The fixed-level saddle-point field for a hydrogen Rydberg level of principal quantum…

PHYSICS · Multi-part · Fermi University Physics · Atomic, Molecular & Optical Physics · One-electron atoms and the origins of atomic structure · Radiative transitions, external fields and collisions

Reading the Spectrum, Ch Atoms in an electric field: the Stark effect and polarizability

Problem

The fixed-level saddle-point field for a hydrogen Rydberg level of principal quantum number

n

F_{\rm fixed} = 3.21\times10^{8}/n^4\,\mathrm{V\,cm^{-1}}

. The reddest extreme state is pushed down by its own linear Stark shift as the field rises. Write the field as

xE_0/n^4

and, for large

n

, replace

n(n-1)

n^2

: the reddest state's energy is then

-\tfrac12 - \tfrac32x

and the saddle's height is

-2\sqrt x

, both in units of

e^2/4\pi\epsilon_0a_0n^2

. ((a)) Calculate

F_{\rm fixed}

for

n=30

. ((b)) Find the value of

x

at which the rising field first brings the saddle down to the reddest state, and hence

F_{\rm red}

for

n=30

and the ratio

F_{\rm red}/F_{\rm fixed}

. ((c)) State, in one sentence, why the reddest state needs a larger field than a level held at fixed energy would suggest.

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