Biomedical Engineering Reference
In-Depth Information
h
2
8π
2
m
e
d
2
P
d
r
2
Ze
2
4πε
0
r
P
−
−
=
ε
P
It is usual in many scientific and engineering applications to try and simplify such com-
plicated equations by use of 'reduced variables'. In this case we divide each variable by its
atomic unit, so for example
r
red
=
r
/
a
0
ε
red
=
ε/
E
h
d
2
d
r
red
=
d
2
d
r
2
a
0
and simplify to give
1
2
d
2
P
red
d
r
red
−
Z
r
red
P
red
=
−
ε
red
P
red
It gets a bit tedious writing the subscript 'red', somost authors choose to state their equations
in dimensionless form such as
1
2
d
2
P
d
r
2
Z
r
P
−
−
=
ε
P
(13.20)
with the tacit understanding that the variables are reduced ones and therefore dimensionless.
Sometimes authors say (incorrectly) that the equation is written 'in atomic units', and
occasionally we come across statements of the impossible (e.g.
m
e
=
1).
We therefore have to solve the differential Equation (13.20) and we follow standard
mathematical practice by first examining the limiting behaviour for large
r
:
d
2
P
d
r
2
1 and
h
/2π
=
=−
2ε
P
If we limit the discussion to bound states (for which ε is negative), we note that
P
(
r
)
≈
exp (
−
β
r
)
(13.21)
where β is a positive real constant. This suggests that we look for a wavefunction of the
form
P
(
r
)
=
Φ(
r
) exp (
−
β
r
)
(13.22)
where Φ(
r
) is a polynomial
in
r
. Substitution of Equation (13.22) into the radial
Equation (13.20) then gives
d
2
Φ
d
r
2
β
2
Φ
2β
dΦ
2
Z
r
−
d
r
+
+
2ε
+
=
0
(13.23)
Suppose for the sake of argument we choose
Φ (
r
)
=
α
2
r
2
+
α
1
r
+
α
0
(13.24)
where the coefficients α have to be determined. Substitution of Equation (13.24) into the
differential Equation (13.23) gives
β
2
×
α
2
r
2
α
0
=
2
Z
r
2α
2
−
2β (2α
2
r
+
α
1
)
+
+
2ε
+
+
α
1
r
+
0
