Timeline for Prove that $J_z=J_{1z}+J_{2z}$
Current License: CC BY-SA 4.0
7 events
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| Dec 2, 2019 at 3:17 | comment | added | James Johns | Hi Frobenius, Apologies for taking so long to get back to you, I was traveling. I agree with everything in that answer, it's standard theory for product spaces, but there you also use that $S_z^{tot} = S_{1z}+S_{2z}$, which comes from the simple vectorial addition of the spin angular momentum along the z axis. So I don't see the conflict. I think the original question was from someone who was confused about why $J_z=J_{1z}+J_{2z}$ which doesn't require sophisticated treatment to get at the right answer | |
| Nov 30, 2019 at 22:03 | comment | added | Voulkos | I suggest you to see my answers here : Total spin of two spin-1/2 particles. These are my efforts to understand all this stuff in the past. | |
| Nov 30, 2019 at 21:59 | comment | added | James Johns | Even in product spaces, you're still talking about vector quantities that add. The allowed values of $J_z$ depend on the value of j, but the form of the operator doesn't. $J_{1z}$ only acts on the angular momentum space of particle 1, so I disagree that it makes no sense. Sorry, but I'd be happy to discuss further and revise or delete my answer | |
| Nov 30, 2019 at 21:58 | comment | added | Voulkos | I didn't downvote your answer. | |
| Nov 30, 2019 at 19:12 | comment | added | Voulkos | .... Think for a moment this : if $j_{1}$ and $j_{2}$ are (nonnegative) integers or half-integers representing angular momenta living in the $\;\left(2j_{1}+1\right)-$ dimensional and $\;\left(2j_{2}+1\right)-$ dimensional spaces $\;\mathcal{H}_{\boldsymbol{1}}\;$ and $\;\mathcal{H}_{\boldsymbol{2}}\;$ respectively, expressions like this \begin{equation} J_{z}=J_{1z}+J_{2z} \tag{01} \end{equation} have no sense since $J_{1z}$ and $J_{2z}$ are operators acting on different spaces and if $j_{1}\ne j_{2}$ of different dimensions too. | |
| Nov 30, 2019 at 19:12 | comment | added | Voulkos | Welcome to PSE. I think your answer does not help. There exists a whole theory about product states, product (Hilbert) spaces, product transformations (operators) etc under which the addition of two angular momenta in QM is well-defined without contradictions.... | |
| Nov 30, 2019 at 14:34 | history | answered | James Johns | CC BY-SA 4.0 |