Spin, Rotation, and Phase Conventions

Rotation signs and basis phases affect intermediate formulas, even though consistently calculated probabilities are unchanged. Use this page to compare angular-momentum tables and translate between active rotations, spinor bases, and spherical-harmonic conventions.

Helpful background. The general conventions and bra–ket notation explain the inner product and state normalization.

Hilbert-Space and Inner-Product Convention

We use the standard physics convention:

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

Normalized kets differing by an overall phase represent the same physical ray (with real $\alpha$):

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

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 squared 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 an irreducible angular-momentum representation, $j=0,\tfrac12,1,\ldots$. 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,

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

Positive ladder coefficients fix the relative phases within a multiplet, but leave an overall phase for each multiplet. Spherical harmonics and coupled bases need the additional conventional phase choices stated below.

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. Here $|+\rangle$ and $|-\rangle$ mean the $S_z$ eigenbasis (often written $|+z\rangle,|-z\rangle$). Quantum-information texts also use these signs for the $x$ basis, so check the stated axis. We use

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

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

For integer $0\leq m\leq\ell$, define the phase-free associated Legendre function

$$P_\ell^m(x)=(1-x^2)^{m/2}\frac{d^mP_\ell(x)}{dx^m}.$$

The Condon–Shortley phase then appears explicitly in the normalized spherical harmonic:

$$Y_\ell^m(\theta,\phi)=(-1)^m \sqrt{\frac{2\ell+1}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} P_\ell^m(\cos\theta)e^{im\phi}.$$

Define negative orders by $Y_\ell^{-m}=(-1)^m(Y_\ell^m)^*$ for $m\geq0$. The normalization is $\int |Y_\ell^m|^2\,d\Omega=1$, with $d\Omega=\sin\theta\,d\theta\,d\phi$. In particular, $P_1^1(x)=\sqrt{1-x^2}$ and $Y_1^1=-\sqrt{3/(8\pi)}\sin\theta\,e^{i\phi}$.

Many references include $(-1)^m$ inside $P_\ell^m$ instead. Multiplying by it again would give the wrong odd-$m$ sign. The associated-Legendre page gives the explicit translation to that convention.

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

$$m_1+m_2=M.$$

The allowed totals are $J=|j_1-j_2|,\ldots,j_1+j_2$. We use the Condon–Shortley choice: at $M=J$, choose the coefficient with $m_1=j_1$ positive, then obtain lower $M$ with positive total-ladder coefficients. Together with orthonormality, this fixes the standard real tables. Reversing the order of the coupled factors gives

$$\langle j_2m_2,j_1m_1|JM\rangle =(-1)^{j_1+j_2-J}\langle j_1m_1,j_2m_2|JM\rangle.$$

Changing this phase convention consistently changes basis coordinates, not measurable predictions. See Clebsch–Gordan tables and conventions before combining tables from different sources.

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:

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

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.

References

• D. M. Brink and G. R. Satchler, Angular Momentum, 3rd ed., Oxford University Press, 1993.
• 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.