Particle- and wave-like properties

Question

Is wave-particle duality a real concept or a pedagogical tool? In a less opinion-based way: what are particle properties and wave properties of a particle (that we speak of particle properties of particles nicely encompasses the problem).

Judging by the recurrent themes in the questions about the basics of quantum mechanics in this community, many difficulties arise from not being able to distinguish the particle properties from the classical properties (there is less confusion about waves, perhaps because people new to QM rarely have much intuition from wave theory).

Let me give two examples:

Two-slit experiment
We say that, if the size of slits is comparable to the de Broglie wave length, the electron behaves as a wave. In the other limit, if the slits are very wide, it will behave as a particle. This is the expression of the duality.

However, if we were to approach this situation rigorously, we would treat the electron as a wave: solve the Schrödinger equation and demonstrate that for wide slits the coarse-grained probability distribution is particle like. Particle-like behavior here refers to a classical trajectory, and is an approximation to "true" wave-like behavior.

Photons
Many questions arise from naively associating photons with particles, having momentum and trajectory. In reality, what makes the photons particle-like is their countability.

Reprise
Thus, I would say that the main particle-like property is the countability, while the wave-like properties are due to the mode structure described by a partial differential equation (Schrödinger, Dirac, Maxwell, etc.) On the other hand, having simultaneously measured velocity and position is not a particle-like, but a classical particle property. Am I still missing something?

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Related

• Is the wave-particle duality a real duality?
• How does light 'choose' between wave and particle behaviour?

Answer 1

Countability is not a property that can be exclusively assigned to particles- one can count waves too.

The characteristics of waves are an ability to interfere, to spread spatially, and to be decomposed into normal modes of vibration. The characteristics we classically associate with particles include a small and unchanging spatial extent and mass.

The wave particle duality is a shorthand way of referring to the fact that fundamental particles exhibit properties of classical particles- they have mass, a small spatial extent, etc- but their behaviour in certain circumstances is analogous to the behaviour of waves. For example, the path of a neutron can be diverted by placing a glass prism in front of it.

The problem with the term 'wave particle duality' is that it can be, and sometimes is, interpreted as meaning that under certain circumstances a particle actually becomes spread out like a wave, so that its mass is no longer confined to a point.

I suspect it is prone to misinterpretation because we are used to tangible physical waves, like water waves. If everybody studied the Heisenberg picture of QM instead of the Schrodinger picture, then the misconceptions might be less common- after all, the 'matrix particle duality' doesn't have quite the same ring to it.