The (quoted) normalised hydrogen ground state is

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · One-electron atoms and the origins of atomic structure

Reading the Spectrum, Ch The Coulomb spectrum from wave mechanics: quantum numbers and degeneracy

Problem

The (quoted) normalised hydrogen ground state is ψ1s(r)=1πa03e−r/a0\psi_{1s}(r) = \dfrac{1}{\sqrt{\pi a_0^3}}e^{-r/a_0}, where a0a_0 is the Bohr radius. ((a)) Verify the normalisation by showing that ∫0∞4πr2∣ψ1s(r)∣2 dr=1\int_0^\infty 4\pi r^2 |\psi_{1s}(r)|^2\,dr = 1, given that ∫0∞r2e−2r/a0 dr=a03/4\int_0^\infty r^2 e^{-2r/a_0}\,dr = a_0^3/4. ((b)) Calculate the ratio of the probability density ∣ψ1s∣2|\psi_{1s}|^2 at r=1.5a0r=1.5a_0 to its value at r=4a0r=4a_0.

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