Spin Operator

The spin operator is written $\mathbf S$ or by components $S_x,S_y,S_z$. It represents intrinsic angular momentum, not literal rotation of a small extended body.

Default Meaning

Spin components obey the same angular momentum algebra as other rotation generators:

$$[S_i,S_j] = i\hbar\sum_k\epsilon_{ijk}S_k.$$

For spin-$1/2$ systems, the physical spin operators are related to Pauli matrices by

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

Thus the $S_z$ eigenvalues for a spin-$1/2$ particle are $\pm\hbar/2$, while the corresponding $\sigma_z$ eigenvalues are $\pm1$.

Mathematical Type

Spin operators act on an internal finite-dimensional Hilbert space for a fixed spin. A spin-$s$ irreducible representation has dimension $2s+1$ and basis states usually written $\lvert s,m\rangle$.

Units

The operator $\mathbf S$ has units of angular momentum. Pauli matrices $\sigma_i$ are dimensionless, so the factor $\hbar/2$ is not optional when the physical spin observable is meant.

Common Variants

• $S_i$: spin component operator.
• $S_\pm=S_x\pm iS_y$: spin raising and lowering operators.
• $\boldsymbol\sigma$: Pauli-matrix vector for spin-$1/2$ or two-level notation.
• $\mathbf J=\mathbf L+\mathbf S$: total angular momentum when orbital and spin parts are both present.

Convention Warnings

• Do not use $\sigma_i$ and $S_i$ interchangeably without the factor $\hbar/2$.
• The $S_z$ basis is a spin basis, not automatically the Hamiltonian eigenbasis.
• $S$ may also mean action, entropy, or scattering matrix; typography and context decide.
• Spin and orbital angular momentum share an algebra but have different physical origins.

Canonical Links

• What Spin Is
• Spin and Pauli Matrix Conventions
• Pauli Matrices
• Spin-1/2 Hilbert Space

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai, Advanced Quantum Mechanics, Addison-Wesley, 1967.