Is there any connection between the eigenvalues of spin up and spin down of particles that is measured in x versus the -x direction?. Ie if we instead decide the measure the basis of the same (prepared say in y) particle which was measured in x; but such that we measured it in the -x basis instead (whilst still in prepared in state y?; is there any relationship between the measurement results or rather what the possible eigenvalues of either measurement mean?.
Ie if the orientation of the apparatus setting or the magnetic field (in the gerlach device) was rotated 180 degrees around the x-y plane from the x direction, which established the basis on which the particle is to be measured/and thus collapse. Is this just the same measurement as a measurement conducted along the x basis ?, or just another distinct basis like any other. And if it is not completely distinct, does it just correspond to a case where spin down in the negative x plane corresponds to spin up in the x basis?
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