Two power series

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

Two power series, two quantisations. In Chapter 7 the one-dimensional oscillator's series for g(ξ)g(\xi) had to stop, and stopping it at jmax⁡=nj_{\max}=n gave E=(n+12)ℏωE=(n+\tfrac12)\hbar\omega with exactly one eigenfunction for each level. In this chapter the Coulomb series has to stop too, and stopping it at jmax⁡=nrj_{\max}=n_r gives ρ0=2n\rho_0=2n with n=nr+l+1n=n_r+l+1 and nr≥0n_r\ge0, so En=−13.6 eV/n2E_n=-13.6\ \mathrm{eV}/n^2. Take a one-dimensional oscillator with ℏω=0.500\hbar\omega=0.500 eV, and a hydrogen atom with a fixed point-charge proton. ((a)) For the oscillator whose series stops at jmax⁡=6j_{\max}=6, give the energy in eV and the number of independent eigenfunctions at that energy. ((b)) For hydrogen, how many pairs (nr,l)(n_r,l) give n=7n=7? ((c)) Give the energy of the hydrogen n=7n=7 level in eV. ((d)) How many independent states lie at that hydrogen energy, ignoring spin and then counting spin?

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