Tensor Product Properties

Classical Mechanics

Two spin state are given by $|\Psi_1\rangle = |\uparrow\rangle$ and $|\Psi_2 \rangle = \frac{1}{\sqrt{2}} (|\uparrow\rangle + |\downarrow\rangle)$ .

Which gives the correct tensor product state $|\Psi_1 \rangle \otimes |\Psi_2\rangle$ ?

Note: The notation $|\uparrow\downarrow\rangle = |\uparrow\rangle \otimes |\downarrow\rangle$ is a common shorthand for tensor products of spin states.

\frac{1}{\sqrt{2}} \vert\uparrow\uparrow\rangle

\frac{1}{\sqrt{2}} (\vert\uparrow\uparrow\rangle + \vert\downarrow\uparrow\rangle)

\frac{1}{\sqrt{2}} \vert\uparrow\downarrow\rangle

\frac{1}{\sqrt{2}} (\vert\uparrow\uparrow\rangle + \vert\uparrow\downarrow\rangle)