Apply the Rydberg scaling laws to a hydrogen atom excited to (rather than the worked…

PHYSICS · Multi-part · 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

Reading the Spectrum, Ch Radial functions, orbital size and Rydberg states

Problem

Apply the Rydberg scaling laws to a hydrogen atom excited to

n=30

(rather than the worked

n=50

example). Take

a_0 = 0.052918\,\mathrm{nm}

hcR_{\rm H} = 13.606\,\mathrm{eV}

R_{\rm H} = 109677.58\,\mathrm{cm^{-1}}

, and the nuclear-field estimate for the ionising field

F_{\rm ion} = 5.14\times10^9/n^4\,\mathrm{V\,cm^{-1}}

. ((a)) Compute the mean radius

\langle r\rangle

for

l=29

(the circular, highest-

l

orbit) and for

l=0

, using

\langle r\rangle = \tfrac{a_0}{2}[3n^2-l(l+1)]

. ((b)) Compute the binding energy of the

n=30

level. ((c)) Estimate the wavenumber spacing to

n=31

using

2hcR_{\rm H}/n^3

(treating

n

as continuous), and compare it (as a frequency in GHz) with the exact interval

hcR_{\rm H}(1/30^2-1/31^2)

. ((d)) Evaluate the nuclear-field estimate

F_{\rm ion}

for this state. (It fixes the

n^{-4}

power, not the measured ionising field.)

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