Question

The radial wave function for the hydrogen atom in three dimensions is given by
Rnl(r) = 1
r
ρ
l+1e
−ρ
v(ρ)
where v(ρ) = P∞
j=0 cjρ
j
is a polynomial of degree jmax = n−l−1 in ρ whose coefficients
are determined by the recursion formula
cj+1 =
2(j + l + 1 − n)
(j + 1)(j + 2l + 2)cj
.
(a) For n = 2 write down the allowed values of ml and jmax.

Hence by using the fact that ρ can be defined in terms of the Bohr radius a i.e.,
ρ = r/an, show that (don’t normalize)
R20(r) = c0
2a

1 −
r
2a

e
−r/2a
.

Write all spherical harmonics up to l = 2 (there are nine of them) in Cartesian form,
i.e. give expressions in terms of x, y, z, and r. You can either use the Rodrigues formula
for the Legendre polynomials or start with the given expressions for Y
m
l
in terms of θ
and φ. In any event you must show your work.

Answer 1