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If we have a two electron system we can either have a singlet or triplet state.
The singlet state looks like
$$\chi=\frac{1}{\sqrt 2}(\chi_{+-}-\chi_{-+})$$
but we also have a triplet state in which
$$\chi=\frac{1}{\sqrt 2}(\chi_{+-}+\chi_{-+})$$
What is the physical difference between these two states due to the different phase factor?
If you rotate the singlet it will remain the same. It has a total spin of 0.
If you rotate the triplet it will become a mixture of $\chi_{++}$, $\chi_{--}$, and ${1\over \sqrt 2}(\chi_{+-}+\chi_{-+})$. It has a total spin of 1.
The singlet state is spin-0 when measured from any angle. In contrast, the triplet state will be seen as spin-0 when measured along the $z$-axis, but will be measured as spin-1 along an orthogonal axis (like the $x$-axis).
