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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?

Qmechanic
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  • "the electron spins around its axis..." I beg you, tell me you know it's wrong. – Jeanbaptiste Roux Sep 27 '21 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. – Markoul11 Sep 27 '21 at 20:12
  • Reading list: 1.https://physics.stackexchange.com/questions/365784/spin-angular-momentum-question 2.https://physics.stackexchange.com/questions/296000/how-is-the-spin-angular-momentum-h-4-pi-determined/ 3.https://physics.stackexchange.com/questions/614941/what-is-the-difference-between-angular-momentum-of-electron-by-bohr-and-orbital –  Sep 28 '21 at 02:51

1 Answers1

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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 $-\frac 12 \hbar$.

Thomas Fritsch
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