The spin components of a beam of atoms prepared in the state

The spin components of a beam of atoms prepared in the state are measured and the following experimental probabilities are obtained: From the experimental data, determine the input state. The spin components of a beam of atoms prepared in the state are measured and the following experimental probabilities are obtained:

Solution

the linear combination I got is 1/(2)^1/2 for |+>z and -i/(2)^1/2 for |->z