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What is the difference between the orbital angular momentum operator $\hat{L}$ and the angular momentum operator $\hat{J}$ ? Is it that $\hat{J}$ is a combination of a particles spin and its classically derived angular momentum $\hat{L}$ ?

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$\vec{J}$ is commonly used for "total" angular momentum. The vast majority of cases, it is "orbital angular momentum plus spin", in other words:

$$\vec{J}=\vec{L}+\vec{S}$$

But it could be used for any sum of angular momenta.


That was the short answer; but how do you actually add angular momenta?

When you work with spins, you're no longer working on a Hilbert Space, but on a more complex space, given by

$$\mathcal{H}\otimes\mathbb{C}^n$$

n is the spin dimension. For the typical $s=½$ case, $m_s$ has two possibilities ($\pm ½$, or "up" and "down", so $n=2).

  • So $\vec{L}$ acts on the Hilbert space.
  • $\vec{S}$ acts on $\mathbb{C}^2$.
  • $\vec{J}$ acts on $\mathcal{H}\otimes\mathbb{C}^2$

The way this works is "extension with the identities":

$$\vec{J}=\vec{L}\otimes \mathbb{I}_S + \mathbb{I}_H\otimes \vec{S}$$

So this is basically "J means appliying $\vec{L}$ to the Hilbert part, leaving the spin alone; and then acting with $\vec{S}$ on the spin part, without touching the other one". That's why we're lazy to write all that, and we just use

$$\vec{J}=\vec{L}+\vec{S}$$

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