Hi, had a question to try out on a lecture slide but I'm a bit confused as the lecturer didn't expand the explanation of the -1 in this example :

"

Imagine you have a particle in state

1/√2 (1 (-1 is under positive 1.) -1)

You send this particle through a Stern-Gerlach apparatus of which the magnetic field is oriented in z direction. "

If it were a normal 1 0, then I'd understand the answer to be 0.5 and 0.5 as, (1/sqrt(2) x 1)^2.

Does the calculating just like above also mean the answer for -1 would in fact also be 0.5?

Which means for the positive 1, Particle being in state |1>z is prob 0.5 and it being in |0>z is also prob 0.5. Is -1 here state |0>z? if so why?

Also how comes it's a superposition of spin-up and spin-down in both directions x?

Thank you.