Wave function | Quantum Mechanics

A wavefunction is a mathematical quantity that represents the state of a quantum mechanical such as an electron or a photon. 

It is represented by:

$\psi$

In other words,

Let’s imagine you have a particle that’s dancing to a rhythm, and its dance moves are described by something called a wave function.

What a wave function can do?

It provides information of different outcomes of measurements of that system.

And that information is in the form of probabilities, as shown below:

As $\psi$ is a complex quantity, thus only $|\psi|^{2}$ has a physical significance.

Where, 

$|\psi|^{2}= \psi*\psi$

Position Probability Density:

$|\psi(x,t)|^{2}$   is called “position probability density”, i.e. probability of finding the particle at position ‘x’ at time ‘t’.

Probability of Finding the Particle in a Region:

$|\psi(x,t)|^{2}dx$, this quantity represents “probability of finding the particle in a region of space” lies between ‘x’ to ‘x+dx’ at time ‘t’.

The probability of finding the particle in a region of space lies between $x_{1}$ and $x_{2}$ at time ‘t’ is calculated by the following integral:

$\intop_{x_{1}}^{x_{2}}|\psi(x,t)|^{2}dx$

And in whole space (from +$\infty$ to -$\infty$):

$\intop_{-\infty}^{+\infty}|\psi(x,t)|^{2}dx$