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.