I'm a beginner in quantum mechanics, and I'm a bit confused about states and the probability to measure certain values. I would like to understand at least the following simplified situation:
Suppose the operator $S_z$ is measuring the spin in the $z$ direction of a free particle. Let $$\{e_s\},\quad s = -L, -L+1, \dots, L-1, L$$ be a basis of eigenstates for $S_z$, with $S_z(e_s) = s e_s$.
If at time $t<0$ the system is constantly in the pure state $P_{\psi}$, where $\psi$ is one basis vector, how can I compute the probability that the measuring the spin in the $z$ direction has result $-L/2$?