What is physical meaning of state getting by acting of Hermitian operator to non-eigenstate?

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 $$\{\left| Q_n \right>\}$$.

• Read: PRINCIPLES QUANTUM MECHANICS by Paul Dirac. – Sahil Dec 9 '18 at 19:15
• A measurement is when you find $\langle Q_m | \Phi \rangle$, then provided the basis states are orthogonal you will only select those $Q_n$ such that $m=n$ – Triatticus Dec 9 '18 at 20:49

$$\def\ket#1{|#1\rangle}$$
Except for calculating averages, what physical meaning of $$\ket\Phi$$? Is the real state, or some ghost? Because we know the measurement of physical quantity should collapsed $$\ket\Psi$$ to some of {$$\ket{Q_n}$$}.
In your example $$\ket\Phi$$ is no "ghost". It's a bona fide representative of a possible state, apart for not being normalized. This doesn't mean that you know how and when it can be reached starting from $$\ket\Psi$$. To know this you need more information on your system and its dynamics.