Take hydrogen ( ) and write for its reduced-mass Bohr radius

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 (

Z=1

) and write

a

for its reduced-mass Bohr radius. A small added potential

\delta V(r)

shifts a level by its average over the state,

\Delta E=\langle\delta V\rangle

, the first-order shift rule held from the first course. ((a)) From the chapter's radial-moment formulas, find

\langle r\rangle

\langle r^2\rangle

and

\langle r^{-2}\rangle

for the

2s

and the

2p

states. ((b)) For

\delta V=\epsilon r^2

with

\epsilon>0

, which of

2s

and

2p

is pushed up more, and by how much more, in units of

\epsilon a^2

? ((c)) For

\delta V=-b/r^2

with

b>0

, find the ratio of the

2s

shift to the

2p

shift. ((d)) For

\delta V=\epsilon(r^2-12ar)

, show that

2s

and

2p

shift by the same amount. Does this contradict the rule that a departure from

1/r

lifts the

l

degeneracy?

Related theory

Quantum Mechanics

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