Use the derived hydrogen levels

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

Use the derived hydrogen levels En=−13.6 eV/n2E_n = -13.6\ \mathrm{eV}/n^2, with hc=1240 eV nmhc = 1240\ \mathrm{eV\,nm} and a0=0.0529 nma_0 = 0.0529\ \mathrm{nm}. The electron has spin one-half, with two values of msm_s. The proton is held fixed; fine structure and the reduced-mass correction are ignored, and each transition emits one photon whose energy equals the difference of the two levels. The n=2n=2, l=0l=0 radial function is R20(r)=12a0−3/2(1−r2a0)e−r/2a0.R_{20}(r) = \frac{1}{\sqrt2}a_0^{-3/2}\left(1 - \frac{r}{2a_0}\right)e^{-r/2a_0}. ((a)) Find the wavelength, in nm, of the line emitted in the transition n=4→n=3n = 4 \to n = 3. ((b)) Find the radius, other than r=0r=0 or r→∞r\to\infty, at which R20(r)=0R_{20}(r) = 0, as a multiple of a0a_0 and in nm. ((c)) How many states does the n=5n = 5 level contain, without spin and then with spin?

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