A moving particle such as a proton or an electron can be described as a matter wave. Because it also exhibit wave-like properties according to de Broglie. Its wavelength called the de Broglie wavelength is given by λ=h/p where p is the momentum of the particle. The displacement of a matter wave is given by its wave function ψ which gives us the distribution of the particle in space. We can use the wave function ψ for a particle to give us information about the state of that particle.

Because a particle can behave like a wave, a wave equation can be used to explain the behaviour of these atomic particles. In 1925, Schrodinger proposed the first wave equation, a differential equation in which one form of it is written as

$ – \frac{\hbar^{2}}{2m} \frac{d^{2} \psi}{dx^{2}} + U \psi = E \psi$

for a particle of mass m moving along the x axis in a system of total energy E and potential energy U.

In the above equation, the particle in a system is represented by a wave function ψ(x) and its solution describes the quantum properties of the system.

**Wave function ψ(x,y,z,t) of a particle is the amplitude of matter wave associated with particle at position and time represented by (x,y,z) and t.**

**Some properties of wave function ψ:**

- ψ is a continuous function
- ψ can be interpretated as the amplitude of the matter wave at any point in space and time.
- ψ allows for the calculation of allowed energy values and momentum through the Schrodinger equation.
- ψ allows for the calculation of a probability distribution in 3-dimensional space.

Max Born proposed a probability interpretation of the physical significance of the wave function ψ. He suggested that the square of the absolute value of the wave function |ψ|^{2} is the probability of finding the particle at a point, which is called the probability density.

Hence, the probability of detecting a particle in a small volume of space ΔV

P = |ψ|^{2}ΔV