Chemistry Reference
In-Depth Information
whereupon eq. (B.11) simplifies to
1
ρ
R
d
d
2
d
d
+
β
ρ
1
4
−
l
(
l
+
1
)
ρ
−
(ρ)
=
0
(B.14)
2
ρ
ρ
2
ρ
It can be shown that solutions to this equation can be written in the form
F
(ρ)
exp
(
−
ρ/
2
)
, where the polynomial
F
satisfies the differential equation
2
d
2
F
d
d
F
d
2
ρ
2
+
(
ρ
−
ρ
)
+[
ρ(β
−
)
−
(
+
)
]
=
2
1
l
l
1
F
0
(B.15)
ρ
ρ
k
, substitution into
eq. (B.15) shows that the leading power of the differential equation is
In this case, if the leading term of the polynomial
F
is
ρ
k
+
1
,
ρ
with coefficient
1. This term cannot cancel against lower order
terms in the polynomial, as all these have powers
β
−
k
−
k
or less; so to satisfy
ρ
eq. (B.15) we must have
β
=
k
+
1,
k
=
0, 1, 2,
...
, and also
k
≥
l
. This is
more usually written as
β
=
n
,
n
=
1, 2, 3,
...
(B.16)
with
n
l
. Combining eqs (B.13) and (B.16) we see that the hydrogen atom
energy levels are then given by
>
me
4
E
=
E
n
=−
(B.17)
2
2
n
2
2
(
4
πε
)
0
To summarise, we have thus deduced three quantum numbers associated
with each allowed energy state in the hydrogen atom, namely
n
,
l
and
m
.
The energy of each allowed hydrogen state depends only on
n
, which is
therefore referred to as the
principal
quantum number, with
l
referred to as
the angular momentum or
orbital
quantum number and
m
as the
magnetic
quantum number. In addition, each electron has an intrinsic
spin
, which
can take the values of
s
z
1
2
. The quantum number names, symbols,
and allowed values are summarised in Table B.1, along with the physical
property to which they are related.
The energy levels in eq. (B.17) do not depend on
l
,
m
or
s
z
. Therefore,
each energy level in the hydrogen atom has a multiple degeneracy, with
the number
N
of degenerate states depending on the number of allowed
values of
l
and
m
for each principal quantum number
n
:
=±
n
−
1
2
n
2
N
=
2
0
(
2
l
+
1
)
=
(B.18)
l
=
where the factor 2 comes from the existence of two spin states. Because all
states with the same principal quantum number
n
have the same energy,
we therefore say those states belong to the
n
th
shell
of states.
