How does one know whether, in treating a certain problem, one should consider particles as waves or as point-like objects? Are there certain guidelines regarding this?
Answer
It will depend on the results of your experiment, the results will tell you whether you are seeing the wave nature or the particle nature.
Take the scattering of an electron on a proton producing an electron and a proton and a pi0 meson. Your experiment measures "particle" interactions, in the form of classical particles, you can see the trajectories of the individual particles with your instruments.
If you measure a lot of scatters and plot the crossection versus energy, then the interpretation uses the quantum mechanical wavefunctions which by construction carry the wave nature of the "particles".
This two slit experiment of electrons one at a time
build up in time
shows clearly both natures. Each individual electron is a dot, i.e. a particle interacting with the screen. The accumulation though shows the probability distribution due to the wave function of the electron with the boundary condition of two slits, the wave form of the duality.
The classical type particle nature appears at a specific (x,y,z). The probability of appearing at the specific (x,y,z) has a wave nature.
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