Question

What are matter waves?

Answer 1

Behavior of all microscopic particles can be described as a wave function. Microscopic particles are matter. The relationship between momentum and wavelength for matter waves is given by $p=\dfrac{h}{\lambda }$. The $\lambda =\dfrac{h}{p}$ relationship energy and frequency is $E=h\times v$.

Complete answer:
In 1924, Louis deBroglie proposed that a wave function is associated with all microscopic particles. Positions where the wave functions have non-zero amplitude; it is more probable for us to find the particle there. So the intensity of the wave function of a particle at any point is directly proportional to the probability of finding the particle at that point. The wavelength obtained from the wave functions can be used to find the most likely momentum of the particle and the uncertainty in the momentum. A material particle has a wave function which is often known as matter wave. The wavelength $\lambda =\dfrac{h}{p}$ is called the de Broglie wavelength. The relations $\lambda =\dfrac{h}{p}$ and $v=\dfrac{E}{h}$ are called the de Broglie relations. These are the same relations we have for the photon, but for particles $E=0.5m{{v}^{2}}=\dfrac{{{p}^{2}}}{2m}$ . The relationship between λ and E is different for particles than for photons. It is considered that all matter exhibits wave-like behavior. So all matter can have a wave equation.

Note: You should not be confused with microscopic and macroscopic particles. A matter wave exists for microscopic particles and macroscopic particles but it is difficult to be noticed. Photons are the particles of light. Matter is made of atoms, and atoms are made protons, neutrons and electrons.