The Born Rule: Difference between revisions
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==Normalizing the Wavefunction== | ==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>x</math> and time <math>t</math>, <math> \left | \Psi(x,t)^2 \right |</math> is the probability density (the physical importance is found in the square of <math>\ | 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>x</math> and time <math>t</math>, <math> \left | \Psi(x,t)^2 \right |</math> is the probability density (the physical importance is found in the square of <math>\Psi</math>. By definition, an entire probability density function must have an area equal to one. Hence it follows that <br> | ||
<math>\int_{-\infty}^{\infty}{ \left | \Psi(x,t)^2 \right |dx}=1 </math>, | <math>\int_{-\infty}^{\infty}{ \left | \Psi(x,t)^2 \right |dx}=1 </math>, | ||
<br>as a given particle in question must be located somewhere between <math> -\infty < x <\infty </math> | <br>as a given particle in question must be located somewhere between <math> -\infty < x <\infty </math> | ||
Revision as of 02:03, 5 October 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.
[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]