# Addition of two angular momenta

In addition of angular momenta, does both the relations depict the same thing?

$$\vec{J} = (\vec{J_1}\otimes 1 +1\otimes \vec{J_2})$$

$$\vec{J} = (\vec{J_2}\otimes 1 +1\otimes \vec{J_1})$$

Physically yes but technically the order matters in the sense that the phase of the state of "good" $$J$$ actually depend on the ordering. Writing \begin{align} \vert J M_J\rangle =\sum_{m_1m_2} C_{j_1m_1;j_2m_2}^{JM_J}\vert j_1m_1\rangle \vert j_2m_2\rangle \end{align} where $$C_{j_1m_1;j_2m_2}^{JM_J}$$ is a Clebsch-Gordan coefficient, and using the symmetry property \begin{align} C_{j_1m_1;j_2m_2}^{JM_J}=(-1)^{j_1+j_2-J}C_{j_2m_2;j_1m_1}^{JM_J} \end{align} shows that inverting the role of $$j_1$$ and $$j_2$$ may introduce an overall phase $$(-1)^{j_1+j_2-J}$$ in the construction of $$\vert JM_J\rangle$$, i.e. \begin{align} \vert J M_J\rangle =\sum_{m_1m_2} C_{j_1m_1;j_2m_2}^{JM_J}\vert j_1m_1\rangle \vert j_2m_2\rangle = (-1)^{j_1+j_2-J}\sum_{m_1m_2} C_{j_1m_1;j_2m_2}^{JM_J}\vert j_2m_2\rangle\vert j_1m_1\rangle \end{align}