Notation and Conventions

This page fixes the default conventions for the symmetry, spin, and geometry volume. Pages may choose specialized conventions when needed, but they should say so explicitly and explain how to translate.

Hilbert-Space and Inner-Product Convention

We use the standard physics convention:

$$\langle\phi|\psi\rangle \quad \text{is conjugate-linear in }\phi \text{ and linear in }\psi.$$

Kets differing by an overall nonzero complex phase represent the same physical ray:

$$\lvert\psi\rangle \sim e^{i\alpha}\lvert\psi\rangle.$$

Relative phases remain physically meaningful.

Commutators and Levi-Civita Symbol

Commutators and anticommutators are

$$[A,B]=AB-BA, \qquad \{A,B\}=AB+BA.$$

The Levi-Civita symbol is defined by

$$\epsilon_{xyz}=+1,$$

with complete antisymmetry.

Angular Momentum Algebra

Angular momentum components satisfy

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

The total angular momentum operator is

$$J^2=J_x^2+J_y^2+J_z^2.$$

Simultaneous eigenstates of $J^2$ and $J_z$ are written $\lvert j,m\rangle$:

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

and

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

For fixed $j$,

$$m=-j,-j+1,\ldots,j.$$

Ladder Operators

The ladder operators are

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

With the standard Condon–Shortley phase convention,

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

This convention fixes many signs in spherical harmonics, Clebsch–Gordan coefficients, and Wigner symbols.

Spin-1/2 Conventions

For spin-$1/2$ systems,

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

where the Pauli matrices follow the convention in the Pauli Matrices table. The $S_z$ basis is usually written

$$\lvert+\rangle = \begin{pmatrix}1\\0\end{pmatrix}, \qquad \lvert-\rangle = \begin{pmatrix}0\\1\end{pmatrix},$$

with

$$S_z\lvert\pm\rangle = \pm\frac{\hbar}{2}\lvert\pm\rangle.$$

Rotation Operators

For an active rotation by angle $\theta$ about a unit vector $\hat{\mathbf n}$, the quantum rotation operator is

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

For spin-$1/2$,

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

The sign convention matters. Passive coordinate rotations can introduce the inverse transformation; pages that use passive language should state it. See Active and Passive Transformations for the canonical convention.

Spherical Harmonics and Phase Convention

Unless stated otherwise, spherical harmonics use the Condon–Shortley phase convention. This is the convention most standard quantum mechanics and angular-momentum tables use:

$$Y_\ell^m(\theta,\phi) \propto (-1)^m P_\ell^m(\cos\theta)e^{im\phi}.$$

The proportionality includes the standard normalization. When comparing references, always check the phase convention for associated Legendre functions and spherical harmonics.

The Toolkit page Associated Legendre Functions records the convention used here and how to translate references that absorb the Condon–Shortley phase into $P_\ell^m$.

Clebsch–Gordan Coefficients

Coupled and uncoupled angular momentum bases are related by

$$\lvert j_1j_2;JM\rangle = \sum_{m_1,m_2} \langle j_1m_1,j_2m_2|JM\rangle \lvert j_1m_1\rangle\lvert j_2m_2\rangle.$$

The coefficient is zero unless

$$m_1+m_2=M.$$

Detailed tables depend on phase conventions; this volume uses the standard Condon–Shortley convention unless a page explicitly declares otherwise.

Units

Most pages keep $\hbar$ explicit. Pages using $\hbar=1$ should declare that convention near the top. Keeping $\hbar$ visible is especially useful when comparing generators:

$$U(\alpha)=e^{-i\alpha G/\hbar}.$$

Cross-Links

• Conventions Overview
• Concept Map
• Active and Passive Transformations
• Bra-Ket Notation
• Commutators
• Angular Momentum Algebra
• SU(2)
• Wigner D-Matrices
• Wigner 3j, 6j, and 9j Symbols
• Spherical Harmonics
• Pauli Matrices
• Commutator Table

References

• A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.
• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.
• D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, 1988.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Using the convention for $J_\pm$, compute $J_+\lvert j,j\rangle$.

Solution

The coefficient contains

$$\sqrt{j(j+1)-j(j+1)}=0,$$

so $J_+\lvert j,j\rangle=0$. The state $\lvert j,j\rangle$ is the highest-weight state in the multiplet.

2. What is the spin-$1/2$ rotation operator for a rotation by $2\pi$ about the $z$ axis?

Solution

For spin-$1/2$,

$$U(\hat z,2\pi) = \exp(-i\pi\sigma_z).$$

Since the eigenvalues of $\sigma_z$ are $\pm1$, both eigenvalues of $U$ are $-1$. Thus $U(\hat z,2\pi)=-I$ on the spinor Hilbert space.