The Born Rule

The Born Rule is an important result of quantum mechanics that describes the probability density of a measured quantum system. In particular, it states that the square of the wavefunction is proportional to the probability density function. [math]\displaystyle{ \left | \Psi(x,t)^2 \right |=P }[/math]

Normalizing the Wavefunction

At the position [math]\displaystyle{ x }[/math] and time [math]\displaystyle{ t }[/math], [math]\displaystyle{ \left | \Psi(x,t)^2 \right | }[/math] is the probability density. By definition, an entire probability density function must have an area exactly equal to one. Hence it follows that
[math]\displaystyle{ \int_{-\infty}^{\infty}{ \left | \Psi(x,t)^2 \right |dx}=1 }[/math]