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
!
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
π
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 (
??
) 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
=
"
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
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Verify 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 differential equation, we will postulate a series solution for
v
:
v
(
ρ
) =
∞
X
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(
j
+
‘
+ 1)

ρ
0
(
j
+ 1)(
j
+ 2
‘
+ 2)
#
c
j
.
(11)
e) Let us explore what happens if the series goes on forever. Write down the large
j
limit of
the recursion formula, and demonstrate that if this approximate form were exact, it would
imply
c
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 STEVEPOLLOCK
 mechanics, Hilbert space, radial equation, u. d2

Click to edit the document details