PROPERTIES OF THE GENERAL SOLUTION
The solutions to the Schrödinger equation for the hydrogen atom should
in principle allow us to calculate all of the properties of molecules. We
will now use the solutions to address three related questions: what are
the physical properties of the orbitals, how do they change when there
is more than one electron, and how do they change when there is more
244
PART 2
QUANTUM MECHANICS AND SPECTROSCOPY
Then
(db12.37)
(db12.38)
Each of these two terms must be zero. The second term yields:
(db12.39)
so
(db12.40)
The first term yields:
(db12.41)
This is the solution for a specific set of quantum numbers (n = 1, l = 0, m l = 0). In general,
the solutions for the Laguerre equation, L s
k (x), can be expressed as in the following form:
(db12.42)
where s is the index with integral values starting with 0, and k is a second index greater
than −1.
L x
e x
s
x
e x
s
k
x
k
s
x s k
( )
!
(
)
=
−
−
+
d
d
E
me
= −
4
2
0
2
32π ε Z
R r
e
m e
r
( )
exp
=
=
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
−α
πε
Z
2
2
0
4
α
πε
=
m e
Z
2
2
0
4
re
mE
e
e
r
−
+
⎛
⎝
⎜ ⎜
⎞
⎠
⎟ ⎟ +
− +
α
r
−α
α
α
πε
2
2
2
0
2
2
4
2
Z
m m
Z
2
0
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=
α
α
πε
α
α
2
2
2
0
2
2
4
re
e
m e
r
E r
r
r
−
−
−
+
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
Z
e e
r
−
=
α
0
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