Definition Radial distribution function

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The radial distribution function can be defined in two ways: 4πr²R(r)² and r²R(r)², with the former including a factor of 4π. This discrepancy arises from whether the calculation is for volume or area; the 4π factor is necessary for volume calculations, while the simpler form is used for area. The normalization of the wavefunction, which ensures that the total probability equals one, supports the inclusion of the 4π factor in volume contexts. Understanding the context of the calculation is crucial for determining which expression to use. Proper application of these definitions is essential for accurate quantum mechanical computations.
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In some books the radial distribution function is defined as 4pi*r^2*R(r)^2 and in others as r^2*R(r)^2. Thus, a factor of 4pi is differing. Which expression is correct and why?
 
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This factor may come from the normalization of the wavefunction ##\int_{\mathbb{R}^3}drd\theta d\varphi \left| \Psi(r,\theta,\varphi) \right|^2 = 1##.
 
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA

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