# Spin angular momentum?

An electron spins around its axis and magnetic field is produced. It can spin either in clockwise $$\left(\frac{1}{2}\right)$$ or in counterclockwise $$\left(\frac{-1}{2}\right)$$ direction.

The spin angular momentum is given by $$S=\sqrt{s\left( s+1\right) }\cdot \dfrac{h}{2\pi }$$.

If $$s$$ is $$\frac{1}{2}$$, then the spin angular momentum is $$\sqrt{\dfrac{1}{2}\left( \dfrac{1}{2}+1\right) }\cdot \dfrac{h}{2\pi }=\dfrac{\sqrt{3}}{2}\cdot \dfrac{h}{2\pi }$$

and If $$s$$ is $$\frac{-1}{2}$$, then the spin angular momentum is $$\sqrt{\dfrac{-1}{2}\left( -\dfrac{1}{2}+1\right) }\cdot \dfrac{h}{2\pi }=\dfrac{i}{2}\cdot \dfrac{h}{2\pi }$$.

Why is the spin angular momentum of $$s=\frac{-1}{2}$$ imaginary value, is this possible? What is the meaning of this; what does it mean physically when particles have spin half and negative half integer and their spin angular momentum real or imaginary?

• "the electron spins around its axis..." I beg you, tell me you know it's wrong. Commented Sep 27, 2021 at 17:59
• Intrinsic total spin angular momentum of the electron at rest is always ~0.866ℏ independent of its spin z-axis component direction, spin up +1/2 or spin down -1/2. Commented Sep 27, 2021 at 20:12
• – user279106
Commented Sep 28, 2021 at 2:51

You are confusing the square of the spin vector ($$\vec{S}^2$$) with its 3 vector components ($$S_x, S_y, S_z$$).
The square of the spin $$\vec{S}^2$$ always has the value $$s(s+1)\hbar^2$$, which is positive. And the electron has $$s=\frac 12$$ (no negative sign).
The components $$S_x$$, $$S_y$$ and $$S_z$$ each have two possible values (a positive and a negative). These are $$+\frac 12 \hbar$$ and $$-\frac 12 \hbar$$.