The Born Rule: Difference between revisions
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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. <br> | 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. <br> | ||
Discrete Probability<math> P_{\psi}(A = a) = |\bra{\phi_{a}}\ket{\psi}|^2 </math> <br> | Discrete Probability<math> P_{\psi}(A = a) = |\bra{\phi_{a}}\ket{\psi}|^2 </math> <br> | ||
Continuous Probability Density Function <math> P = \left | \Psi(x,t) | Continuous Probability Density Function <math> P = \left | \Psi(x,t) \right |^2 </math> | ||
==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) | 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) \right |^2 </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) | <math>\int_{-\infty}^{\infty}{ \left | \Psi(x,t) \right |^2 dx}=1 </math>, | ||
<br>as a given particle in question must be located somewhere between <math> -\infty < x <\infty </math> <br> | <br>as a given particle in question must be located somewhere between <math> -\infty < x <\infty </math> <br> | ||
Or more formally: | Or more formally: | ||
<math>P_{x\in[-\infty,\infty]}=1 </math> | <math>P_{x\in[-\infty,\infty]}=1 </math> | ||
Revision as of 21:00, 27 November 2022
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.
Discrete Probability[math]\displaystyle{ P_{\psi}(A = a) = |\bra{\phi_{a}}\ket{\psi}|^2 }[/math]
Continuous Probability Density Function [math]\displaystyle{ P = \left | \Psi(x,t) \right |^2 }[/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) \right |^2 }[/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) \right |^2 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]