Calculate the probability of an electron in the ground state of the hydrogen atom being inside the region of the proton. (For purposes of calculation, use a proton radius r = 1.09 × 10^(-15) m. Hint: Note that r ≪ a0.) P = ?
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This is given by: ψ(r) = (1/πa0^(3/2)) * e^(-r/a0) where a0 is the Bohr radius, approximately 5.29 x 10^(-11) m. The probability of finding the electron in a small volume element dV = 4πr^2 dr at a distance r from the nucleus is given by |ψ(r)|^2 dV. Show more…
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An electron in a hydrogen atom is in the ground state (1s). Calculate the probability of finding the electron within a Bohr radius $\left(a_{0}=0.05295 \mathrm{nm}\right)$ of the proton. The ground state wave function for hydrogen is: $\psi_{1 s}(r)=A_{1 s} e^{-r / a_{0}}=e^{-r / a_{0}} / \sqrt{\pi a_{0}^{3}}$.
The radius of the proton is about $R_{0}=10^{-15} \mathrm{~m} .$ The probability that the electron is inside the volume occupied by the proton is given by $$ P=\int_{0}^{R_{0}} P(r) d r $$ where $P(r)$ is the radial probability density. Compute $P$ for the hydrogen ground state. (Hint: Show that $e^{-2 r / a_{0}} \approx 1$ for $r \ll a_{0}$ is valid for this calculation.)
Find the probability that the electron in the hydrogen ground state will be found in the radial-distance range $r=a_{0} \pm 0.1 a_{0}$.
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