He ( ) has one electron

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 Fine structure and the Lamb shift

Problem

^+

(

Z=2

) has one electron, so its

1S

and

2S

levels follow the same fine-structure formula as hydrogen's. Ignore the difference between its reduced mass and hydrogen's, take

E_n=-Z^2hcR_{\rm H}/n^2

with

cR_{\rm H}=3.28805\times10^6\,\mathrm{GHz}

, and take

\alpha=1/137.036

. ((a)) Using the shift of an

s

level,

\frac{E_n^2}{2mc^2}(3-4n)

, and

E_n/mc^2=-(Z\alpha)^2/2n^2

, find how far fine structure moves the

1S

and

2S

levels of He

^+

, in GHz. ((b)) Hence find how much fine structure widens the

1S

2S

interval, and compare with the

30.1\,\mathrm{GHz}

by which it widens hydrogen's. ((c)) A non-relativistic treatment was expected to miss a fraction of order

(Z\alpha)^2

of this interval,

\tfrac34Z^2cR_{\rm H}

. What fraction of that scale does the fine-structure widening actually reach?

Related theory

Quantum Mechanics

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