Hopefully someone can clear up a basic misconception I am having about the nature of spin state vectors.
According to the book i am reading, The basis vectors of up/down spin are orthogonal to each other. Same goes for in/out and left/right. I understand this because if the spin is measured as up it precludes it from being down. Thus , the inner product of the two basis vector, is 0.
But where I'm confused is, aren't all spin states orthogonal? Cant there only be one spin at a time? As in, if you have an apparatus oriented along the z-axis, and measure a +1 (up spin), then doesnt that preclude the spin from being left, right, in, or out, unless you shift the apparatus/remeasure?
I understand that if you were to measure along the x or y axis at this point you would have a 50/50 chance of getting an up or down spin, whereas if you measured via the negative z axis you would have 100% chance of getting a down spin.
So then what does orthogonality represent in spin systems? My guess is that after the initial apparatus direction is set, and the original spin value is measured, orthogonality means having a 0% chance of achieving that same measured spin after adjusting the apparatus along a different axis and remeasuring. Thus if you measured spin +1 with the apparatus in the +z direction, and then shifted the apparatus to -z to perform a second measurement, you would have a 0% chance of it being +1, making it orthogonal. But if you shifted the apparatus along the +/- x/y axis, you would have a 50% chance of getting the same measured spin?
Is my reasoning correct, or way off base?