Spin angular momentum?

Question

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?

Answer 1

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 $-\frac 12 \hbar$.