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I wonder what is the number of different possible values that the total angular momentum quantum number $j$ can take for fixed values of the angular quantum number $\ell$ and the spin quantum number $s$.

If $j$ can take the following range of values, jumping only in integer steps,

$$|\ell-s| \leq j \leq \ell+s$$

is there any formula that gives the number of different values it can take?

Qmechanic
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Invenietis
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1 Answers1

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This depends on which one is bigger.

  • If $\ell$ is bigger than $s$, then $|\ell-s|=\ell-s$, so $j$ goes from $\ell-s$ to $\ell+s$ in unit steps, which means $2s+1$ different values.
  • If $\ell$ is smaller than $s$, you get the same story with swapped symbols, so you get $2\ell+1$ different values.
  • If they're equal, then both options apply.
Emilio Pisanty
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