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Apr 8, 2020 at 14:10 comment added Akoben I've updated the answer. Momentum conservation just tells you how to shift the other particle (so that they cancel in momentum conservation, as you correctly pointed out) and $\hat{p}_i^2 = 0$ just tells you what $q$ has to be.
Apr 8, 2020 at 14:08 history edited Akoben CC BY-SA 4.0
edited to expand answer a bit.
Apr 7, 2020 at 13:30 comment added user256673 The expansion makes perfect sense to me, but I don't understand how q was chosen based on the momentum conservation rules, because if that was the case wouldn't it be that: $$\sum p_i = \sum \hat{p}_i = p_1 + p_2 = \hat{p}_1 + \hat{p}_2 $$ but then because the value of $zq$ changes sign in the equations: $$\hat{p_1}=p_1 -zq \hspace{5mm}; \hspace{5mm} \hat{p_n} = p_n +zq$$ don't they cancel out? And I don't understand how $(\hat{p}_i^2 = 0)$ helps either. I know this is based on $p_1 = |1\rangle|1]$ but could you help me understand the value of q?
Mar 28, 2020 at 11:57 vote accept user7077252
Mar 28, 2020 at 11:56 vote accept user7077252
Mar 28, 2020 at 11:57
Mar 28, 2020 at 7:57 history answered Akoben CC BY-SA 4.0