Introduction: A Strange Kind of Vector

Spin states transform under rotations in a peculiar way: a 360° rotation gives a minus sign, not identity. Objects with this property are called spinors.

Rotation Operator for Spin

Rotation by angle \(\theta\) about axis \(\hat{n}\):

\[R(\hat{n}, \theta) = e^{-i\theta\hat{n}\cdot\vec{\sigma}/2} = \cos\frac{\theta}{2}I - i\sin\frac{\theta}{2}(\hat{n}\cdot\vec{\sigma})\]

The 2π Rotation

\[R(\hat{n}, 2\pi) = \cos(\pi)I - i\sin(\pi)(\hat{n}\cdot\vec{\sigma}) = -I\]

Rotating by 360° gives a minus sign! It takes 720° to return to the original state.

Physical Consequence

This sign can be measured! In neutron interferometry, rotating one path by 360° shifts the interference pattern—the minus sign has physical effects.

The Quantum Connection

Spinor behavior underlies the distinction between fermions and bosons. Particles with half-integer spin (1/2, 3/2, ...) are spinors obeying Fermi-Dirac statistics—they can't occupy the same state. This leads to the Pauli exclusion principle and the structure of atoms.