# Product states - Addition of angular momentum

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

$$|j_1m_1j_2m_2\rangle=|j_1m_1\rangle^2|j_2m_2\rangle^2$$

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.