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[tex]\textbf{J}_1[/tex] and [tex]\textbf{J}_2[/tex] are angular momentum (vector-)operators.

In many textbooks [tex]\left[\textbf{J}_1,\textbf{J}_2\right] = 0[/tex] is stated to be a condition to show that [tex]\textbf{J}=\textbf{J}_1+\textbf{J}_2[/tex] is also an angular momentum (vector-)operator. But what is meant with [tex]\left[\textbf{J}_1,\textbf{J}_2\right] = 0[/tex]. When i show that [tex]\textbf{J}[/tex] is an angular momentum operator (i.e. [tex]\left[J_x,J_y\right]=iJ_z[/tex] ...) i always need the condition [tex]\left[(\textbf{J}_1)_x,(\textbf{J}_2)_x\right][/tex] and the like. So the components of [tex]\textbf{J}_1[/tex] and [tex]\textbf{J}_2[/tex] should mutually commute. Is this the meaning of [tex]\left[\textbf{J}_1,\textbf{J}_2\right] = 0[/tex]? For me it looks like [tex](\textbf{J}_1)_x(\textbf{J}_2)_x+(\textbf{J}_1)_y(\textbf{J}_2)_y+(\textbf{J}_1)_z(\textbf{J}_2)_z-(\textbf{J}_2)_x(\textbf{J}_1)_x-(\textbf{J}_2)_y(\textbf{J}_1)_y-(\textbf{J}_2)_z(\textbf{J}_1)_z=0[/tex] and this does not imply the conditions i need (as far as i see).

I know Operators acting on different spaces commute and this fact is often used but i want to know how to treat the situation above only with the formal condition [tex]\left[\textbf{J}_1,\textbf{J}_2\right] = 0[/tex].

thanks and greetings tommy.

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# Addition of angular momenta

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