PHY4604 Fall 2007
Problem Set 7
Department of Physics
Page 1 of 3
PHY 4604 Problem Set #7
Due Friday November 30, 2007 (in class)
(Total Points = 115, Late homework = 50%)
Reading:
Griffiths Chapter 5 (Sections 5.1 and 5.2).
Problem 1 (30 points):
The state of a two particle system is described by the wave function
)
,
,
(
2
1
t
r
r
r
r
Ψ
.
The time evolution is given by Schrödinger’s equation
t
i
H
∂
Ψ
∂
=
Ψ
h
with
)
,
,
(
2
2
2
1
2
2
2
2
2
1
1
2
t
r
r
V
m
m
H
r
r
h
h
+
∇
−
∇
−
=
where
2
2
2
2
2
2
2
k
k
k
k
z
y
x
∂
∂
+
∂
∂
+
∂
∂
=
∇
for k = 1,2.
For timeindependent potentials, we obtain a complete
set of solutions of the form
h
r
r
r
r
/
2
1
2
1
)
,
(
)
,
,
(
iEt
e
r
r
t
r
r
−
=
Ψ
ψ
, where
)
,
(
2
1
r
r
r
r
ψ
satisfies the time
independent Schrödinger equation
ψ
ψ
ψ
ψ
E
r
r
V
m
m
=
+
∇
−
∇
−
)
,
(
2
2
2
1
2
2
2
2
2
1
1
2
r
r
h
h
.
Typically the interaction potential depends only on the vector
2
1
r
r
r
r
r
r
−
=
(
i.e.
the separation
between the two particles).
Suppose
)
(
)
,
(
2
1
r
V
r
r
V
r
r
r
=
and we change variables from
1
r
r
and
2
r
r
to
2
1
r
r
r
r
r
r
−
=
and
)
/(
)
(
2
1
2
2
1
1
m
m
r
m
r
m
R
+
+
=
r
r
r
(
i.e.
the centerofmass vector).
(a) (5 points)
Show that
r
m
R
r
r
r
r
)
/
(
1
1
μ
+
=
and
r
m
R
r
r
r
r
)
/
(
2
2
μ
−
=
, where
)
/(
2
1
2
1
m
m
m
m
+
=
μ
is
the “reduced mass”.
(b) (5 points)
Show that
r
R
m
∇
+
∇
=
∇
r
r
r
)
/
(
2
1
μ
and
r
R
m
∇
−
∇
=
∇
r
r
r
)
/
(
1
2
μ
, where
)
/(
2
1
2
1
m
m
m
m
+
=
μ
is the “reduced mass”.
(c) (5 points)
Show that the timeindependent Schrödinger equation becomes
ψ
ψ
ψ
μ
ψ
E
r
V
M
r
R
=
+
∇
−
∇
−
)
(
2
2
2
2
2
2
r
h
h
, where
2
1
m
m
M
+
=
and
)
/(
2
1
2
1
m
m
m
m
+
=
μ
is the
“reduced mass”.
(d) (5 points)
Separate the variables by letting
)
(
)
(
)
,
(
r
R
r
R
r
R
r
r
r
r
ψ
ψ
ψ
=
and show that
)
(
R
R
r
ψ
satisfies the oneparticle Schrödinger equation with mass M = m
1
+m
2
, potential zero, and
energy E
R
(
i.e.
the centerofmass moves like a free particle).
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 Spring '07
 FIELDS
 mechanics, Work, ground state, Department of Physics

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