Chapter 6 Concept Tests
6-1 In a gas of hydrogen atoms at room temperature, what is the
ratio of atoms in the 1st excited energy state (n=2) to atoms in the
ground state(n=1). (Actually H forms H2 molecules, but let’s
pretend.)
P(n = 2)
=?
P(n = 1)
A) less than 1, but not extremely small, between 0.01 and 0.5
B) much less than 1, but not infinitesimal, between 0.001 and 10-6
C) infinitesimal, less than 10-100
6-2
A quantum system has the following energy level diagram. Notice
that the temperature is indicated
E
g(E)
6
3
kT
5
2
4
1
1
0
5
6
7
8
1
2
3
4
i=0
Which is true?
A) P(i=3) > P(i=2)
B) P(i=3) < P(i=2)
C) P(i=3) = P(i=2)
A) P(i=5) > P(i=2)
B) P(i=5) < P(i=2)
C) P(i=5) = P(i=2)
A) P(E=1) > P(E=0)
B) P(E=1) < P(E=0)
C) P(E=1) = P(E=0)
6-3 A quantum system has the following energy level diagram.
The ground state energy is set at zero. The first excited energy
level has energy . Notice that the temperature is indicated.
g(E)
E
6

5

4

1

kT
5
6
7
8
1
2
3
4
9
i=0
Roughly, what is the value of Z, the partition function?
A) 2.8  B) 3
C) 10
D) 20
E) impossible to tell without more information
Roughly, what is the probability that the system will be found in
the ground state?
A) 1
B) 1/3
C) 1/10 D) 1/100
[Exact Answers: 7.6, 0.13]
As the temperature is increased, the partition function will
A) increase
B) decrease
C) remain constant
6-4 A particle (moving in 1D) has position and momentum
indicated. Where is the particle in phase space?
x
0
p
(D)
(A)
(B)
x
(E)
(C)
6-5 A 1D simple harmonic oscillator is oscillating back and forth.
As the system evolves in time, its phase space points traces out
which trajectory?
x
0
p
(A)
(B)
x
(C)
6-6
+¥
- x2
The “gaussian integral”
òe
- ¥
dx
is
A) about 1
B) about 10
C) about 0.1
D) about 100
Answer:
p ; 1.77
6-7. The distribution of particle kinetic energies in an ideal gas is
given below. The rms average kinetic energy is most nearly
A
B
0.5
C
0.4
D ( x)
0.2
0
0
0
0
1
2
3
4
x
E(units
of kT)
4
6-8
The number of states in the volume of the shell shown of thickness
dn and radius n is
nz
dn
n
ny
nx
1
2
A) 4p n dn
8
C) something else
Exam I was…
A) too hard
B) too easy
C) about right
B)
14 3
p n dn
83
6-9 What is the probability that a molecule in an ideal gas has an
energy between 0 and 3kT
A) about 0.1
B) about 0.5
C) about 0.99 D)None of these
0.5
0.4
D ( x)
D(E)
0
0.2
0
0
1
2
0
x of kT)
E(units
3
4
4
6-10 How would you compute the average speed (mean speed) of
a molecule in an ideal gas?
¥
v D(v)dv
A) ò
0
B)
3kT
m
C) Answers A and B give the same answer
6-11 This is the energy-level spectrum of a
A) 1D simple harmonic oscillator
B) a particle in a 1D box
C) a hydrogen atom
D) a two-state paramagnet
E) a quantum rotor
E
All levels
Non-degenerate
6-12 True(A) or False(B): the entropy of a system in thermal
contact with a heat bath cannot decrease.
6-13 True(A) or False(B): the entropy of an isolated system
cannot decrease.
6-14 A ball rolls back and forth in a valley and eventually comes
to rest at the bottom of the valley.
As the ball rolled to a stop, the Helmholtz free energy F = U – TS
of the ball
A) increased
B) decreased
C) remained constant
D) impossible to tell without more information
6-15 A quantum system has non-degenerate energy levels
described by E(n) = C n2 where C is a constant and n = 0, 1, 2, ....
This is the energy-level spectrum of a
A) 1D simple harmonic oscillator
C) a hydrogen atom
E) a quantum rotor
B) a particle in a 1D box
D) a two-state paramagnet
6-16 Consider the function Ae-E/kT, A some constant. What’s the
¥
area under the curve
ò Ae
- E / kT
dE ? (Pick closest answer.)
0
A
kT
2kT
A) AkT B) 3AkT C) (1/3) AkT
D) All these are off by a factor of 2 or more
E
6-17
¶F
¶F
dF
=
dT
+
dV and dF = - S dT + - p dV
Recall
¶T V
¶V T
True (A) or False (B) :
¶S
¶p
=
¶V T ¶T V
6-18 A quantum system has degenerate energy levels described by
C
E(n) = - 2 where C is a constant and n =
n
+1, 2, .... This is the energy-level spectrum of a
A) 1D simple harmonic oscillator
C) a hydrogen atom
E) a quantum rotor
B) a particle in a 1D box
D) a two-state paramagnet
6-19 True(A) or False(B):
for x, y = 0, 1, 2..
å
x,y
e
x+ y
ö
æ
öæ
x ֍
y÷
ç
= çå e ÷ç
e ÷
å
÷
÷
ç
çè x
÷
øçè y
ø
6-20 As the temperature decreases, wavepackets describing
particles in a gas tend to get
A) larger B) smaller
C) stay constant in size
6-21. A 1D SHO quantum system is in the “high-temperature”
regime kT>>. What is the thermal average energy of the system?
E

kT


0
A) kT
B) (3/2)kT
C) (1/2)kT
6-22. A 1D SHO quantum system is in the “low-temperature”
regime kT<<. The thermal average energy of the system is
closest to :
A) 0
B) kT
C) 
E



0
kT