Q:

How do we physically apply the operators of quantum mechanics on a particle?

What do we have to perform physically that is equivalent to applying those quantum mechanical operators on a state $|\psi\rangle$? 
Edit: I have removed the part I was asking regarding measurement because it takes us away from the real question.

A:

Perhaps I misunderstand your question but I would like to make clear that operating on a state with say, the momentum operator is not meant to be the equivalent of measuring the momentum of the system in that state.
Consider, for example, a state that is a superposition of two momentum eigenstates:
$$|\psi\rangle = \frac{1}{\sqrt{2}}\left(\,|p_1\rangle + |p_2\rangle \,\right)$$
If we operate on this state with the momentum operator, we get a different state:
$$\hat p |\psi\rangle = \frac{1}{\sqrt{2}}\left(\,p_1|p_1\rangle + p_2|p_2\rangle\, \right)$$
But note that $\hat p |\psi\rangle$ is a superposition of momentum eigenstates, i.e., operating on the state with the momentum operator did not 'collapse' the state to one or the other momentum eigenstate.
However, if we measure the momentum of the system in this state,  we will measure either $p_1$ or $p_2$ and, further, the state of the system, immediately after the measurement, will be the associated eigenstate.

A measurement always causes the system to jump into an eigenstate of
  the dynamical variable that is being measured, the eigenvalue this
  eigenstate belongs to being equal to the result of the measurement.
P.A.M Dirac in "The Principles of Quantum Mechanics"

Thus, the 'momentum measurement operator' (whatever that is) is not the momentum operator.
Put another way, the result of operating on the state with the momentum operator is determined by the state; the result of the operation is certain.
However, the result of measuring the momentum of the system in this state is not determined.  The result will be either $p_1$ or $p_2$ but which value will be measured is not determined by the state.

