0
$\begingroup$

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!

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.