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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?

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possible duplicate of Is the wave-particle duality a real duality? –  Kyle Kanos May 16 at 17:05
    
@KyleKanos It's not my answer, not duplicated. My Answer is how to know / define when a duality behaves as a wave or as a particle. –  Only a Curious Mind May 16 at 17:07
    
@LucasAbilidebob Does my edit accurately reflect what you were really trying to ask? –  Danu May 16 at 17:19
    
@Danu yes! thanks! my english is bad :/ –  Only a Curious Mind May 16 at 17:31
    
Since the path integral interpretation uses no waves just particles, you can always consider them particles. –  jinawee May 16 at 21:23

3 Answers 3

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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

electron double slit

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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So before the experiment I can't determine the side? (wave or particle) –  Only a Curious Mind May 16 at 20:30
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Correct. Now that we have a mathematical theory for quantum mechanics you might design an experiment with the objective of determining what type will manifest, you may posit a hypothesis that you will find particle or wave behavior, but it is observation that will validate your hypothesis. –  anna v May 17 at 3:19

As a general rule the wave model is most useful when you're looking at the propagation of light and the particle model is most useful when you're looking at the light wave exchanging energy with something else.

If you take the good old Young's slits experiment as an example, the wave model well describes how the light diffracts at the slits, but you need the photon model to explain how the light interacts with the CCD or photographic plate recording the diffraction pattern.

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If you are doing QM calculations then you do not choose. An experiment is described by a quantum state which encodes both aspects.

If you want a classical approximation then one guide are the De Broglie wavelength $\lambda = h/p$ of various entities relative to the sizes of components of the experimental setup. Entities with wavelength much shorter than components of the experimental setup will display primarily particle like properties. If the wavelength are comparable to the components of the experiment or larger then wavelike properties will tend to dominate. In the middle is where classical approximations are the most unsatisfactory since either aspect can manifest depending on the measurement being made.

There are two reasons for this. On the one hand waves with very short wavelength compared to the experiment display wavelike effects on small scales that may be hard to measure. For example the interference pattern of a hologram is not really discernable directly on the human scale. On the other hand De Broglie wavelength is inversely proportional to momentum, so long wavelength entities must have small momenta, limiting their particle like effects.

There is some subtlety as to what is the relevant scale of the experiment. If you are doing crystal diffraction then the scale is the inter atomic distances of the crystals rather than the size of the crystals.

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