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with $J=L+S$

Nope. In general, it can be shown that the permitted values of the $J$ quantum number for the total angular momentum that arise from a two sources characterized by quantum numbers $J_1$ and $J_2$ are given by $$ J = J_1 + J_2, J_1 + J_2 - 1, \dotsc, |J_1 - J_2| \, . $$ In this particular case, $J_1 = L$ and $J_2 = S$, so $J$ ranges from $L + S$ to $|L - S|$ in integral steps and not necessarily equal to $L + S$.

with $J=L+S$

Nope. In general, it can be shown that the permitted values of the $J$ quantum number for the total angular momentum that arise from a two sources characterized by quantum numbers $J_1$ and $J_2$ are given by $$ J = J_1 + J_2, J_1 + J_2 - 1, \dotsc, |J_1 - J_2| \, . $$ In this particular case, $J_1 = L$ and $J_2 = S$, so $J$ ranges from $L + S$ to $|L - S|$ in integral steps and not necessarily equal to $L + S$.

with $J=L+S$

Nope. In general, it can be shown that the permitted values of the $J$ quantum number for the total angular momentum that arise from two sources characterized by quantum numbers $J_1$ and $J_2$ are given by $$ J = J_1 + J_2, J_1 + J_2 - 1, \dotsc, |J_1 - J_2| \, . $$ In this particular case, $J_1 = L$ and $J_2 = S$, so $J$ ranges from $L + S$ to $|L - S|$ in integral steps and not necessarily equal to $L + S$.

Source Link
Wildcat
Wildcat
  • 19.1k
  • 2
  • 78
  • 107

with $J=L+S$

Nope. In general, it can be shown that the permitted values of the $J$ quantum number for the total angular momentum that arise from a two sources characterized by quantum numbers $J_1$ and $J_2$ are given by $$ J = J_1 + J_2, J_1 + J_2 - 1, \dotsc, |J_1 - J_2| \, . $$ In this particular case, $J_1 = L$ and $J_2 = S$, so $J$ ranges from $L + S$ to $|L - S|$ in integral steps and not necessarily equal to $L + S$.