# 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

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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:$$\lambda_{dB} = \sqrt{\dfrac{2 \pi \hbar^2}{m k_B T}}.$$ 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.

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Michael Brown: I absolutely agree with you about the fact that wave/particle duality never takes a vacation and that when we work in the classical approximation, it is not because our electrons have somehow become particles rather than waves. My answer was really to illuminate the fact that one needs to determine which character is more dominant in order to effectively model systems. Also, can you please accept my answer so that I get >50 rep points, I am new and would like to be able to leave comments!! –  MaxGraves Sep 30 '13 at 17:43
Thanks for the information @MaxGraves , I did some calculation for de Broglie wavelength of electrons with temperature of 300 K which gave 6 nm approximately. The latest double slit experiment was done at University of Nebraska-Lincoln where they used a slit width of 62 nm. Don't you think that having a slit width 10 times larger than electron wavelength should make the electron behave classically? –  Gotaquestion Sep 30 '13 at 20:29
Hmm, no this is only one order of magnitude larger. Besides that point, this goes back to what @Michael Brown said, which was that you never have strictly one or the other. You are forced by nature to realize that there are wavelike properties and particle like properties to everything. As far as I'm aware, one should expect the diffraction from the double slit to occur regardless of temperature. My statement was more about in what regime can you treat particles as classical and get somewhat reasonable results from calculations of expected physical observables such as energy, etc... –  MaxGraves Sep 30 '13 at 21:33