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Two spin state are given by ∣Ψ1⟩=∣↑⟩|\Psi_1\rangle = |\uparrow\rangle∣Ψ1⟩=∣↑⟩ and ∣Ψ2⟩=12(∣↑⟩+∣↓⟩)|\Psi_2 \rangle = \frac{1}{\sqrt{2}} (|\uparrow\rangle + |\downarrow\rangle)∣Ψ2⟩=21(∣↑⟩+∣↓⟩).
Which gives the correct tensor product state ∣Ψ1⟩⊗∣Ψ2⟩|\Psi_1 \rangle \otimes |\Psi_2\rangle∣Ψ1⟩⊗∣Ψ2⟩?
Note: The notation ∣↑↓⟩=∣↑⟩⊗∣↓⟩|\uparrow\downarrow\rangle = |\uparrow\rangle \otimes |\downarrow\rangle∣↑↓⟩=∣↑⟩⊗∣↓⟩ is a common shorthand for tensor products of spin states.
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