Question 39.46: Calculate the probability that the electron in the hydrogen ......

Fundamentals of Physics Extended – Instructor’s Solutions Manual [1360823]

Calculate the probability that the electron in the hydrogen atom, in its ground state, will be found between spherical shells whose radii are a and 2a, where a is the Bohr radius.

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From Sample Problem — “ Probability of detection of the electron in a hydrogen atom,” we know that the probability of finding the electron in the ground state of the hydrogen atom inside a sphere of radius r is given by

$p(r)=1-e^{-2 x}( 1+2 x+2 x^2 )$

where x = r/a. Thus the probability of finding the electron between the two shells indicated in this problem is given by

$\begin{aligned}p(a<r<2 a) & =p(2 a)-p(a)=\left[1-e^{-2 x}\left(1+2 x+2 x^2\right)\right]_{x=2}-\left[1-e^{-2 x}\left(1+2 x+2 x^2\right)\right]_{x=1} \\& =0.439\end{aligned}$

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