How to tell theoretically whether an electron behaves as wave or particle

Question

I have seen many questions on SE on the dual nature of electrons behaving in certain circumstances as particles and as waves in some other circumstance. There is one thing I couldn't get a clear answer on.

When making double slit experiment, we all agree that the electrons behave as waves. The same is true in atoms, where electron levels are described by Schrödinger equation. However, if we speak about a field like plasma physics (my field of work) and maybe beam physics, electrons are treated classically as particles with applying Newton's equation to describe their motion. The models built on particle treatment of electrons show an excellent agreement with experimental results.

From experimental results and testing, we know that electrons behave like waves (in double slit experiment) or as particles (gas discharge models). My question is, is experimenting the only way to decide which model (wave/particle) describes electrons better in particular circumstances? Isn't there any theoretical frame that decides whether electrons will behave as particles or wave in particular circumstance??

For the record, in plasma physics the strongest type of theoretical models is called Particle In Cell models (PIC). In those models Newton equation of motion is solved for a huge number of particles including electrons. Then the macroscopic properties are determined by averaging. This method although it treats electrons classically it is very successful in explaining what happens in experemints

Answer 1

When we treat quantum mechanical objects as if they are particles, this is often referred to as a classical treatment. Intuitively, this is going to be valid based on a simple argument related to the de Broglie wavelength:\begin{equation} \lambda_{dB} = \sqrt{\dfrac{2 \pi \hbar^2}{m k_B T}}.\end{equation} Most often, when this wavelength is on the order of interatomic (or inter-'object') spacing, then quantum mechanical effects become quite relevant and one must consider the wave-like nature of matter. For wavelengths much smaller than the distance between atoms (or molecules, elementary particles, etc..) quantum effects will be negligible and the classical treatment works just fine. You can notice that $\lambda_{dB}$ is a function of both the mass of the object and the temperature, so making either of these larger while the other is constant will decrease the deBroglie wavelength.

You work in plasma physics so this wavelength will most often be very small due to the high temperatures even for very 'light' entities such as electrons. As such you need not consider the wave-like properties of the electron to make accurate calculations of certain physical properties of the system. Electrons are negatively charged and because of the Coulomb repulsion, I would suspect that no matter how much energy they have they will not be a distance apart that is on the order of this wavelength. I study low-temperature condensed matter though most often, so I may be wrong about this spacing.

Hope this helps give some intuitive picture of when the classical treatment is acceptable without having to refer to empirical evidence.