In 1920s, Pauli faced the problem to solve properties of electrons:

1. Magnetic behavior like a tiny bar magnet
2. Two-valued “intrinsic” angular momentum (spin up / spin down)
3. Spectral line splittings (Zeeman effect) that orbital angular momentum alone could not explain

So Pauli needed a mathematical object that:

• Acts like angular momentum
• But has only two internal states
• Works inside Schrödinger’s quantum mechanics

Experiments of measuring spinor S on z axis, i.e. Sz yield two outputs; +z and -z, half and half:

Note Pauli matrices describe spin, but the physical spin operator includes the factor $\hbar/2$.

Spinor matrix combined with h/2 is spinor operator S!

It’s essential to understand the concept of spinor operator, spinor quantum state/electron, and <Svector>, with the following concrete example.

the spinor/electron is in this quantum state: