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Quantum Mechanics
Multiple Choice
1.
A 15-kg mass, attached to a massless spring whose force constant is 2500 N/m,
has an amplitude of 4 cm. Assuming the energy is quantized, find the quantum
number of the system, n, if En = nhf.
a.
b.
c.
d.
e.
2.
Find the kinetic energy (in terms of Planck’s constant) of a baseball (m = 1 kg)
confined to a one-dimensional box that is 25 cm wide if the baseball can be
treated as a wave in the ground state.
a.
b.
c.
d.
e.
3.
3 h2
2 h2
h2
4h2
0.5 h2
Calculate the ground state energy (in eV) for an electron in a box (an infinite
well) having a width of 0.050 nm.
a.
b.
c.
d.
e.
4.
1.5 × 1033
3.0 × 1033
4.5 × 1033
5.4 × 1033
1.0 × 1033
10
75
24
150
54
A particle is in the ground state of a one-dimensional box of length 1 m. What is
the minimum value of its momentum (in kg · m/s)?
a.
b.
c.
d.
e.
9.9 × 10–34
6.6 × 10–34
3.3 × 10–34
13.2 × 10–34
cannot be solved unless mass of particle is known.
333
5.
A particle is in the first excited state of a one-dimensional box of length 1 m.
What is the minimum value of its momentum (in kg · m/s)?
a.
b.
c.
d.
e.
6.
A particle is in the second excited state of a one-dimensional box of length 1 m.
What is its momentum (in kg · m/s)?
a.
b.
c.
d.
e.
7.
1
4
2
8
16
What is the quantum number n of a particle of mass m confined to a onedimensional box of length L when its energy is 2 h2/mL2?
a.
b.
c.
d.
e.
9.
6.6 × 10–34
3.3 × 10–34
9.9 × 10–34
13.2 × 10–34
cannot be solved unless mass of particle is known
What is the quantum number n of a particle of mass m confined to a onedimensional box of length L when its momentum is 4h/L?
a.
b.
c.
d.
e.
8.
3.3 × 10–34
6.6 × 10–34
9.9 × 10–34
13.2 × 10–34
22.3 × 10–34
2
8
4
1
16
The wave function for a particle confined to a one-dimensional box located
between x = 0 and x = L is given by Ψ(x) = A sin (nπ x/L) + B cos (nπ x/L) . The
constants A and B are determined to be
a.
2/ L , 0
b.
1/ L ,
c.
0,
d.
e.
2/ L ,
2/L, 0
1/ L
2/ L
2/ L
Quantum Mechanics
10.
Classically, the concept of “tunneling” is impossible. Why?
a.
b.
c.
d.
e.
11.
a positive constant.
exponentially increasing.
oscillatory.
exponentially decreasing.
none of the above.
The ground state energy of a harmonic oscillator is
a.
b.
c.
d.
e.
14.
P < 0.
P = 0.
P = 1.
P > 0.
P=∞.
A particle has a total energy that is less than that of a potential barrier. When the
particle penetrates the barrier, its wave function is
a.
b.
c.
d.
e.
13.
The kinetic energy of the particle would be negative.
The velocity of the particle would be negative.
The total energy of a particle is equal to the kinetic and potential energies.
The kinetic energy must be equal to the potential energy.
The total energy for the particle would be negative.
When a particle approaching a potential step has a total energy that is greater
than the potential step, what is the probability that the particle will be reflected?
a.
b.
c.
d.
e.
12.
335
E = hω
E = hω/2
E = (2/3)hω
E=0
E = hω/4
Particles in degenerate energy levels all have the same
a.
b.
c.
d.
e.
momentum.
quantum numbers.
energy.
all of the above.
velocity.
15.
 nπx 
The wave function for a particle in a one-dimensional box is ψ = A sin 
.
 L 
Which statement is correct?
a.
This wavefunction gives the probability of finding the particle at x.
2
b.
Ψ(x) gives the probability of finding the particle at x.
c.
Ψ(x) ∆x gives the probability of finding the particle between x and x + ∆x.
2
L
d.
∫ Ψ(x)dxgives the probability of finding the particle at a particular value
0
of x.
L
e.
∫
2
Ψ(x) dxgives the probability of finding the particle between x and x + ∆x.
0
16.
The fact that we can only calculate probabilities for values of physical quantities
in quantum measurements means that
a.
b.
c.
d.
e.
17.
When the potential energy of a system is independent of time, the wave function
of the system
a.
b.
c.
d.
e.
18.
radiation and matter are not described by mathematical relations between
measurements.
the probabilities cannot be calculated from mathematical relationships.
the results of physical measurements bear no relationship to theory.
the average values of a large number of measurements correspond to the
calculated probabilities.
the average of the values calculated in a large number of different theories
corresponds to the results of a measurement.
is a constant.
is directly proportional to the time.
cannot be normalized.
depends only on the center of mass, R , of the system.
depends on the vector positions, ri , of each particle in the system.
A physically reasonable wave function, ψ (x), for a one-dimensional system must
a.
b.
c.
d.
e.
be defined at all points in space.
be continuous at all points in space.
be single-valued.
obey all the constraints listed above.
obey only (b) and (c) above.
Quantum Mechanics
19.
337
The average position, or expectation value, of a particle whose wave function
ψ (x) depends only on the value of x, is given by < x > =
a.
∫
∫
+∞
−∞
b.
+∞
−∞
c.
xψ (x)dx .
xψ (x)dx.
+∞
x 2
ψ (x)dx.
2
+ ∞ x2
ψ (x)dx.
−∞ 2
∫
−∞
d.
e.
∫
∫
+∞
−∞
20.
