4604_ProblemSet7_fa07 - PHY4604 Fall 2007 Problem Set 7 PHY...

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 time-independent 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 center-of-mass 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 time-independent 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 one-particle Schrödinger equation with mass M = m
1
+m
2
, potential zero, and
energy E
R
( i.e. the center-of-mass moves like a free particle).