The definition of an operator is $$ \hat{Q}\left|\Psi\right> = \left|\Phi\right>, $$ the thing that convert one state vector $\left|\Psi\right>$ to another $\left|\Phi\right>$.
For example, the one of such operators is time evolution operator $\hat{U}(t_1,t_2) = e^{-\frac{\hat{H}}{\hbar}(t_2 - t_1)}$: $$ \left|\Psi(t_2)\right> = \hat{U} \left|\Psi(t_1)\right>. $$
Here we know, the result of acting $\hat{U}$ to $\left|\Psi(t_1)\right>$ lead to some another state $\left|\Psi(t_2)\right>$.
Now, let's consider some Hermitian Operator $\hat{Q}$, with the discrete specrum: $$ \hat{Q}\left| Q_n \right> = Q_n\left| Q_n \right>, $$ acting on some non-eigenstate $\left| {\Psi} \right\rangle = \sum\limits_n \left| {Q_n}\right\rangle a_n = \sum\limits_n \left| {Q_n} \right\rangle \left\langle {Q_n} | {\Psi} \right\rangle $.
We get $$ \hat{Q}\left|\Psi\right> = \left|\Phi\right>,$$ where $$ \left|\Phi\right> = \sum\limits_n Q_n \left| {Q_n}\right\rangle a_n = \sum\limits_n Q_n\left| {Q_n} \right\rangle \left\langle {Q_n} | {\Psi} \right\rangle $$
Except for calculating averages, what physical meaning of $\left|\Phi\right>$? Is the real state, or some ghost? Because we know the measurement of physical quantity should collapsed $\left|\Psi\right>$ to some of $\{\left| Q_n \right>\}$.