Expectation of radii of hydrogen atom wave function derivation
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The average or "expectation value" of the radius for the electron in the ground state of hydrogen is obtained from the integral
This requires integration by parts. The solution is
All the terms containing r are zero, leaving
It may seem a bit surprising that the average value of r is 1.5 x the first Bohr radius, which is the most probable value. The extended tail of the probability density accounts for the average being greater than the most probable value.
This requires integration by parts. The solution is
All the terms containing r are zero, leaving
It may seem a bit surprising that the average value of r is 1.5 x the first Bohr radius, which is the most probable value. The extended tail of the probability density accounts for the average being greater than the most probable value.
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