de Broglie Equation and Matter Waves
Description
The question of whether light is a wave or a particle has been a major interest in the history of physics. In the early 20th century, several experiments revealed that light possesses both particle and wave properties.
$$ \begin{align} E=\sqrt{p^2c^2+m_{0}^{2}c^{4}} \\ E=h\nu= \frac{hc}{\lambda} \end{align} $$
From the equation $(1)$ that expresses the relativistic energy of a particle and the equation $(2)$ derived from the photoelectric effect, it is understood that the wavelength of a photon with mass $0$ can be expressed in terms of momentum and Planck’s constant as follows.
$$ \begin{align} \lambda=\frac{h}{p} \end{align} $$
At this point, de Broglie proposed an outstanding idea: not only light, but all matter possesses duality. Therefore, particles also have wave properties and one of the characteristics of waves is that they have a wavelength. This wavelength is given by $(3)$. This theory is called de Broglie’s matter-wave theory. It was demonstrated by the experiments of Davisson and Germer, which observed the diffraction of electrons. Consequently, the wavelength of a particle is expressed as follows and is called the de Broglie relation.
De Broglie Relation
The wavelength of a particle is as follows.
$$ \lambda=\frac{h}{p}=\frac{h}{mv} $$
Moreover, by using the relation between wavenumber and wavelength $k=\frac{2\pi}{\lambda}$, wavenumber can also be expressed in terms of Planck’s constant and momentum.
$$ k=\frac{p}{\hbar} $$
Of course, in reality, this applies practically only to particles with very small masses, like electrons. If it applied to all particles, we would be able to walk through walls, but in actuality, we just bump into them, indicating that this does not hold true in the macroscopic world.