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For Tutorial-I, you need the Postulate 3 of quantum mechanics.
\(\newcommand{\ket}[1]{\vert#1\rangle}\)
The relevant parts of the third postulate tell us:
The first part of the third postulate says that the only outcome of a
measurement of a dynamical variable $A$ is one of the eigenvalues of the
corresponding operator $\hat{A}$. In particular, an answer different,
from every eigenvalue, cannot be the outcome of measurement of $A$.\\
The next part of the postulate tells that if a system is represented by one of
the eigenvectors $\ket{u_n}$, a measurement of the dynamical variable $A$ will
give the corresponding eigenvalue $\alpha_n$.
This also means that if a system has a definite value \(\alpha_n\) for observable \(A\),
the state vector( wave function) is eigenvector of operator \(\hat{A\).
So for spins \[(\hat{n}\cdot \vec{S}) \ \chi = m \hbar \chi\] where \(\hat{n}\) is the unit vector in direction in which
spin has definite value \(m\hbar\). Solve for wave function \(\chi\) if direction and projetion value \(m\) are given.
Click on the link below to go to lecture notes of Postulate-III .
