14.6 The hydrogen atom 401
(the radial function depends only on the modulus of n). The symmetries of these wave
functions, using the real forms (12.71) of the angular functions, are illustrated in
Figure 14.3.
We note that these diagrams also represent the normal modes of motion, the standing
waves, of a circular membrane.
0 Exercise 11
14.6 The hydrogen atom
A hydrogen-like atom is a system of two charges, a nucleus with charge+Ze(Z 1 = 11
for the hydrogen atom itself ) and an electron with charge −e, interacting through a
Coulomb force. The potential energy of the system is (see Examples 1.21 and 5.17)
(14.48)
where ris the distance between the charges. Let the nucleus be fixed at the origin of a
coordinate system, with the electron at position (x, 1 y, 1 z). The Schrödinger equation
for the motion of the electron about the stationary nucleus is then
(14.49)
where∇
2is the three-dimensional Laplacian and, in terms of cartesian coordinates,
the potential energy function is
(14.50)
We first simplify the equation by expressing all physical quantities in atomic units
(see Section 1.8); the Schrödinger equation ‘in atomic units’ is
− (14.51)
∂
∂
∂
∂
∂
∂
−
++
1
2
222222222ψψψ
xyz
Z
()xyz
112 /ψψ=E
Vxyz
Ze
xyz
()
()
,, =
−
++
/20222124 πε
−∇+=
222 m
VE
eψψ ψ
V
Ze
r
=
−
204 πε
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...k=1 k=1 k=1 k=2
n=0 n=± 1 n=± 2 n=0
Figure 14.3