The Born Rule

Claimed - Rehaan Naik 11/27/2022

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. This result is a continuous form of the norm squared of the inner product being the probability that a known state takes the value of another.
[math]\displaystyle{ P_{\psi}(A = a) = |\bra{\phi_{a}}\ket{\psi}|^2 }[/math]
[math]\displaystyle{ \left | \Psi(x,t)^2 \right |=P }[/math]

Normalizing the Wavefunction

As the physical meaning of complex numbers is nonsense, we must normalize the wavefunction to conduct a meaningful analysis of quantum systems. 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 (the physical importance is found in the square of [math]\displaystyle{ \Psi }[/math]). By definition, an entire probability density function must have an area equal to one. Hence it follows that
[math]\displaystyle{ \int_{-\infty}^{\infty}{ \left | \Psi(x,t)^2 \right |dx}=1 }[/math],
as a given particle in question must be located somewhere between [math]\displaystyle{ -\infty \lt x \lt \infty }[/math]
Or more formally: [math]\displaystyle{ P_{x\in[-\infty,\infty]}=1 }[/math]