1. A one-electron atom has atomic number ?, mass number ? and a spherical nucleus of...

1. A one-electron atom has atomic number ?, mass number ? and a spherical nucleus of radius ? . Assume electric charge +?? is uniformly distributed throughout the volume of
the nucleus. Ignoring spin, use first order non-degenerate perturbation theory and the
hydrogenic wave functions adapted to the one-electron atom to determine the dependence
of the ground state energy of the atom on ?

Solutions

Expert Solution

The ground state of the hydrogenic atom with a point nucleus is non-degenerate. The potential energy of an electron outside a uniformly charged sphere of radius r0 and total charge Zqe is , with .
Inside the sphere the potential energy is
U(r) = [Ze2/(2r0)][(r/r0)2 - 3].
From Gauss' law we know that E(r < r0) = qinside/(4πε0r2) = Zqer/(4πε0r03) (SI units).
Φ(r) = Φ(r0) + [Zqe/(4πε0r03)]∫rr0rdr = Zqe/(4πε0r0) + Zqe(r02 - r2)/(8πε0r03).
U(r) = -qeΦ(r) = [Ze2/(2r0)][(r/r0)2 - 3].

Let H0 = p2/(2μ) - Ze2/r, H = p2/(2μ) + [Ze2/(2r0)][(r/r0)2 - 3] = H0 + H' (r < r0).
H' = Ze2r2/(2r03) - 3Ze2/(2r0) + Ze2/r.
E10 = <100|H'|100> = ∫02πdφ∫0πsinθ dθ Y00(θ,φ)∫0r0r2dr |R10(r)|2 H'
≈ ∫0r0r2dr |R10(0)|2 H',
as long as r0 << a.
R10(r) = (2/a3/2)exp(-r/a), R10(0) = 2/a3/2, with a = a0/Z.
E10 = (4Z/a3)[ e2r05/(10r03) - 3e2r03/(6r0) + e2r02/2] = (2/5)Z(e2/a)(r0/a)2
= (4/5)Z2EI(r0/a)2.

Hence the ground state energy depends upon r0

genius_generous answered 1 year ago

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