When we say electron has spin of $\frac{1}{2}$, is that the value of the total spin of electron, or the projection on z axis, or the spin quantum number?
When we say "electron has spin of $\frac{1}{2}\hbar$", is that the value of the total spin or projection? Also, sometimes people say just "spin 1/2" without $\hbar$.
Is spin quantum number $s$ analogous to the l (total orbital angular momentum) or to the $m_s$ (projection of l).
I am confused because when I am trying to learn addition of angular momeneta (eg. in j-j coupling) where we use formula:
$$\vec{j}=\vec{l}+\vec{s}$$
to get total angular momentum for the particle and then we sum all into:
$$\vec{J}=\sum \vec{j}$$
What is the $s$ in this context? I mean in the equation: $\vec{j}=\vec{l}+\vec{s}$ since we are summing it with $\vec{l}$ then it must be spin projection on z axis right?
Best Answer
When we say that the electron has "spin half," we mean half of the quantum of angular momentum, $\hbar$. A good quantum mechanics text or other reference will help you derive that the Laplacian operator transforms into spherical coordinates like \begin{align} \nabla^2 &= \left(\frac\partial{\partial x}\right)^2 + \left(\frac\partial{\partial y}\right)^2 + \left(\frac\partial{\partial z}\right)^2 \\ &= \frac 1{r^2}\frac\partial{\partial r}\left(r^2\frac\partial{\partial r}\right) + \frac 1{r^2 \sin^2\theta}\left(\frac\partial{\partial\theta}\right)^2 + \frac 1{r^2 \sin\phi}\frac\partial{\partial\phi}\left(\sin\phi\frac\partial{\partial\theta}\right) \end{align} The angular parts of this operator act on the spherical harmonics to give eigenvalue $\ell(\ell+1)$ for integer $\ell$. This means that the effective form of the kinetic energy operator is \begin{align} \frac{\hbar^2}{2m}\nabla^2 &= \frac{\hbar^2}{2mr^2}\frac\partial{\partial r}\left(r^2\frac\partial{\partial r}\right) + \boxed{\frac{\hbar^2}{2mr^2}{\ell(\ell+1)}} \end{align} In the limit of large $\ell$, the term in the box is the same as the orbital kinetic energy for a point mass $m$ rotating some $r$ from the center of motion with angular momentum $L\sim\ell\hbar$.
This argument is what lets us say things like "$\hbar$ is the quantum of angular momentum," or "angular momentum comes in lumps, and the size of each lump is $\hbar$." Since $\hbar$ is the only quantum of angular momentum, sometimes we only count quanta and leave the unit off. Same as when someone quotes you a price and gives the value but not the currency ("I'll take your car off this tow truck for fifty-five").
Spin angular momentum falls naturally out of the Dirac question in a surprisingly elegant way. You get the same quantum, $\hbar$. However the Dirac equation describes objects whose intrinsic angular momentum is $\hbar/2$. Therefore the projection $m_s$ of the electron spin along any axis can be $\pm\frac12\hbar$, but never zero.
I think this might clarify your search for guidance on the rules for summing vector angular momenta.