How does one actually measure the position or momentum of a quantum object? How is the position or momentum of a Quantum particle is measured experimentally in laboratory? Suppose we want to know the position or momentum of quantum particle which is kept in a box i.e. an infinite square well, then how to perform the experiment? In some books it is said that we have to use photons to illuminate the quantum particle and then visualise its position but simultaneously it is written that this is just a thought experiment to make students understand the concept.
 A: I want to clear up that the original post has suffered a number of edits, and not by the OP, and I am starting my answer to the original post which had explicit reference to the Heisenberg uncertainty principle. I.e. it stated:

Heisenberg uncertainty says that the position and momentum or the energy and time of a quantum particle can't be measured simultaneously with great accuracy.

Which is the reason for my  discussing the HUP below.
The Heisenberg uncertainty principle, HUP,

was a  precursor  of the final theory formulated that describes the behavior of quantum entities, elementary particles and their composites, the standard model,SM.
What does it say? It describes a limiting volume of the two variables under consideration, so that the greater the accuracy of simultaneous measurement of p will give a large uncertainty to the value of x and vice verso. If one substitutes  laboratory numbers, for example microns for space, the momentum will have to be larger than a very small number, as h_bar is a very small number. 
It is a relationship due to the commutator relations of the quantum mechanical operators of the corresponding variables and which are  axiomatic in the quantum mechanical theory that is used in the SM.
The  SM has been validated innumerable times. In this convoluted sense the good agreement of the model with innumerable data is enough to validate the HUP. The HUP relationship also can be derived from the basic theory of quantum mechanics, so all measurements in the particle physics can be considered to validate the HUP.
This, a recent measurement, may interest you

have performed measurements on photons (particles of light) and showed that the act of measuring can introduce less uncertainty than is required by Heisenberg’s principle. The total uncertainty of what can be known about the photon's properties, however, remains above Heisenberg's limit.

Edit after comment by OP:

I just want to know the experiment by which,say,i will be able to measure the position of a quantum particle in 1d box,or its momentum

The real world is three dimensional in space, and your thought experiment cannot materialize in the lab.The positions and momenta of elementary particles are measured with complicated detector systems and the measurement errors are such that the HUP constraint is always fulfilled. It needs special setups that can go to accuracies which challenge the dimensional constraints of the HUP, as in the link above:

Steinberg's group does not measure position and momentum, but rather two different inter-related properties of a photon: its polarization states. In this case, the polarization along one plane is intrinsically tied to the polarization along the other, and by Heisenberg’s principle, there is a limit to the certainty with which both states can be known.

they use another set of conjugate variables in order to check the HUP.
