How to measure the wave-function experimentally? Do experimental physicists really measure the wave function of a system?
How do they do it?
Do they make many identically-prepared systems and measure the position of the particle(s) over and over and find the probability density from the graph?
How do they actually prepare identical systems and do they first calculate the wave function and find the momentum by momentum operator?
sorry if this is too naive ,I have started reading Griffiths and have a curiosity of knowing how physicists do QM but I am confused and google search didn't help.
 A: The wave function is connected to an experimentally observable value through its complex conjugate square, which gives the probability of an interaction or a decay happening; from this a crossection for the interaction can be predicted, number of events versus some variable in appropriate units.
Example:  the experiment can measure very many decays of the pi0, decaying in a bubble chamber;  not an interesting experiment to spend money and effort  on, there are other experimental ways of identifying pi0s, but good as an example of an "event".

The two photons are seen in the bubble chamber by their creating an electron positron pair in the field of the nucleus of the liquid. This is called an event. An accumulation of such events will allow to measure the probability distributions in angular and energy variables and compare them with the quantum mechanical prediction, not of the wave function, but of its complex conjugate squared.  
The interesting experiments are the experiments that accumulate scattering events, for example e+ on e-  and measure the crossection ( essentially number of events with appropriate units) as in this e+e- interactions versus energy (sqrt(s)) from the particle data book,  fig 49.5 :

We see resonances at fixed sqrt(s), the mass of the resonance.
The "identical systems"  are the scatterings of an electron against a positron in a collider, i.e. the e+ and e- have equal and opposite momenta,they are prepared in the center of mass. 
Many of  the interesting experimental measurements that led to  the quark model are in this plot. The recent discovery of the Higgs was established in a  similar manner: number of events versus mass plots in appropriate crossection units.
A: Experimental physicists are very visual people. They are more interested in seeing the wavefunction. Calculating it is the work of a theoretical physicist. "Catching sight of the elusive wavefunction", would be a good read for you at this point.
As you proceed with Griffiths, you will see that in experiments we measure certain quantities, called the observables. The verification of a quantum mechanical analysis of a system is justified by confirming that the observations are in the spectrum of the observable in question.
A: I could be wrong, but I remember reading that scanning tunneling microscopy can be used to measure $|\psi|^2$. A quick Google search brought me to this paper, "http://www.ncbi.nlm.nih.gov/pubmed/19090685." I'm sure you can find more information by looking into STM online.
A: In the paper Measuring the quantum mechanical wave function they go through experiments in atomic, molecular and optical physics which allowed them to partially reconstruct the original wavefunction based on real measurements of position and momentum.
