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

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

• MaxGraves' answer is pretty much what I was going to write. Just to add a conceptual/terminological point that always annoys me personally: electrons always behaves as quantum particles, which is never the same thing as a classical wave or classical particle. It is not a wave one day and a particle the next. It is always one thing, but that thing is not perfectly analogous to anything classical. – Michael Brown Sep 30 '13 at 17:08
• So the whole wave/particle duality thing is approximate at best and highly misleading at worst. The rules of quantum mechanics simply work without any extra input about whether today is a "wave day" or a "particle day."[/rant over] Classical mechanics arises as the so called "geometrical optics" approximation to quantum mechanics, if you want to look that up. – Michael Brown Sep 30 '13 at 17:09

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.