Atomic Structure Theory

Evaluate norm and scalar product.
assume(Z>0);
for n1 from 2 to 4 do
for n2 from n1 to 4 do
sp := int(P(1,n1,1,r)*P(1,n2,1,r),r=0..infinity);
print( scal-prod(n1,n2) = sp)
od;
od;
2.2 Obtain an expression for the expectation value
1
r2
,
in the 3d state of a hydrogen-like ion with nuclear charge Z:
3d
1
r
2
3d
= 2Z2
135 .
This reduces to 0.1333 for hydrogenlike lithium, Z=3.
2.3 From Gauss’s law, the radial electric field of the nucleus is:
E(r) = Z|e|
4πɛ0
= Z|e|
4πɛ0
r
, r ≤ R
R3 1
, r > R.
r2 The corresponding electrostatic potential Φ(r) is
Φ(r) = − Z|e|
4πɛ0
Z|e|
=
4πɛ0
r 2
+ C, r ≤ R
2R3 1
, r > R,
r
Solutions 269
where C =3Z|e|/8πɛ0 is a constant chosen to make Φ continuous at r = R.
The potential energy V (r) =−|e|Φ(r) (in atomic units) is:
V (r) = − Z
R
3 1
−
2 2
r2 R2
, r ≤ R
= − Z
, r > R.
r
Let us use mathematica to evaluate the energy shift. First expand the
wave function in powers of r, then integrate the difference between the finite
nuclear potential and the Coulomb potential in 0 ≤ r ≤ R: