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I have just begun studying spin and there are two spinors mentioned: The main spinor $\chi $ and the spin-up spin down spinors (eigenspinors) $\chi_+ ,\chi_- $.

I learned that the main spinor is a linear combination of the two spin spinors and I understand that the main spinor must be normalized. I don't understand however why the spin-up and spin-down spinors must also be normalized. As far as I understand, they are not involved in probability calculations

for example: $\chi_+^{(x)} = \left( \begin{array}{ccc} \frac{1}{\sqrt2} \\ \frac{1}{\sqrt2} \\ \end{array} \right)$ instead of just $\chi_+^{(x)} = \left( \begin{array}{ccc} 1 \\ 1 \\ \end{array} \right)$

What am I missing?

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up vote 2 down vote accepted

As you correctly notice, this does not follow from strictly physical considerations, and it is mostly for convenience that we do this. Essentially, this simplifies quite a bit the calculations of the coefficients of your state in a given basis.

Suppose, for example, that you have a basis $\{\chi_+,\chi_-\}$ which is orthogonal but not necessarily normalized. Then you can always write a given state vector $\chi$ as $$\chi=a\chi_++b\chi_-.$$ To find out the coefficients $a$ and $b$, you take the inner product of $\chi$ with the basis states: $$⟨\chi_+,\chi⟩=a⟨\chi_+,\chi_+⟩$$ (since $⟨\chi_+,\chi_-⟩=0$), and $$⟨\chi_-,\chi⟩=b⟨\chi_-,\chi_-⟩.$$ If your states are not normalized, then you need to have the norms of the basis states in the denominator of these coefficients. However, if you set them to one, then the coefficients are exactly the overlap between the two vectors, which is a lot simpler to use: $$\chi=⟨\chi_+,\chi⟩\chi_++⟨\chi_-,\chi⟩\chi_-.$$ Additionally, this lets you directly interpret basic coefficients as direct products, which gives them a direct physical interpretation.

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