar/2$.

Basis order

Fixed spin matrices in this reference use descending $m$ unless a page states otherwise:

$$\lvert j,j\rangle, \lvert j,j-1\rangle, \ldots, \lvert j,-j\rangle.$$

Reversing this order changes matrix placement. It does not change the abstract operator if every state and operator is transformed consistently.

Phase convention

The ladder coefficients are chosen real and positive, compatible with the Condon–Shortley convention. Clebsch–Gordan coefficients, spherical harmonics, and Wigner symbols inherit phase choices from their basis states. An overall phase for one multiplet is conventional; mixing relative phases across tables is not.

Active rotations

Active state rotations use

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

Passive coordinate rotations or reversed conjugation order produce inverse three-dimensional rotations. State the convention before comparing signs.

Spherical components

For a vector,

$$V_0=V_z, \qquad V_{\pm1} = \mp\frac{V_x\pm iV_y}{\sqrt2}.$$

The component phases must match the Wigner–Eckart and coefficient convention.

Reduced matrix elements

This compendium uses

$$\begin{aligned} \langle j'm'\rvert T_q^{(k)}\lvert jm\rangle &= \frac{ \langle j,m;k,q\vert j',m'\rangle }{\sqrt{2j'+1}} \\ &\qquad\times \langle j'\lVert T^{(k)}\rVert j\rangle. \end{aligned}$$

Some sources absorb the square-root factor into the double-bar symbol. The defining equation, not the notation alone, determines the convention.

Choosing a basis

Dominant structure	Usually convenient basis
Strong field resolving separate projections	uncoupled $\lvert j_1m_1;j_2m_2\rangle$
Isotropic scalar coupling	coupled $\lvert j_1j_2;JM\rangle$
Spin-half directional measurement	eigenbasis of $\boldsymbol n\cdot\boldsymbol\sigma$
Axially symmetric Hamiltonian	basis with definite conserved projection
Fully rotationally invariant Hamiltonian	total-$j,m$ multiplets
Tensor transition amplitude	angular-momentum basis plus spherical components

In an intermediate-field regime, neither limiting basis may diagonalize the Hamiltonian. A basis remains useful for matrix construction even when its labels are not exact conserved quantum numbers.

Distinguish the physical realizations

The same $SU(2)$ algebra does not make all angular momenta physically interchangeable:

• orbital angular momentum is generated by spatial rotations and, in ordinary three-dimensional scalar wave mechanics, has integer $\ell$;
• intrinsic spin acts on internal spinor degrees of freedom and may have half-integer labels;
• total angular momentum generates simultaneous rotations of all coupled factors;
• pseudospin is a two-state or multiplet label that may transform algebraically like spin without being literal mechanical angular momentum;
• nuclear, electronic, rotational, and hyperfine angular momenta carry different magnetic moments and coupling constants.

Before applying a magnetic-field formula, identify the physical magnetic moment or gyromagnetic tensor rather than inferring it from the algebra alone.

Reliability checks

• A $j$ multiplet contains exactly $2j+1$ states.
• The physical magnitude is $\hbar\sqrt{j(j+1)}$, not $\hbar j$.
• $J_+$ annihilates $m=j$ and $J_-$ annihilates $m=-j$.
• In an orthonormal basis, $J_-=(J_+)^\dagger$.
• Coupled-sector dimensions sum to the tensor-product dimension.
• Every nonzero Clebsch–Gordan coefficient satisfies $M=m_1+m_2$.
• A Wigner–Eckart amplitude satisfies $m'=m+q$ and the triangle rule.
• Parity and other discrete symmetries are checked separately from rotations.
• A result imported from a table carries that table’s phase, basis-order, and normalization conventions.

Related reference layers

• Spin Matrices and Pauli Matrices provide fixed entries.
• Clebsch–Gordan Coefficients and Wigner Symbols provide convention-fixed coupling tables.
• Formula Sheet gives a compressed volume-level scan.
• Angular-Momentum Problems and Selection-Rule Problems provide worked practice.
• Notation and Conventions records the volume-wide choices.

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.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• R. N. Zare, Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics, Wiley, 1988.
• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.