# What do the $j$ and $m$ stand for in $|j,m\rangle$ for angular momentum in quantum mechanics?

I'm assuming it is a jth state with m value as total angular momentum?

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Wikipedia discusses quantization of angular momentum e.g. here and here. – Qmechanic♦ Jan 28 at 1:45
– Chris White Jan 28 at 2:13

## 1 Answer

The states $|j,m\rangle$ are simultaneous eigenstates of the total angular momentum squared operator $\mathbf J^2$ and the $z$-component of the total angular momentum operator $J_z$. The letter $j$ is related to the eigenvalue of the operator $\mathbf J^2$, while the letter $m$ gives the eigenvalue of the operator $J_z$. Specifically $$\mathbf J^2|j,m\rangle =\hbar^2 j(j+1)|j,m\rangle, \qquad J_z|j,m\rangle = \hbar \,m|j,m\rangle$$ Given these considerations, $j$ is called the total angular momentum quantum number.

Hope that helps! Let me know if you'd like more details.

Cheers!

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But does J here include spin or not?  If it doesn't, isn't it notationally more common to denote the total ang. momentum by $\ell$ and not $j$?  I thought $j$ was for the total ang. momentum including spin... – daaxix Jan 29 at 18:30
@daaxix The letter J is generically used for all forms of angular momentum in quantum.  What $J_i$ denotes depends on the context.  As long as the components of $\mathbf J$ satisfy the angular momentum commutation relations $[J_i, J_j] = i\hbar\epsilon_{ijk}J_k$, one can show that the analysis of its eigenvalues and eigenvectors leads to states labeled in this way.  So $J_i$ could denote the components of orbital angular momentum, or spin, or any combination of both, and the notation will carry through. – joshphysics Jan 29 at 18:40
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