Spin Operator

Operator

Spin operators obey the angular-momentum algebra

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

For spin $1/2$,

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

where $\sigma_i$ are the Pauli matrices.

The spin projection along a unit vector $\hat{\mathbf n}$ is

$$S_{\hat{\mathbf n}} = \hat{\mathbf n}\cdot\mathbf S.$$

Assumptions

• Spin acts on an internal spin Hilbert space.
• The spin representation, such as $s=1/2$, $1$, or another value, is specified.
• If spin is coupled to spatial motion, the total Hilbert space is a tensor product of spatial and spin factors.
• Magnetic coupling conventions depend on charge, $g$ factor, and sign conventions.

Common Mistakes

• Treating spin as literal rotation of a small classical body.
• Forgetting that spin operators act on internal components, not on the coordinate argument by themselves.
• Mixing Pauli matrices $\sigma_i$ with spin operators $S_i$ without the factor $\hbar/2$.
• Assuming every spin system is spin $1/2$.

Canonical Links

• What Spin Is
• Pauli Matrices
• Spin Rotations
• Spin Operator Symbol

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.