The ion has a single electron bound to an alpha particle,

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

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

Problem

The ion

\mathrm{He}^+

has a single electron bound to an alpha particle,

Z=2

. Its Rydberg constant, with the alpha particle's reduced mass included, is

R_{\mathrm{He}^+} = 109\,722.27\,\mathrm{cm^{-1}}

; hydrogen's is

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

. Use

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

and

\alpha = 1/137.036

. (a) Calculate

Z\alpha

for the

1s

electron of

\mathrm{He}^+

and hence the fractional scale,

(Z\alpha)^2

, of what a non-relativistic treatment of the binding energy misses. (b) The non-relativistic

1S

-to-

2S

analogue interval of

\mathrm{He}^+

\Delta E/h = \tfrac34 Z^2 R_{\mathrm{He}^+}c

. Compute this interval in GHz, and the error that using

R_{\rm H}

in place of

R_{\mathrm{He}^+}

would make in it. Then use the fractional scale from part (a) to estimate the absolute size of the relativistic miss expected in this interval, and decide whether non-relativistic hydrogenic scaling is adequate to predict this interval to within

1\,\mathrm{GHz}

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