CHEM 120A Problem Set 1 Solutions
David Hoffman
,
Tara Yacovitch
,
Doran Bennett
1/27/2010
These solutions are intended to be a guide, or roadmap, to help
you
answer the homework
questions, they are
not
intended to be stepbystep instructions. For this reason, the majority
of the mathematical manipulations have been intentionally left out so as to force you to work
through the questions yourselves. If you find that this guide is not enough then please email
us or come to our office hours or the discussion sections where we will be more than happy
to help you.
1. The solution can be found in the McQuarrie Solutions manual.
2. The solution can be found in the McQuarrie Solutions manual.
3.
(a) We have the following equation for the total energy
E
=
N
X
i
=1
p
2
i
/
2
m
i
+
V
(
x
1
, x
2
, . . . , x
N
)
(1a)
We can differentiate both sides with respect to time
d
E
d
t
= 0 =
N
X
i
=1
v
i
˙
p
i
+
∂V
∂x
i
(1b)
In order for the right hand side to be equal to zero the term in parentheses must
equal zero. This because the
v
i
’s can be chosen arbitrarily for an arbitrary system,
yet the equality must hold. Thus,
˙
p
i
=

∂V
∂x
i
,
∀
i
(1c)
(b) Note that
p
=
ˆ
ı
p
x
+
ˆ
p
y
+
ˆ
k
p
z
.
Thus
˙p
=
∇
V
follows directly.
1
4.
(a) We start with the expression for total energy in normal coordinates and then
transform to center of mass (
COM
) coordinates.
1
We will use bolded items for vector quantities and nonbold items for either the magnitude of the vector
or a scalar. Ex:
p
is the momentum
vector
whereas
p
=

p

is the magnitude of the momentum, which is a
scalar
. Also, the dot above a variable indicates its time derivative, i.e.
˙p
=
d
d
t
p
1
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The total energy is,
E
tot
=
1
2
m
1
v
2
1
+
1
2
m
2
v
2
2
+
V
(
r
)
(2a)
Now we can begin to perform some algebra.
2
After multiplying by
M/M
and
completing the square you’ll find that indeed,
E
tot
=
M
2
v
2
cm
+
μ
2
v
2
+
V
(
r
)
(2b)
QED
3
(b) Now we need to prove that the
COM
momentum is conserved.
There are two
ways of solving this problem:
i. First we could realize that the Coulomb potential is proportional to 1
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 Spring '10
 DAVIDCHANDLER
 Energy, Force, Kinetic Energy, Mass, Potential Energy, Total Energy

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