In quantum mechanics, how does the probability amplitude relate to the measurement of an observable in a system described by its eigenstates?

In quantum mechanics a system’s state can be written as a sum of eigenstates of an observable, and each term in this sum carries a complex coefficient called the probability amplitude. When a measurement is performed, the probability of obtaining a particular eigenvalue equals the squared magnitude of the corresponding amplitude, because the amplitude tells us how much of that eigenstate is present in the overall state. Thus the amplitude itself does not give a probability directly, but its absolute square does. For example, if a spin‑½ particle is in the state \( \psi = \tfrac{1}{\sqrt{2}}\bigl|+\! z\rangle + \tfrac{1}{\sqrt{2}}\bigl|-\!