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272 8 ATOMIC STRUCTURE AND SPECTRA
and ϕ = 0 to 2π; the volume element is r2 sin θ dr dθ dϕ.�e required integral
is therefore
N2 ∫
∞
0
∫
π
0
∫
2π
0
(2 − r/a0)2r2e−r/a0 sin θ dr dθ dϕ
�e integrand is a product of functions of each of the variables, and so the
integral separates into three
N2 ∫
∞
0
(2 − r/a0)2r2e−r/a0 dr∫
π
0
sin θ dθ ∫
2π
0
dϕ
�e integral over r is found by �rst expanding the bracket to give
∫
∞
0
(2 − r/a0)2r2e−r/a0 dr = ∫
∞
0
(4r2 − 4r3/a0 + r4/a20)e−r/a0 dr
= 4 2!
(1/a0)3
− 4
a0
3!
(1/a0)4
+ 1
a20
4!
(1/a0)5
= 8a30 − 24a30 + 24a30 = 8a30
�ese integrals are evaluated using Integral E.3 with the appropriate value of n
and k = 1/a0.�e integrals over the angles are straightforward
∫
π
0
sin θ dθ ∫
2π
0
dϕ = (− cos θ)∣π0 × ϕ∣2π
0 = 4π
�e complete integral evaluates to N2 × 4π × 8a30 ; setting this equal to 1 gives
N = (32πa3)−1/2 .
E8A.3(b) �e wavefunction is given by [8A.12–307], ψn , l ,m l = Yl ,m l (θ , ϕ)Rn , l(r); for the
state with n = 3, l = 0, m l = 0 this is
ψ3,0,0 = Y0,0(θ , ϕ)R3,0(r) = (4π)−1/2(243)−1/2(Z/a0)3/2(6 − 6ρ + ρ2)e−ρ/2
where the radial wavefunction is taken fromTable 8A.1 on page 306, the angular
wavefunction (the spherical harmonic) is taken from Table 7F.1 on page 286,
and ρ = 2Zr/na0.�e probability density is therefore
P3,0,0 = ∣ψ3,0,0∣2 = (4π)−1(243)−1(Z/a0)3(6 − 6ρ + ρ2)2e−ρ
�e probability density at the nucleus, ρ = 0, is then (4π)−1(243)−1(Z/a0)3(6)2 =
Z3/(27πa3) .
E8A.4(b) �e radial wavefunction of a 3s orbital is taken from Table 8A.1 on page 306,
R3,0(r) = N(6 − 6ρ + ρ2)e−ρ/2, where ρ = 2Zr/na0; for n = 3, ρ = 2Zr/3a0.
�e extrema are located by �nding the values of ρ for which dR3,0/dρ = 0; the
product rule is required
dR3,0
dρ
= N d(6 − 6ρ + ρ2)
dρ
e−ρ + N(6 − 6ρ + ρ2)de
−ρ/2
dρ
= (2ρ − 6)e−ρ/2 − 1
2 (6 − 6ρ + ρ2) e−ρ/2
= −(ρ2 − 10ρ + 18)e−ρ/2