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I am not sure if I understand spin operators correctly. Given a two spin system in state $|++\rangle$ and an operator $S = S^{(1)} + S^{(2)}$

Then I have

$$ S_z |++\rangle = (S^{(1)}_z + S^{(2)}_z) (|\tfrac{1}{2}, \tfrac{1}{2}\rangle \otimes |\tfrac{1}{2}, \tfrac{1}{2}\rangle) = (S_z|\tfrac{1}{2}, \tfrac{1}{2}\rangle \otimes S_z|\tfrac{1}{2}, \tfrac{1}{2}\rangle) = (\tfrac{\hbar}{2} |\tfrac{1}{2}, \tfrac{1}{2}\rangle \otimes \tfrac{\hbar}{2} |\tfrac{1}{2}, \tfrac{1}{2}\rangle) = \tfrac{\hbar}{2} (|\tfrac{1}{2}, \tfrac{1}{2}\rangle \otimes |\tfrac{1}{2},\tfrac{1}{2}\rangle) = \tfrac{\hbar}{2} |++\rangle $$

But everywhere I read, I see $$ S_z |++\rangle = \hbar |++\rangle $$

What did I do wrong?

iblue
  • 642

1 Answers1

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Clearly, calling $\mathbb{I}^{(j)}$ the identity matrix acting on subspace $j$ of the tensor space,

$$S_z|++\rangle= \left(S^{(1)}_z\otimes \mathbb{I}^{(2)} + \mathbb{I}^{(1)} \otimes S^{(2)}_z\right) |\frac{1}{2},\frac{1}{2}\rangle \otimes |\frac{1}{2},\frac{1}{2}\rangle = \left( S_z^{(1)}|\frac{1}{2},\frac{1}{2}\rangle\otimes \mathbb{I}^{(2)} |\frac{1}{2},\frac{1}{2}\rangle \right) + \left( \mathbb{I}^{(1)} |\frac{1}{2},\frac{1}{2}\rangle\otimes S_z^{(2)} |\frac{1}{2},\frac{1}{2}\rangle \right) = \left( \frac{\hbar}{2}|\frac{1}{2},\frac{1}{2}\rangle\otimes |\frac{1}{2},\frac{1}{2}\rangle \right) + \left( |\frac{1}{2},\frac{1}{2}\rangle\otimes \frac{\hbar}{2} |\frac{1}{2},\frac{1}{2}\rangle \right) = 2 \left( \frac{\hbar}{2}|\frac{1}{2},\frac{1}{2}\rangle\otimes |\frac{1}{2},\frac{1}{2}\rangle \right) = \hbar |++\rangle$$

perplexity
  • 489