Take hydrogen with and

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

Deriving the Spectrum, Ch The exact Coulomb problem and Rydberg states

Problem

Take hydrogen with

R_{\rm H}=109\,677.58\,\mathrm{cm^{-1}}

and

1\,\mathrm{cm^{-1}}=29.9792\,\mathrm{GHz}

, and use the electric-dipole selection rule

\Delta l=\pm1

. The classical orbital frequency at level

n

\nu_{\rm orb}=2R_{\rm H}c/n^3

. ((a)) The circular state

n=100

l=99

emits a single electric-dipole photon. Predict which lower

n

it can reach, find the frequency of that line in GHz, and compare it with

\nu_{\rm orb}

n=100

. ((b)) The state

n=100

l=1

can instead jump to

n=98

. Find the frequency of that line and compare it with

2\nu_{\rm orb}

. ((c)) Ionised gas in the Galaxy shows both

\Delta n=1

and

\Delta n=2

recombination lines over a wide range of

n

. Explain why the

\Delta n=2

lines appear although a circular state never emits one, and why the comb of

\Delta n=1

lines does not need any single electron to pass through every level.

Related theory

Quantum Mechanics

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