Homework 11 Solution
Prob. 11.1
Consider two canonical systems that are initially in respective equilibrium at
different temperature
T
1
and
T
2
and
µ
1
and
µ
2
.
They are brought to contact at
t
= 0.
(a)
How does the entropy of the total system change at
t
= 0? (This is actually one of the
definition of time)
(5 pts)
With the interaction, the entropy will increase.
When the system migrates towards
T
1
=
T
2
and
µ
1
=
µ
2
, the total entropy will reach a maximum (if the two systems are identical, then
this is towards the center of the Pascal tree).
Time can actually be defined as the direction of
increasing entropy when the system has a constant energy (or no energy input).
From here,
you can derive the second law of Thermodynamics.
(b)
If
T
1
>
T
2
and
µ
1
=
µ
2
for
t
< 0, describe the exchange between the two systems for
t
≥
0.
When will the exchange stop?
(5 pts)
Since
µ
1
=
µ
2
for all
t
, the exchange is to make
T
1
goes towards
T
2
for
t
> 0.
The particle on
system 1 with the higher
T
side has a higher kinetic energy, and hence higher probability of
going towards system 2.
To maintain
µ
1
=
µ
2
, some particle with lower kinetic energy will
flow back to system 1.
This phenomenon is often called thermal diffusion.
(c)
If
T
1
=
T
2
and
µ
1
>
µ
2
for
t
< 0, describe the exchange between the two systems for
t
≥
0.
When will the exchange stop?
(5 pts)
Since
T
1
=
T
2
for all
t
, the exchange is to make
µ
1
goes towards
µ
2
for
t
> 0.
The particle on
system 1 has a higher concentration
(higher µ
with fixed
V
and
T
), and hence higher
probability of going towards system 2.
This phenomenon is often called particle diffusion.
Prob. 11.2
For a 3D particle in the 3D simple harmonic oscillator described by:
(
29
(
29
(
29
(
29
(
29
(
29
z
y
x
r
m
z
y
x
m
z
y
x
z
y
x
m
z
y
x
m
z
y
x
H
o
o
,
,
2
1
,
,
2
,
,
2
1
,
,
2
,
,
ˆ
2
2
2
2
2
2
2
2
2
2
ψ
ϖ
+
∇

=
+
+
+
∇

=
The energy eigenvalues are:
+
=
+
+
+
=
2
3
2
3
0
0
n
n
n
n
E
z
y
x
n
, where
0
2
3
is
referred to as the “zeropoint energy” (okay, I know this name was twisted in the movie of “The
Incredibles”)
(a)
For
n
= 2, what are the possible eigenstates? How many states are there?
(5 pts)
(
n
x
, n
y
, n
z
) = (2, 0, 0) have three possible eigenstates.
There are another three in (1, 1, 0).