Repeat the scaling of the Coulomb radial equation for a one-electron ion of nuclear charge

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

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The hydrogen atom

Problem

Repeat the scaling of the Coulomb radial equation for a one-electron ion of nuclear charge ZeZe, whose potential energy is V(r)=−Ze2/4πϵ0rV(r) = -Ze^2/4\pi\epsilon_0r, with the energy zero at infinity and the nucleus held fixed; the electron has mass mem_e, is treated non-relativistically, and spin is ignored. As in the chapter, κ=−2meE/ℏ\kappa = \sqrt{-2m_eE}/\hbar, ρ=κr\rho=\kappa r, the quantisation condition is ρ0=2n\rho_0 = 2n, and a0=4πϵ0ℏ2/mee2=0.0529a_0 = 4\pi\epsilon_0\hbar^2/m_ee^2 = 0.0529 nm with ℏ2/2mea02=13.6\hbar^2/2m_ea_0^2 = 13.6 eV. Take hc=1240hc=1240 eV nm. ((a)) Show that the dimensionless equation keeps the chapter's form but with ρ0=2Z/κa0\rho_0 = 2Z/\kappa a_0, so that κ=Z/(na0)\kappa = Z/(na_0) and En=−13.6Z2/n2E_n = -13.6Z^2/n^2 eV. ((b)) Calculate the wavelength, in nm, of the photon emitted in the n=2→n=1n=2\to n=1 transition of Li2+^{2+} (Z=3Z=3). ((c)) Find the most probable radius of the ground state of Li2+^{2+}, in nm.

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