17.4 The Orbitals of the Hydrogen-Like Atom 743
Table 17.3 Real Hydrogen-Like Energy Eigenfunctionsψ 10 ψ 1 s1
√
π(
Z
a) 3 / 2
e−Zr/aψ 20 ψ 2 s1
4
√
2 π(
Z
a) 3 / 2 (
2 −
Zr
a)
e−Zr/^2 aψ 21 xψ 2 px1
4
√
2 π(
Z
a) 3 / 2 (
Zr
a)
e−Zr/^2 asin(θ) cos(φ)ψ 21 yψ 2 py1
4
√
2 π(
Z
a) 3 / 2 (
Zr
a)
e−Zr/^2 asin(θ) sin(φ)ψ 210 ψ 2 pz1
4
√
2 π(
Z
a) 3 / 2 (
Zr
a)
e−Zr/^2 acos(φ)ψ 300 ψ 3 s1
18
√
3 π(
Z
a) 3 / 2 [
6 −
4 Zr
a+
(
2 Zr
3 a) 2 ]
e−Zr/^3 aψ 310 ψ 3 pz√
2
81
√
π(
Z
a) 3 / 2 (
6 Zr
a−
Z^2 r^2
a^2)
e−Zr/^3 acos(θ)ψ 31 xψ 3 px√
2
81
√
π(
Z
a) 3 / 2 (
6 Zr
a−
Z^2 r^2
a^2)
e−Zr/^3 asin(θ) cos(φ)ψ 31 yψ 3 py√
2
81
√
π(
Z
a) 3 / 2 (
6 Zr
a−
Z^2 r^2
a^2)
e−Zr/^3 asin(θ) sin(φ)ψ 320 ψ 3 dz 2 1
81
√
6 π(
Z
a) 3 / 2 (
Zr
a) 2
e−Zr/^3 a[3 cos^2 (θ)−1]ψ 3 dxz√
2
81
√
π(
Z
a) 3 / 2 (
Zr
a) 2
e−Zr/^3 asin(θ)cos(θ)cos(φ)ψ 3 dyz√
2
81
√
π(
Z
a) 3 / 2 (
Zr
a) 2
e−Zr/^3 asin(θ)cos(θ) sin(φ)ψ 3 dx (^2) −y 2
1
81
√
2 π(
Z
a) 3 / 2 (
Zr
a) 2
e−Zr/^3 asin^2 (θ)cos(2φ)ψ 3 dxy1
81
√
2 π(
Z
a) 3 / 2 (
Zr
a) 2
e−Zr/^3 asin^2 (θ)sin(2φ)The Qualitative Properties of the Hydrogen-Like Orbitals
It is important to have a grasp of the qualitative properties of the hydrogen-like orbitals
in three-dimensional space and to realize that they represent three-dimensional de
Broglie waves. The real orbitals that we have obtained correspond to standing waves,
with stationary nodes. We can visualize these waves by considering where they vanish.
A three-dimensional wave can vanish at a surface (anodal surface). Since each orbital
is a product of three factors, the orbital vanishes if any one of the factors vanishes.