Chemistry Reference
In-Depth Information
number of terms. Here we need only note that allowed solutions will only
exist if the coefficient of each power of
z
in the polynomial is identically
zero when substituted into eq. (B.8).
If the leading term of the polynomial
P
is
z
n
, substitution into eq. (B.8b)
shows that the leading power in the differential equation is
z
n
+
2
, with
coefficient
n
2
. Thismust vanish, so
(
n
−
1
)
−
λ
λ
=
n
(
n
−
1
)
. It is conventional
to put
n
=
l
+
1, and so to write
λ
=
(
+
)
l
l
1
(B.9)
Further detailed analysis shows that allowed solutions can only exist when
l
m
2
, which requires
(
l
+
1
)>
−
l
≤
m
≤
l
(B.10)
so that there are 2
l
1 allowed values of
m
for each value of
l
.
We now substitute eq. (B.9) for
+
λ
back into eq. (B.4) to derive the radial
Schrödinger equation
R
r
2
d
R
d
r
2
2
1
d
d
r
l
(
l
+
1
)
r
2
−
+
+
V
(
r
)
(
r
)
=
ER
(
r
)
(B.11)
2
m
2
m
r
2
The
l
-dependent termmaybewrittenas
Q
2
2
mr
2
, and is the quantumcoun-
terpart of the classical 'centrifugal' potential barrier
Q
2
/
2
mr
2
encountered
for example in the Kepler problem of planetary orbits, where
Q
/
mr
2
is
the angular momentum of the orbiting particle. We have thus shown that
the angular momentum is quantised in a spherically symmetric potential,
with the magnitude squared of the angular momentum
Q
2
=
ω
2
l
,
where
l
is an integer. Further analysis reveals that if we quantise along
a particular direction (e.g. along the
z
-axis) then the angular momen-
tum component along that axis is also quantised, with the component
Q
z
projected onto that axis equal to
=
(
l
+
1
)
Q
z
=
m
,
|
m
|≤
l
(B.12)
We note that eq. (B.11) depends on the total angular momentum through
the term containing
l
, but does
not
depend on
m
, the angular momen-
tum component along the quantisation axis. This is to be expected. The
energy should not depend on the orientation of the
z
-axis in a spherically
symmetric potential.
Equation (B.11) can be solved for the hydrogen atom using standard
mathematical techniques, again described in many quantum mechanics
texts. To avoid having to work with a large number of constants (
e
,
m
,
(
l
+
1
)
,
etc.) we introduce a change of variables
2
me
2
=
−
8
mE
2
ρ
=
α
r
;
α
;
β
=
(B.13)
2
2
4
πε
α
0
