# 'Multiplication' of Spinors

I am struggling to understand what is meant when spin eigenstates are 'multiplied' together. For example, Brandsen and Joachain's Quantum Mechanics says that there are four spin eigenfunctions for a two particle system:

$$\alpha(1)\alpha(2), \alpha(1)\beta(2), \beta(1)\alpha(2), \beta(1)\beta(2)$$

Where $$\alpha$$ and $$\beta$$ are the simultaneous eigenfunctions of the single-particle spin operators $$S^2$$ and $$S_z$$.

If a spinor can be represented by $$[a \space b]^T$$, what does it mean to 'multiply' two spinors together?

And further, how should I then try to do something simple like take the magnitude squared of such a state? I.e.:

$$\alpha^*(1)\alpha^*(2)\alpha(1)\alpha(2)$$

He means the tensor product. In quantum mechanics, when one handles two systems in different Hilbert spaces, $$\mathcal{H}_1$$ and $$\mathcal{H}_2$$, the joint state that considers the two systems lives in $$\mathcal{H}_1\otimes\mathcal{H}_2$$. So, if you have two spin states $$|\alpha(1)\rangle\in\mathcal{H}_1$$ and $$|\alpha(2)\rangle\in\mathcal{H}_2$$ ,the state that describes the two states will be $$|\alpha(1)\rangle\otimes|\alpha(2)\rangle, \:\:\text{usually denoted by} \:\:|\alpha(1)\alpha(2)\rangle.$$
$$\langle\alpha(1)\alpha(2)|\alpha(1)\alpha(2)\rangle = \langle\alpha(1)|\alpha(1)\rangle\langle\alpha(2)|\alpha(2)\rangle,$$