z-axis

y-axis

y

x-axis

x

Spin observable:

Project the state vector (black arrow) onto one of the axes. The projections relate to the probability amplitudes of the measurement outcomes of the spin observable in that direction.

z

The spin state (black arrow) is projected onto the z-axis, and the length of the projection is related to the probability of finding an "up" measurement according to:

K

Reset

The spin state (black arrow) is projected onto the x-axis. Since the arrow is in the yz-plane, this projection is K = 0. 

The spin state (black arrow) is projected onto the y-axis, and the length of the projection is related to the probability of finding an "up" measurement in the y-direction: