Now that the History and base of Quantum Physics has been established, we can start the fun stuff. The first question of any good scientific theory is this, How can I prove this?
I'll start with my personal favorite, the double-slit experiment, also known as Young's experiment. A double slit is basically a diffraction grating. When light goes through a diffraction grating, well a basic double slit one, it has a banded pattern, the same thing happens with any wave. When a particle Is shot through a double slit, there should just be two lines the same shape and size of the slits. An electron, the smallest particle we know of, should have acted like any other particle and produced two lines the same shape and size of the slits, but as you can guess, that didn't happen. Instead, the electrons produced a banded pattern, like that of a wave. This proved that both particles and waves share common traits. This experiment was also completed successfully with larger particles which raises the question, could we eventually diffract ourselves? How big can we go?
Also, through experimental discovery, we have found that photons have momentum. For many this doesn't make sense, how can something with no mass have momentum? Well, scientists have found out that An objects momentum is also proportional to its Energy thusly explain why a photon has momentum. But this further shows the particle-wave duality of matter.
There is also black body radiation. Black body radiation is an electro magnetic radiation that is within or surrounds a body that is in thermodynamic equilibrium. Planck explained this as energy distribution and this could also be discrete. This is in part what helped him create the theory of the quanta, which is a building block of quantum physics and one of the things it explains.
As you can see, many experiments have been done to prove the duality of matter and have allowed Quantum physics to be universally accepted and be very helpful in understanding what happens when classical laws of physics break down.