Biomedical Engineering Reference
In-Depth Information
where
m
e
is the electron mass,
M
the nuclear mass and
U
the mutual electrostatic
potential energy of the nucleus and the electron, given by
=
−
Ze
2
4πε
0
1
U
(
x
(13.12)
X
)
2
Y
)
2
Z
)
2
−
+
(
y
−
+
(
z
−
I have temporarily added a subscript 'tot' to show that we are dealing with the total atom,
nucleus plus electron, at this stage. The equation is not at all simple in this coordinate system
and it proves profitable to make a change of variable. We take the coordinate origin as the
centre of gravity of the atom (coordinates
x
c
,
y
c
,
z
c
) and use spherical polar coordinates for
the nucleus and the electron. If the spherical polar coordinates of the electron are
a
, θ , φ
and those of the nucleus
b
, θ , φ where
M
a
=
m
e
r
M
+
m
e
b
=−
m
e
r
M
+
then, if μ is the reduced mass
m
e
M
m
e
+
=
μ
M
μ
M
r
sin θ cos φ
μ
m
e
r
sin θ cos φ
X
=
x
c
−
x
=
x
c
+
(13.13)
μ
M
r
sin θ sin φ
μ
m
e
Y
=
y
c
−
y
=
y
c
+
r
sin θ sin φ
μ
M
r
cos θ
μ
m
e
r
cos θ
Z
=
z
c
−
z
=
z
c
+
With this change of variable we find
∂
2
Ψ
tot
∂
x
c
h
2
8π
2
(
m
e
+
∂
2
Ψ
tot
∂
y
c
∂
2
Ψ
tot
∂
z
c
−
+
+
(13.14)
M
)
1
r
2
r
2
∂Ψ
tot
∂
r
sin θ
∂Ψ
∂θ
∂φ
2
h
2
8π
2
μ
∂
2
Ψ
tot
∂
∂
r
1
r
2
sin θ
∂
∂θ
1
r
2
sin
2
θ
−
+
+
+
U
Ψ
tot
=
ε
tot
Ψ
tot
(13.15)
and also in this coordinate system
1
4πε
0
Ze
2
r
U
=−
It is apparent that we can separate the wave equation into two equations, one referring to the
centre of mass and involving the total mass of the atom, and the other containing
r
, θ and
φ We put Ψ
tot
=
ψ
e
ψ
c
and follow through the separation of variables argument to obtain
