What does it mean when people (physicists) say electron has a wavelength of $x$ unit of length physically?

Question

When we discuss about the wavelength of em(electromagnetic) wave's wavelength, It is meant we are talking about the tip to tip of the oscillation of electrical and magnetic field in physical space. Now, In case of Davisson-Germer experiment, we have experimentally find the evidence of the wavelength of electron. What is it mean by the wavelength electron in this context. is it probabilities of position, oscillation of electron charge or something else?

Answer 1

The electron is a quantum mechanical entity and in the theory it is a point particle, so cannot have a wavelength in space. The wavelength seen in experiments is from an accumulation of data, and the theory fits it with the probability of finding the electron at a specific (x,y,z,t)

Before the theory of quantum mechanics developed the fact that experiments saw interference effects in electron experiments ,it was called the De Broglie wavelength

The de Broglie wavelength is the wavelength, λ, associated with a massive particle (i.e., a particle with mass, as opposed to a massless particle) and is related to its momentum, p, through the Planck constant, h:

$λ = h/p$

This can now be shown to be the consequence of the quantum mechanical theory.

So it is a probability distribution wavelength. The double slit experiment with single electrons at a time gives an intuition of how the wavelength interference effect is seen in the accumulations of data.

Answer 2

In quantum theory we find that the behaviours of microscopic objects can be accurately modelled by equations that assume each object has an associated mathematic function (we call it a wave function) which changes over time. The wave function is correlated in a probabilistic way with various properties of the object. For example, the probability of the object being detected at a particular point in space is linked to the amplitude of the wave function at that point, so there is more chance of finding the object where the wave function has a large amplitude, and a lesser chance where it has a small amplitude.

When we say that an electron has a particular wavelength, what we really mean is that its wave function resembles a wave with a given wavelength. In calculations, which are necessarily idealised to some extent, we might assume that an electron's wave function is a plane wave with an exact wavelength, which might be unrealistic in some ways but good enough to yield results that agree with experiment.

In electron diffraction experiments, for example, by assuming an incident electron's wave function is a plane wave with a given wavelength, you can show that the diffraction pattern arises from the wave function interfering with itself. In regions where the wave function interferes destructively, almost cancelling itself out, its amplitude is small so there is a correspondingly small chance of finding the electron there. In regions where the wave function reinforces itself there is a higher change of finding the electron. You can do experiments with different slit geometries and different wavelengths and find the results always fit quite well with the assumed nature of the wave function.

The idea that every particle has an associated wave was put forward by Louis de Broglie, who showed how the wavelength could be linked to the particle's momentum. He later won the Nobel prize for that.

Answer 3

Well, we use the Schrodinger wave equation quite a bit in solid state physics where the electron is the particle of with some effective mass and has a de Broglie wavelength. This is quite useful when making quantum wells for LEDs. A quantum well is like an artificial atom, except that the different layers of semiconductors have different bandgaps, and depositing a few atomic layers of a lower band gap material between a two higher bandgap materials you have a two dimensional potential well. If you small enough piece of semiconductor you can also make a quantum dot, which is even more like an artificial atom.

The point is that an electron is the conduction band that falls into the quantum well, or quantum dot is confined and has a discrete energy level. And the solutions to the wave equation if you plot them against the potential are Sine or Cosines, with decaying exponentials to match the boundary conditions.

and using analytical tools measure the dimensions of the layers of atoms deposited and know the dimensions very exactly. From a practical perspective if we chance the width of the quantum well, we change the allowed energy level of electron and that is useful in getting the color of the emitted photon for the LED we want.

There is a little more to it, such as the electron in the material has a different effective mass than an electron in vacuum, but I think it is pretty convincing that the electron has wave properties, because as you change materials and the effective mass of the electron is different you have different size quantum wells and that is all connected by the de Broglie wavelength.