the probabilistic nature of the wavefunction are features of the quantum mechanics and have no
classical analog. In subsequent chapters we discuss the implications of quantization on nanomaterial
properties and the methods used to characterize them.
End of chapter questions
1. Consider an electron free to move along a line
of length 200 nm. What is the ground state
energy of the electron? What is the energy of
the electron in the n = 3 state? What is the
effect on the spacing between neighboring
energy levels if the length of the line increased
from 200 nm to 300 nm?
2. Calculate the energy of the n = 2 and n = 6 level
of an electron constrained to move along a line
of length 100 nm. Determine the energy corresponding to an electronic transition between
these two levels. What wavelength of light is
required to affect this transition? Draw an
energy level diagram to illustrate this transition.
3. Consider the wavefunction ψ m = Ae
imφ
, where A
is a constant and i =
ffiffiffiffiffi ffi
−1
p
. If this wavefunction is
an eigenfunction of the operator − i
h
2π
d
dφ
,
determine the corresponding eigenvalue.
4. Consider the wavefunction ψ(x) =
ffiffiffi
2
L
r
sin
πx
L
.
Explain why this wavefunction is well-behaved.
Which energy state does this wavefunction
correspond to? Either by plotting the function
or by differentiation, find the value of x for
which ψ(x) is a maximum.
5. Consider a ring of diameter 4 nm containing four
free electrons. Draw a diagram showing the electrons occupying the various energy levels when
the system is in its ground state. What wavelength of light is required to excite the electron
from the ground state to the first excited state?
6. A particle on a line is constrained between 0
and L. With the aid of a diagram (a plot of ψ
2
(x)
from x = 0 to x = L), estimate the probability of
finding the particle from L/4 to 3L/4 for the n =
1, 2, 3, 4, and 5 states. Do you see a pattern?
7. Calculate the probability of finding the particle
in a 1D box of length L between the interval
1
10
L
to
1
4
L for the n = 3 state. Sketch the probability
(ψ
2
3 (x) from x = 0 to x = L) for the n = 3 state and
shade in the region from
1
10
L to
1
4
L. Show that
the percentage of the area shaded is in
agreement with the calculated probability.
8. Is there a relationship between the peak
absorption wavelength (λ max ) and the length of
a nanowire? If so, plot the relationship on a
graph. Your answer must be quantitative and
use the particle on a line model to calculate the
relevant information.
9. When sodium dissolves in liquid ammonia, it
reacts with the solvent to form Na
+ and solvated electrons, which are not associated with
any particular atom or molecule but are stabilized by the solvent molecules surrounding
them, just like a typical ion. Solvated electrons
can be used as a powerful reducing agent or to
give a material unusual electronic or optical
properties. The solvated electron can be
treated as a particle in a three-dimensional box.
Assume that the box is cubic with an edge
length of 1.35 nm and suppose that excitation
END OF CHAPTER QUESTIONS 131
classical analog. In subsequent chapters we discuss the implications of quantization on nanomaterial
properties and the methods used to characterize them.
End of chapter questions
1. Consider an electron free to move along a line
of length 200 nm. What is the ground state
energy of the electron? What is the energy of
the electron in the n = 3 state? What is the
effect on the spacing between neighboring
energy levels if the length of the line increased
from 200 nm to 300 nm?
2. Calculate the energy of the n = 2 and n = 6 level
of an electron constrained to move along a line
of length 100 nm. Determine the energy corresponding to an electronic transition between
these two levels. What wavelength of light is
required to affect this transition? Draw an
energy level diagram to illustrate this transition.
3. Consider the wavefunction ψ m = Ae
imφ
, where A
is a constant and i =
ffiffiffiffiffi ffi
−1
p
. If this wavefunction is
an eigenfunction of the operator − i
h
2π
d
dφ
,
determine the corresponding eigenvalue.
4. Consider the wavefunction ψ(x) =
ffiffiffi
2
L
r
sin
πx
L
.
Explain why this wavefunction is well-behaved.
Which energy state does this wavefunction
correspond to? Either by plotting the function
or by differentiation, find the value of x for
which ψ(x) is a maximum.
5. Consider a ring of diameter 4 nm containing four
free electrons. Draw a diagram showing the electrons occupying the various energy levels when
the system is in its ground state. What wavelength of light is required to excite the electron
from the ground state to the first excited state?
6. A particle on a line is constrained between 0
and L. With the aid of a diagram (a plot of ψ
2
(x)
from x = 0 to x = L), estimate the probability of
finding the particle from L/4 to 3L/4 for the n =
1, 2, 3, 4, and 5 states. Do you see a pattern?
7. Calculate the probability of finding the particle
in a 1D box of length L between the interval
1
10
L
to
1
4
L for the n = 3 state. Sketch the probability
(ψ
2
3 (x) from x = 0 to x = L) for the n = 3 state and
shade in the region from
1
10
L to
1
4
L. Show that
the percentage of the area shaded is in
agreement with the calculated probability.
8. Is there a relationship between the peak
absorption wavelength (λ max ) and the length of
a nanowire? If so, plot the relationship on a
graph. Your answer must be quantitative and
use the particle on a line model to calculate the
relevant information.
9. When sodium dissolves in liquid ammonia, it
reacts with the solvent to form Na
+ and solvated electrons, which are not associated with
any particular atom or molecule but are stabilized by the solvent molecules surrounding
them, just like a typical ion. Solvated electrons
can be used as a powerful reducing agent or to
give a material unusual electronic or optical
properties. The solvated electron can be
treated as a particle in a three-dimensional box.
Assume that the box is cubic with an edge
length of 1.35 nm and suppose that excitation
END OF CHAPTER QUESTIONS 131