In the book Quantum Mechanics - Franz Schwabl, in chapter 10, equation (10.4) says

Since all the properties of angular momenta and their eigenstates hold for the total angular momentum J .We can construct the product states: $$|j_1m_1j_2m_2\rangle=|j_1m_1\rangle|j_2m_2\rangle$$

I want to know why multiplication? instead of other relations like: $$|j_1m_1j_2m_2\rangle=|j_1m_1\rangle+|j_2m_2\rangle$$



or some weird relations?

Any help will be greatly appreciated!


The true way of seeing this is that the states are written as a tensor product in that the total Hilbert space is formed of ordered products of states of the form $$\left|j_{1}m_{1}\right> \otimes \left|j_{2}m_{2}\right>.$$ Moreover the so-called addition of angular momentum should really be written as $$J_{T} = J_{1}\otimes \mathbb{I} + \mathbb{I} \otimes J_{2}$$ so that $J_{1}$ acts only on the first element of the tensor product, $\left|j_{1}m_{1}\right>$ and $J_{2}$ on the second element, $\left|j_{2}m_{2}\right>$.

To see this geometrically we note that having fixed an arbitrary direction as $\hat{z}$ the angular momentum in that direction is additive.


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