I am new to quantum mechanics and have some questions regarding expectation values of operators. I have the non-degenerate eigenstates $|\psi_n \rangle $ of an operator $A$. I have the normalized superposition of the eigenstates given by:
$$|\psi\rangle=\alpha|\psi_1 \rangle +\beta|\psi_2\rangle+\gamma|\psi_3 \rangle$$
Where $\alpha,\beta,\gamma$ are numbers. What the expectation value of $A$ is equal to?
Assume $A$ is a self-adjoint operator.
Since $|\psi_n\rangle$ are eigenstates of $A$ then $A|\psi_n\rangle = \lambda_n|\psi_n\rangle$. In particular, we have that \begin{align} \langle \psi \mid A\ \mid \psi\rangle =&\ \langle\alpha\psi_1 + \beta\psi_2 + \gamma \psi_3 \mid A\mid \alpha\psi_1 + \beta\psi_2 + \gamma \psi_3\rangle\\ =&\ \langle\alpha\psi_1 + \beta\psi_2 + \gamma \psi_3 \mid \alpha\lambda_1\psi_1 + \beta\lambda_2\psi_2 + \gamma\lambda_3 \psi_3\rangle\\ =&\ \lambda_1|\alpha|^2\langle \psi_1\mid \psi_1\rangle +\lambda_2|\beta|^2\langle \psi_2\mid \psi_2\rangle +\lambda_3|\gamma|^2\langle \psi_3\mid \psi_3\rangle\\ =&\ \lambda_1|\alpha|^2+\lambda_2|\beta|^2+\lambda_3|\gamma|^2 \end{align}