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Mathematics for Physics

145 Spin

An intrinsic angular momentum that is used to describe a particle’s built-in rotational quantum number, where intrinsic means the angular momentum is not built from \(\mathbf{r}\) and \(\mathbf{p}\).

\(s=1/2\) for the electron. An electron has spin quantum number \(s=1/2\). This principle is used to assign two spin states to every orbital.

The spin eigenvalue equations are

\[ S^{2}|s,m_{s}\rangle = \hbar^{2}s(s+1)|s,m_{s}\rangle \]

\[ S_{z}|s,m_{s}\rangle = m_{s}\hbar|s,m_{s}\rangle \]

where

  • \(s\) is the spin quantum number.
  • \(m_{s}\) is the spin projection.
  • \(\hbar\) is the reduced Planck constant.

The two projections. For \(s=1/2\) the projections are \(m_{s}=\pm 1/2\). This principle is used to write the Stern-Gerlach pair of beams.

The Pauli representation. Spin operators act on two-component spinors, not on functions of \(\mathbf{r}\). A spinor is a two-component object that represents a spin-\(1/2\) state. This principle is used to write \(\mathbf{S}=\dfrac{\hbar}{2}\boldsymbol{\sigma}\).

The Pauli representation is

\[ \mathbf{S} = \dfrac{\hbar}{2}\boldsymbol{\sigma} \]

where

  • \(\mathbf{S}\) is the spin operator.
  • \(\boldsymbol{\sigma}\) are the Pauli matrices.

The sign change under a \(2\pi\) rotation. A \(2\pi\) rotation of a spin-\(1/2\) state changes its sign. This principle is used to distinguish spinors from ordinary vectors.

Note: Also called spin angular momentum. Also called intrinsic angular momentum.

145.1 References

  1. Sakurai, J. J., & Napolitano, J. Modern Quantum Mechanics. Cambridge University Press, 2021. — spin-\(1/2\) and Pauli matrices.
  2. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — electron spin.