3 = 4.3.12 Consider the following problem (Stewart 2006). The hydrogen atom con- sists of one proton in the nucleus and

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3 4 3 12 Consider The Following Problem Stewart 2006 The Hydrogen Atom Con Sists Of One Proton In The Nucleus And 1 (172.59 KiB) Viewed 24 times

3 = 4.3.12 Consider the following problem (Stewart 2006). The hydrogen atom con- sists of one proton in the nucleus and one electron, which moves about the nucleus. The electron does not move in a well-defined orbit, but there is a probability for finding the electron at a certain distance from the nucleus. The PDF is given by p(r) = 4 r- exp(-2 r/ao) / a' for r > 0, where do 5.59 10-11 m is the Bohr radius. The integral over this PDF from zero to r gives the probability that the electron will be found within a sphere of radius r meters centered at the nucleus. a) Calculate the mean distance R of the electron from the nucleus. b) Calculate the variance o of the PDF. c) Show that the PDF is a gamma PDF. Hint: use I(n + 1) = n! for integer values of n. =