If we consider the fact that the square of the wave function is positive everywhere we can understand that it gives us information on the region of space where the wave has the greaest amplitude. We call the square of the wave function the probability amplitude. The integral of the square of the wave function is called the probability. In order for a probability function to be meaningful it must be normalized. This is not just true in quantm mechanics, but in all statistical applications. Normalization means that if we integrate over the square the wave function including all possible space then the value should be equal to one. In this way we define a function that can give information on the fraction probability that the particle is in a region of space. We will consider an example derived from the particle in a box. However, before we do we can can example the figure above to see how normalization works in particle. The red function on the left is a wave function that is not normalized. Above the function we can see the value of its numerical integral. The value is approximately 180. When the same function is multipled by an appropriate constant, called a normalization constant, we see that the integral is equal to 1.0. Now we will examine how this concept is applied to the wave function for the particle in a box problem.
