Answer
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Work Step by Step
The concept of radial probability density, denoted as $ P_r(r) $, represents the likelihood of locating an electron at a certain distance $ r $ from the nucleus. It takes into account the spherical shell's thickness at that distance. The expression for $ P_r(r) $ is $ 4\pi r^2 |R_{nl}(r)|^2 $, where $ R_{nl}(r) $ is the radial wave function.
In the case of the hydrogen atom, the radial wave function for the $1s$ orbital, $ R_{1s}(r) $, reaches its highest value at $ r = 0 $, indicating that the electron is most likely to be found at the origin. However, if we consider the radial probability density itself, the probability at a specific distance $ r = a_B $ (the Bohr radius) is lower at any individual point compared to the origin.
Nonetheless, the number of possible points where $ r = a_B $ is significantly greater than the single point at $ r = 0 $. This larger number of points compensates for the reduced probability density at each individual point, making the overall probability of finding the electron at $ r = a_B $ higher than at the origin. Thus, even though the electron's probability density is higher at the origin, the increased volume at the Bohr radius makes it more likely to find the electron there.
