The Hydrogen Atom
The H atom is one of the problems we discussed two lectures ago. The Schrodinger
equation is
The angular terms are solved identically to last time. The separation constant
β
is
so the radial equation becomes.
)
,
,
(
)
,
,
(
)
(
)
,
,
(
)
(
sin
1
)
sin(
)
sin(
1
1
2
2
2
2
2
2
2
2
2
φ
θ
ψ
φ
θ
ψ
φ
θ
ψ
φ
θ
θ
θ
θ
θ
μ
r
E
r
r
V
r
r
r
r
r
r
r
=
+
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
∂
∂
∂
∂
−
h
)
1
(
+
l
l
(
)
r
e
r
V
r
R
E
r
R
r
V
r
dr
r
dR
r
dr
r
R
d
r
E
r
V
r
dr
r
dR
r
dr
d
r
R
0
2
2
2
2
2
2
2
2
2
2
4
)
(
where
)
(
)
(
)
(
)
1
(
2
)
(
2
)
(
2
or
)
1
(
)
(
2
)
(
)
(
1
ε
π
μ
μ
μ
−
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
+
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
+
−
=
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎝
⎛
−
+
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
l
l
h
h
l
l
h
Since V(r) is a function only of r, the potential energy commutes with both L
2
and L
z
, and
therefore so does the Hamiltonian operator itself. All have simultaneous eigenvalues and
eigenfunctions. The constant
ε
0
= 8.854·10
12
just makes things work in the infuriating
MKS electrical system. Note also that this equation will solve for any one-electron ion if
we replace e
2
by Ze
2
where Z is the atomic number of the nucleus, though answers become
poor much above Kr because the electron motion starts becoming relativistic.

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This is an important equation, so lets choose it to discuss how these things are solved or rather
were solved 160 years ago.
As r becomes large, the terms in 1/r and 1/r
2
become progressively smaller, and at
asymptotically large r we have
whose general solution is
If c is imaginary we find that E is positive. These are not the solutions we want, as they
represent incoming and outgoing spherical wavefronts. If c is real and positive the term with A
blows up as r gets large, which is wrong. So c is real and positive and
As r goes to zero the term with 1/r
2
in it dominates
(even the term in E becomes unimportant) and
whose solution is of the form
We then define a function
v(r):
and try to solve the equation for
v(r)
.
large)
r very
(for
)
(
)
(
2
2
2
2
r
u
E
dr
r
u
d
=
−
μ
h
imaginary
or
real
be
may
c
where
)
(
cr
cr
Be
Ae
r
u
−
+
≈
∞
→
≈
−
r
Be
r
u
cr
as
)
(
0
2
2
2
2
2
2
4
where
)
(
)
(
)
1
(
2
)
(
2
becomes
equation
above
the
and
define
First we
ε
π
μ
μ
e
V
r
u
E
r
u
r
V
r
dr
r
u
d
rR(r)
u(r)
−
=
′
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
′
+
+
+
−
=
l
l
h
h
)
(
)
1
(
)
(
2
2
2
r
u
r
dr
r
u
d
+
=
l
l
r
small
for
)
(
so
0
r
as
up
blows
but
)
(
1
1
+
+
≈
→
+
=
l
l
l
l
Cr
r
u
r
D
r
D
Cr
r
u
)
(
)
(
1
r
v
e
r
r
u
cr
−
+
=
l