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Physics 3220 – Quantum Mechanics 1 – Fall 2008
Problem Set #12
Due Wednesday, December 3 at 2pm
Problem 12.1
: Analytic solution of radial equation for hydrogen. (20 points)
Stationary states for the hydrogen atom that are also eigenstates of
L
2
and
L
z
were found
to take the form
ψ
n±m
(
r, θ, ϕ
) =
R
n±
(
r
)
Y
m
±
(
θ, ϕ
)
≡
u
n±
(
r
)
r
Y
m
±
(
θ, ϕ
)
,
(1)
where the
Y
m
±
are the spherical harmonics, and
u
(
r
) solves the radial equation,

¯
h
2
2
m
e
d
2
u
dr
2
+
±

ke
2
r
+
¯
h
2
2
m
e
±
(
±
+ 1)
r
2
²
u
=
Eu ,
(2)
where we have written
m
e
for the mass of the electron to avoid confusion with the azimuthal
angular momentum quantum number
m
, and
k
= 1
/
4
π²
0
.
We will solve this equation using the method of Frobenius, the same method we explored
for the analytic solution of the harmonic oscillator.
a) We begin by introducing a dimensionless variable to replace
r
, which we’ll call
ρ
. Divide
the radial equation (2) by
E
and deﬁne a variable
ρ
≡
r/
¯
r
, where ¯
r
is for you to determine,
such that the ﬁrst and last terms take the form
d
2
u
dρ
2
+
. . .
=
u .
(3)
What is ¯
r
? Check that it has units of length. What is the sign of
E
appropriate to bound
states, and given this, is ¯
r
real and positive?
Next deﬁne a constant
ρ
0
to absorb most of the remaining constants, so that the equation
can be written
d
2
u
dρ
2
=
³
1

ρ
0
ρ
+
±
(
±
+ 1)
ρ
2
´
u .
(4)
What is
ρ
0
? What are its units?
b) Next we will study the asymptotics of the solution. Unlike the harmonic oscillator case,
where
x
→ ∞
and
x
→ ∞
had the same behavior and could be examined at the same
time, here we will separately consider
r
→ ∞
and
r
→
0.
Explain why in the
r
→ ∞
limit (which implies
ρ
→ ∞
), the radial equation reduces to
d
2
u
dρ
2
≈
u .
(5)
1
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View Full DocumentVerify that the solution in this limit is
u
(
ρ
)
≈
Ae

ρ
+
Be
+
ρ
.
(6)
Explain what constraint must we put on
A
or
B
to make sure the wavefunction can be
normalized.
Now show that in the
r
→
0 limit (implying
ρ
→
0) the radial equation becomes
d
2
u
dρ
2
≈
±
(
±
+ 1)
ρ
2
u .
(7)
This equation has a simple solution:
consider
u
(
ρ
)
≈
Cρ
α
for some number
α
.
What two
values of
α
satisfy the equation? Which one must we throw out to prevent the wavefunction
from blowing up?
c) We will now extract
both
asymptotic behaviors from
u
(
ρ
) to deﬁne a new function to work
with,
v
(
ρ
), as:
u
(
ρ
)
≡
ρ
±
+1
e

ρ
v
(
ρ
)
.
(8)
Verify that the radial equation becomes, in terms of
v
(
ρ
),
ρ
d
2
v
dρ
2
+ 2(
±
+ 1

ρ
)
dv
dρ
+ [
ρ
0

2(
±
+ 1)]
v
= 0
.
(9)
d) To solve this diFerential equation, we will postulate a series solution for
v
:
v
(
ρ
) =
∞
±
j
=0
c
j
ρ
j
,
(10)
where the
c
j
are constants. Show that the result of part c) implies the
recursion relation
for
the constants:
c
j
+1
=
²
2(
j
+
±
+ 1)

ρ
0
(
j
+ 1)(
j
+ 2
±
+ 2)
³
c
j
.
(11)
e) Let us explore what happens if the series goes on forever. Write down the large
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 Fall '08
 STEVEPOLLOCK
 mechanics

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