Suppose we have normalised and orthogonal wavefunctions $\psi_1$ and $\psi_2$ as different solutions to the same equations with eigenvalues of $\lambda_1$ and $\lambda_2$ for the momentum operator $\hat p$. Then, I construct a new wavefunction as $\psi = sin \theta \cdot\psi_1 + cos \theta \cdot \psi_2$.
Now $\psi$ is normalised. On applying the momentum operator, we have $$\hat p\psi = sin \theta \cdot \lambda_1\psi_1 + cos \theta \cdot \lambda_2\psi_2$$ As far as I can tell, this $\psi$ doesn't turn out to be an eigenfunction of $\hat p$. What is happening here? It should have an eigenvalue, right?