ALL 11 CHAPTERS COVERED
SOLUTIONS MANUAL
,Applied Quantum Mechanics
Chapter 1 problems and solutions
Applied quantum mechanics 1
,Problem 1.1
A metal ball is buried in an ice cube that is in a bucket of water.
(a) If the ice cube with the metal ball is initially under water, what happens to the water level
when the ice melts?
(b) If the ice cube with the metal ball is initially floating in the water, what happens to the water
level when the ice melts?
(c) Explain how the Earth’s average sea level could have increased by at least 100 m compared
to about 20 000 years ago.
(d) Estimate the thickness and weight per unit area of the ice that melted in (c). You may wish to
use the fact that the density of ice is 920 kg m-3, today the land surface area of the Earth is about 148
300 000 km2 and water area is about 361 800 000 km2.
Problem 1.2
Sketch and find the volume of the largest and smallest convex plug manufactured from a sphere of
radius r = 1 cm to fit exactly into a circular hole of radius r = 1 cm, an isosceles triangle with base 2
cm and a height h = 1 cm, and a half circle radius r = 1 cm and base 2 cm.
Problem 1.3
An initially stationary particle mass m1 is on a frictionless table surface and another particle mass
m2 is positioned vertically below the edge of the table. The distance from the particle mass m1 to
the edge of the table is l. The two particles are connected by a taught, light, inextensible string of
length L > l.
(a) How much time elapses before the particle mass m1 is launched off the edge of the table?
(b) What is the subsequent motion of the particles?
(c) How is your answer for (a) and (b) modified if the string has spring constant ?
Problem 1.4
The velocity of waves in shallow water may be approximated as = gh where g is the accelera-
tion due to gravity and h is the depth of the water. Sketch the lowest frequency standing water wave
in a 5 m long garden pond that is 0.9 m deep and estimate its frequency.
Problem 1.5
(a) What is the dispersion relation of a wave whose group velocity is half the phase velocity?
(b) What is the dispersion relation of a wave whose group velocity is twice the phase velocity?
(c) What is the dispersion relation when the group velocity is four times the phase velocity?
(d) What is the dispersion relation when the group velocity is the negative of the phase velocity?
Problem 1.6
A stationary ground-based radar uses a continuous electromagnetic wave at 10 GHz frequency to
measure the speed of a passing airplane moving at a constant altitude and in a straight line at 1000
km hr-1. What is the maximum beat frequency between the out going and reflected radar beams?
Sketch how the beat frequency varies as a function of time. What happens to the beat frequency if
the airplane moves in an arc?
2
, Problem 1.7
(a) If classical electromagnetism were described in terms of a single complex field G show that
Maxwell’s equations in free space and in the absence of free charges may written as the complex
equations
G = 0
and
G 1
i = -------------- G
t 0 0
where G = ------- -------- + i ---------
1 D B
2 0 0
(b) Show that the energy flux density in the electromagnetic field given by the Poynting vector
is
–i
S = E H = -------------- G G
*
0 0
(c) If the field G is purely real, what is the value of S?
2
(d) Show that the electromagnetic energy density is U = G .
Problem 1.8
How would Maxwell’s equations be modified if magnetic charge g (magnetic monopoles) were dis-
covered? Derive an expression for conservation of magnetic current and write down a generalized
Lorentz force law that includes magnetic charge. Write Maxwell’s equations with magnetic charge
in terms of a field G = ------- ------ + i ------- .
1 D B
2
Problem 1.9
(a) The capacitance of a small metal sphere in air is C 0 = 1.1 10
–18
F . A thin dielectric film with
relative permittivity r1 = 10 uniformly coats the sphere and the capacitance increases to
2.2 10 F . What is the thickness of the dielectric film and what is the single electron charging
– 18
energy of the dielectric coated metal sphere?
(b) The dielectric coated metal sphere of part (a) is now coated with metal. What is the new
value of the single-electron charging energy for the central metal sphere?
(c) Compare the result in (b) to the charging energy of a metal sphere radius
0.5 nm r 0 10 nm embedded in a dielectric of relative permittivity r = 10 and surrounded by
metal shell of internal radius r 1 = 2r 0 . Plot single-electron charging energy E as a function of
r0 .
Problem 1.10
(a) A diatomic molecule has atoms with mass m1 and m2. An isotopic form of the molecule has
atoms with mass m'1 and m'2. Find the ratio of vibration oscillation frequency / ' of the two mol-
ecules.
12 16
(b) What is the ratio of vibrational frequencies for carbon monoxide isotope 12 ( C O ) and
13 16
carbon monoxide isotope 13 ( C O )?
