402 Chapter 14Partial differential equations
Separation of variables
The first step in the solution of the partial differential equation in three variables is
to separate the variables; that is, to reduce it to three ordinary equations. This is not
possible in cartesian coordinates because the potential function Vcannot then be
expressed as a sum of terms, one in each variable only. There exist a number of other
coordinate systems, however, in terms of which the separation is possible, and one
of these is the system of spherical polar coordinates discussed in Chapter 10. The
Laplacian operator in these coordinates is (Section 10.5)
and the Schrödinger equation is then (after multiplication by − 2 and rearrangement)
(14.52)
We first separate the radial terms from the angular terms by writing the wave function
as the product
ψ(r, θ, φ) 1 = 1 R(r) 1 × 1 Y(θ, φ) (14.53)
Substitution into (14.52), division throughout byψ 1 = 1 RY, and multiplication byr
2then gives
(14.54)
We call the separation constantl(l 1 + 1 1)for reasons that will become clear. Then
or
(14.55)
is the radial equationof the hydrogen atom, and
(14.56)
is the angular equation.
11
1
222sin
sin
sin
()
θθ
θ
θ
θ φ∂
∂
∂
∂
∂
∂
++
YY
ll Y== 0
112
2
222r
d
dr
r
dR
dr
ll
r
Z
r
ER
+−
++
=
()
00
1
22 1
22R
d
dr
r
dR
dr
Zr Er l l
++ =+()
1
22
11
22R
d
dr
r
dR
dr
Zr Er
Y
++ +
siin
sin
sin
θθ
θ
θ
θ φ
∂
∂
∂
∂
+
∂
∂
=
YY 1
22200
11
222r
r
r
r
r
∂
∂
∂
∂
+
∂
∂
∂
∂
ψψ
θ
θ
θ
θ
sin
sin ++
∂
∂
++=
12
20
2222r
Z
r
E
sin θ
ψ
φ
ψψ
∇=
∂
∂
∂
∂
∂
∂
∂
∂
222211
r
r
r
r
r sinsin
θ
θ
θ
θ
∂
∂
1
2222r sin θ φ