I am reading an introductory book about Quantum Mechanics, and I have a problem understanding one concept that is probably very simple. In particluar, I have two questions.
My book says that a spin state can be expressed as a linear combination of two orthogonal ket vectors (in Hilbert space). Let's imagine a 3D space with the three axes $x,y,z$. If we measure spin along $z$ and we get $+1$, then we call that state $|u\rangle$ (as in "up"). Vice versa, if we get -1, we'll call that state |$d\rangle$. We do this with the other axes and we get the states $|r\rangle$ ("right") and $|l\rangle$ ("left") for the x axis and the states $|i\rangle$ ("in") and $|o\rangle$ ("out") for the y axis.
We then pick two of these vectors arbitrarily (although they need to be orthogonal), so my book takes $|u\rangle$ and $|d\rangle$. Any state $|A\rangle$ can be expressed like this: $$ |A\rangle=\alpha_1|u\rangle+\alpha_2 |d\rangle, $$ where $\alpha_1$ and $\alpha_2$ are complex numbers and are numerically equal to the square root of the probability that, measuring the spin along the z axis and having observed $|A\rangle$, one will respectively get the state $|u\rangle$ or $|d\rangle$.
My first question is why is it so? I mean, why are they square roots and not simply the probabilities? Anyway, it is pretty obvious if that's the case that $$ |r\rangle=\frac{1}{\sqrt2}|u\rangle+\frac{1}{\sqrt2} |d\rangle, $$ since the direction "right" is orthogonal (spatially) to the $z$ axis, so the spin $+1$ and $-1$ are equally probable.
My second question is in understanding why my book then says that $$ |l\rangle=\frac{1}{\sqrt2}|u\rangle-\frac{1}{\sqrt2} |d\rangle. $$ Why is that? Why is there a minus? In writing the "in" and "out" states the thing gets even weirder (with the appearance of the imaginary unit), but I don't wanna go into that before clearly understanding this. Thank you for your time!