Problem 1.11
(a) Find the frequency of oscillation of the particle of mass m illustrated in the figure below. The
particle is only free to move along a line and is attached to a light spring whose other end is fixed at
Applied quantum mechanics 3
SOLUTIONS MANUAL
,Applied Quantum Mechanics
Chapter 1 problems and solutions
Applied quantum mechanics 1
,Problem 1.1
A metal ball is buried in an ice cube that is in a bucket of water.
(a) If the ice cube with the metal ball is initially under water, what happens to the water level
when the ice melts?
(b) If the ice cube with the metal ball is initially floating in the water, what happens to the water
level when the ice melts?
(c) Explain how the Earth’s average sea level could have increased by at least 100 m compared
to about 20 000 years ago.
(d) Estimate the thickness and weight per unit area of the ice that melted in (c). You may wish to
use the fact that the density of ice is 920 kg m-3, today the land surface area of the Earth is about 148
300 000 km2 and water area is about 361 800 000 km2.
Problem 1.2
Sketch and find the volume of the largest and smallest convex plug manufactured from a sphere of
radius r = 1 cm to fit exactly into a circular hole of radius r = 1 cm, an isosceles triangle with base 2
cm and a height h = 1 cm, and a half circle radius r = 1 cm and base 2 cm.
Problem 1.3
An initially stationary particle mass m1 is on a frictionless table surface and another particle mass
m2 is positioned vertically below the edge of the table. The distance from the particle mass m1 to
the edge of the table is l. The two particles are connected by a taught, light, inextensible string of
length L > l.
(a) How much time elapses before the particle mass m1 is launched off the edge of the table?
(b) What is the subsequent motion of the particles?
(c) How is your answer for (a) and (b) modified if the string has spring constant ?
Problem 1.4
The velocity of waves in shallow water may be approximated as = gh where g is the accelera-
tion due to gravity and h is the depth of the water. Sketch the lowest frequency standing water wave
in a 5 m long garden pond that is 0.9 m deep and estimate its frequency.
Problem 1.5
(a) What is the dispersion relation of a wave whose group velocity is half the phase velocity?
(b) What is the dispersion relation of a wave whose group velocity is twice the phase velocity?
(c) What is the dispersion relation when the group velocity is four times the phase velocity?
(d) What is the dispersion relation when the group velocity is the negative of the phase velocity?
Problem 1.6
A stationary ground-based radar uses a continuous electromagnetic wave at 10 GHz frequency to
measure the speed of a passing airplane moving at a constant altitude and in a straight line at 1000
km hr-1. What is the maximum beat frequency between the out going and reflected radar beams?
Sketch how the beat frequency varies as a function of time. What happens to the beat frequency if
the airplane moves in an arc?
2
, Problem 1.7
(a) If classical electromagnetism were described in terms of a single complex field G show that
Maxwell’s equations in free space and in the absence of free charges may written as the complex
equations
G = 0
and
G 1
i = -------------- G
t 0 0
where G = ------- -------- + i ---------
1 D B
2 0 0
(b) Show that the energy flux density in the electromagnetic field given by the Poynting vector
is
–i
S = E H = -------------- G G
*
0 0
(c) If the field G is purely real, what is the value of S?
2
(d) Show that the electromagnetic energy density is U = G .
Problem 1.8
How would Maxwell’s equations be modified if magnetic charge g (magnetic monopoles) were dis-
covered? Derive an expression for conservation of magnetic current and write down a generalized
Lorentz force law that includes magnetic charge. Write Maxwell’s equations with magnetic charge
in terms of a field G = ------- ------ + i ------- .
1 D B
2
Problem 1.9
(a) The capacitance of a small metal sphere in air is C 0 = 1.1 10
–18
F . A thin dielectric film with
relative permittivity r1 = 10 uniformly coats the sphere and the capacitance increases to
2.2 10 F . What is the thickness of the dielectric film and what is the single electron charging
– 18
energy of the dielectric coated metal sphere?
(b) The dielectric coated metal sphere of part (a) is now coated with metal. What is the new
value of the single-electron charging energy for the central metal sphere?
(c) Compare the result in (b) to the charging energy of a metal sphere radius
0.5 nm r 0 10 nm embedded in a dielectric of relative permittivity r = 10 and surrounded by
metal shell of internal radius r 1 = 2r 0 . Plot single-electron charging energy E as a function of
r0 .
Problem 1.10
(a) A diatomic molecule has atoms with mass m1 and m2. An isotopic form of the molecule has
atoms with mass m'1 and m'2. Find the ratio of vibration oscillation frequency / ' of the two mol-
ecules.
12 16
(b) What is the ratio of vibrational frequencies for carbon monoxide isotope 12 ( C O ) and
13 16
carbon monoxide isotope 13 ( C O )?
Problem 1.11
(a) Find the frequency of oscillation of the particle of mass m illustrated in the figure below. The
particle is only free to move along a line and is attached to a light spring whose other end is fixed at
Applied quantum mechanics 3