ψ *(x)xψ (x)dx.
The expectations value of a function f(x) of x when the wave function depends
only on x is given by < f(x)> =
a.
∫
∫
∫
∫
+∞
∫
+∞
−∞
b.
+∞
−∞
c.
+∞
−∞
d.
−∞
21.
f(x)ψ (x)dx.
f(x)ψ *(x)dx .
+∞
−∞
e.
f(x)ψ (x)dx .
ψ *(x)f(x)ψ (x)dx .
ψ *(x)
f2(x)
ψ (x)dx.
2
If the interaction of a particle with its environment restricts the particle to a finite
region of space, the result is the quantization of _____ of the particle.
a.
b.
c.
d.
e.
the momentum
the energy
the velocity
all of the above properties
only properties (a) and (b)
22.
When U(x) is infinitely large elsewhere, the wave function of a particle
restricted to the region 0 < x < L where U(x) = 0 , may have the form ψ (x ) =
 nπx 
A sin
.
 L 
 nπx 
B cos
.
 L 
Aenπx/ L .
 nπx 
 nπx 
A sin
 + B cos
.
 L 
 L 
 nπx 
 nπx 
A sin
 − B cos
.
 L 
 L 
a.
b.
c.
d.
e.
23.
A particle in a finite potential well has energy E, as shown below.
I
II
U
III
E
L
0
The wave function in region I where x < 0 has the form ψ I =
24.
a.
b.
c.
Ae−Cx .
AeCx .
F sinkx.
d.
G coskx.
e.
F sinkx+ G coskx.
A particle in a finite potential well has energy E, as shown below.
I
II
U
III
E
0
L
The wave function in region II where x > 0 has the form ψ II =
a.
b.
c.
Ae−Cx .
AeCx .
F sinkx.
d.
G coskx.
e.
F sinkx+ G coskx.
Quantum Mechanics
25.
A particle in a finite potential well has energy E, as shown below.
I
II
U
III
E
L
0
The wave function in region III where x > L has the form ψ III=
26.
a.
b.
c.
Ae−Cx .
AeCx .
F sinkx.
d.
G coskx.
e.
F sinkx+ G coskx.
Quantum tunneling occurs in
a.
b.
c.
d.
e.
27.
nuclear fusion.
radioactive decay by emission of alpha particles.
the scanning tunneling microscope.
all of the above.
only (b) and (c) above.
The graph below represents a wave function ψ (x ) for a particle confined to
−2.00 m ≤ x ≤ +2.00 m . The value of the normalization constant A may be
ψ (x )
A
–2
a.
b.
c.
d.
e.
1
.
4
1
− .
2
1
+ .
2
1
+ .
4
−
either −
1
1
or + .
2
2
+2
x
339
28.
The graph below represents a wave function ψ (x ) for a particle confined to
−2.00 m ≤ x ≤ +2.00 m . The value of the normalization constant A may be
ψ (x )
–2
+2
x
–A
a.
b.
c.
d.
e.
29.
1
.
4
1
− .
2
1
+ .
2
1
+ .
4
−
either −
1
1
or + .
2
2
The graph below represents a wave function ψ (x ) for a particle confined to
−4.00 m ≤ x ≤ +4.00 m . The magnitude of the normalization constant A is
ψ (x )
+A
+4
–4
–A
a.
b.
1
.
4
1
8
c.
d.
e.
.
1
.
2
8.
4.
x
Quantum Mechanics
30.
Frank says that quantum mechanics does not apply to baseballs because they do
not jump from quantum state to quantum state when being thrown. Francine
agrees with him. She says that there is no uncertainty in a baseball’s position or
momentum. Are they correct, or not, and why?
a.
b.
c.
d.
e.
31.
341
They are correct because the first excited state of a baseball is at a higher
energy that any baseball ever receives. Therefore we cannot determine
whether or not there is uncertainty in its position or momentum.
They are correct because the first excited state of a baseball is at a higher
energy that any baseball ever receives. Therefore its position and
momentum are completely uncertain until it is caught.
They are wrong because the baseball goes through so many quantum states
in being thrown that we cannot observe the transitions. The uncertainties in
its position and momentum are too small to observe.
They are wrong because the baseball goes through so many quantum states
in being thrown that we cannot observe the transitions. Because of the
number of transitions its position and momentum are completely uncertain
until it is caught.
Quantum mechanics states that they are correct as long as they do not make
any observations, but wrong as soon as they begin to make observations.
The graph below shows the value of the probability density | ψ ( x )| 2 in the region
−3.00 m ≤ x ≤ +3.00 m . The value of the constant A is
| ψ (x)| 2
A2
x
–3
a.
−
b.
−
c.
+
1
.
3
1
3
1
3
+3
.
.
1
.
3
d.
+
e.
either −
1
3
or +
1
3
.
32.
The wave function ψ (x ) of a particle confined to 0 ≤ x ≤ L is given by ψ (x ) = A x .
ψ (x ) = 0 for x < 0 and x > L . When the wave function is normalized, the
probability density at coordinate x has the value
a.
b.
c.
d.
e.
2
L2
2
L2
2
L3
3
L3
3
L3
x.
x2 .
x2 .
x2 .
x3 .
Quantum Mechanics
Chapter 41
Quantum Mechanics
1.
a
17.
e
2.
b
18.
d
3.
d
19.
e
4.
c
20.
d
5.
b
21.
d
6.
c
22.
a
7.
d
23.
b
8.
c
24.
e
9.
a
25.
a
10.
a
26.
d
11.
d
27.
c
12.
d
28.
b
13.
b
29.
b
14.
c
30.
c
15.
c
31.
e
16.
d
32.
d
343