Take the hydrogenic state with and in the field of a nucleus of charge

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · 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 the hydrogenic state with

n=3

and

l=0

in the field of a nucleus of charge

Z

, and use the dimensionless variable

\rho=2\kappa r

with

\kappa=Z/(na)

. ((a)) State

n_r

, and use the recursion

c_{k+1}/c_k=(k+l+1-n)/[(k+1)(k+2l+2)]

to find the polynomial

w(\rho)

up to its overall constant

c_0

. ((b)) Assemble

u=\rho^{l+1}e^{-\rho/2}w

and write the unnormalised

R_{30}(r)=u/r

as a function of

Zr/a

. ((c)) Find the radii of the nodes of

R_{30}

, in units of

a/Z

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