Widener University
Summer COOP 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 3
Quantum Theory of Light
Exercise #1 (Similar to Problems 2 & 4)
An iron ball of radius 0.10 m is heated to 500 K. Calculate:
a) the wavelength at which the energy distribution is maximum.
b) the total power E radiated per unit area by the iron ball.
c) the total power emitted by the entire surface of the iron ball.
Exercise #2 (Similar to Problems 8 & 9)
A photon has a frequency of 1 GHz. What is its:
a) energy in both J and eV? Note 1 eV = 1.602  10-19 J.
b) wavelength in nm?
Exercise #3 (Problem 12)
A sodium vapor lamp has a power output of 10 W. Using 589.3 nm as the average wavelength of the source, calculate
the number of photons emitted per second.
Exercise #4 (Problem 18)
Light of wavelength 500 nm is incident on a metallic surface. If the stopping potential for the photelectric effect is 0.45
V, calculate:
a) the maximum kinetic energy of the ejected photoelectrons.
b) the work function.
c) the cutoff wavelength
Exercise #5 (Problem 35)
In the original Compton experiment, Compton used photons of wavelength 0.0711 nm. Calculate:
a) the energy of these photons before collision.
b) the wavelength of the photons scattered at an angle 180 (backscattering).
c) the energy of backscattered photons.
d) the recoil energy of the electrons.
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 4
The Particle Nature of Matter
Exercise #1 (Similar to Problems 11 & 12)
a) What are the values of n for the transition corresponding to the first two lines of the Paschen series?
b) Calculate the wavelength for the first two lines of the Paschen series.
c) Calculate the radii for the orbits corresponding to the three different values of n in part a).
Exercise #2 (Similar to Problems 16 & 17)
Consider the hydrogenic ion Li2+, where Z = 3. Calculate:
a) the energies of the first three levels of Li2+.
b) the wavelengths of the first two lines of Li2+.
c) the orbital radius of the first Bohr orbit of Li2+.
d) the ionization energy for Li2+.
Exercise #3 (similar to Problem 20)
a) Calculate the energy of the photon that causes an electronic transition between the n=4 to n=5 states of hydrogen
b) Calculate the energy of the photon that causes an electronic transition between the n=5 to n=6 states of hydrogen.
c) Calculate the wavelengths of the photons in parts a) and b).
Exercise #4 (similar to Problem 23)
A hydrogen atom is in the first excited state (n=2). Calculate from Bohr theory:
a) the radius of the orbit.
b) the linear momentum of the electron.
c) the angular momentum of the electron.
d) the kinetic energy.
e) the potential energy
f) the total energy
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 5
Matter Waves
Exercise #1 (Similar to Problems 1 & 2)
Calculate the de Broglie wavelength for each of the following:
a) an electron (me = 9.1  10-31 kg) moving at 2.0  106 m/s
b) a proton with kinetic energy 100 eV.
c) a 500 kg car moving at a speed 20 m/s.
Exercise #2
A proton (mass 1.67  10-27 kg) has kinetic energy 1.0 MeV. If its momentum is measured with an uncertainty of 5.0%,
what is the minimum uncertainty in its position?
Exercise #3 (Problem 25)
An excited nucleus with lifetime 0.100 ns emits a -ray of energy 2.00 MeV.
a) Calculate the energy width (uncertainty) E of the 2.00 MeV -ray emission line.
b) Can this energy width be directly measured if the best gamma detectors can measure no smaller than  5 eV?
Exercise #4 (Similar to Problem 27)
A monoenergetic beam of electrons is incident on a single slit of width 1.50 nm, and a diffraction pattern is formed on
a screen 20 cm from the slit. The distance between successive minima of the diffraction pattern is 2.1 cm. Calculate
the energy of the incident electrons in eV.
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 6
Quantum Mechanics in One Dimension
Exercise #1 (Similar to Problem 3)
A free electron has wave function given by (x) = A cos (2  1010 x), where is in m. Find the electron's:
a) de Broglie wavelength in nm
b) momentum in kg m/s
c) energy in eV
Exercise #2
The wave function describing a state of an electron confined to move along the x-axis is given at any time zero by
(x,0) = A exp(-x2/42)
Find the probability of finding the electron in a region dx centered on:
a) x = 0 ?
?
most likely to be found?
constant A?
b) x = 
c) x = 2 ?
d) Where is the electron
e) What is the normalization
Exercise #3 (similar to Problem 10)
An electron is contained in a one-dimensional box of width 0.100 nm.
a) Calculate the energy levels En for the electron for levels n = 1, 2, 3, and 4.
b) Write all possible transitions of the electron from the n = 4 state to the n = 1 state
c)
Calculate both shortest and the longest wavelengths of the emitted photons corresponding to the possible transitions
found in part b).
Exercise #4 (Problem 11a)
Consider a particle moving in a one-dimensional box with walls at x = -L/2 and x = L/2.
a) Write the wavefunctions n(x) for the states n = 1, n = 2, and n = 3
b) Write the corresponding probability densities P n(x) for states n = 1, n = 2, and n = 3
Exercise #5 (Problem 16)
An electron is trapped in an infinitely deep potential well 0.300 nm in width.
a) If the electron is in the ground state, what is the probability of finding it within 0.100 nm of the left hand wall; i.e.,
x in the region (0.00, 0.100) nm
b) Repeat part a) for an electron in the 99 th energy state above ground state (i.e., n = 100)
c) Are the above answers consistent with the correspondence principle?
Exercise #6
A particle is in the ground state of an infinite square-well potential given by
U(x)
= 
= 0
= 
x<0
0<x<L
x>L
Find the probability of finding the particle in the interval x = 0.002 L at the following positions:
a) x = L/2
b) x = 2L/3
c) x = L
*Note: since x is very tiny, you do not need to do any integration!
Exercise #7 (Problem 12)
A ruby laser emits light of wavelength 694.3 nm. If this light is due to transitions from the n = 2 state to the n = 1 state
of an electron in a box, find the width L of the box.
Exercise #8
A mass of 10-9 kg is moving with a speed of about 10-3 m/s in a box of length 0.01 m. Treating this as a one-dimensional
infinite square-well potential, calculate the approximate value of the quantum number n.
Exercise #9
A particle is in the first excited state (n = 2) of an infinite square potential. Calculate:
a) <x>
b) <x2>
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 7
Tunneling Phenomena
Exercise #1 (Similar to Problem 5)
When particle energies are well below the top of the barrier (E << U), the probability for transmission is given
approximately by:
P  16 (E/U) e-2L
where
 = [2m(U-E)]1/2 / 
Using  = 1.055  10-34 J.s, calculate the exponential factor e-2L for each of the following cases:
a) a proton (m = 1.67  10-27 kg) with U - E = 10 eV and L = 10-12 m
b) a baseball (m = 0.10 kg) with U - E = 5 J and L = 100 m
Exercise #2 (similar to Problem 16)
An alpha particle is confined to the nucleus of an unspecified radioactive element. Assume a semiinfinite square well
with infinitely high wall at r = 0 and wall of height 22 MeV at the nuclear radius R = 7.0 fm.
Take  = 197.3 MeV.fm/c
a) Use the interative method of Example 5.9 to estimate lowest (ground state) energy permitted for the alpha particle.
b) Calculate the smallest value of velocity of the alpha particle from the result of part a).
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 8
Quantum Mechanics in Three Dimensions
Exercise #1 (Problem 5)
Assume that the nucleus of an atom can be regarded as a three-dimensional box of width 2  10-14 m. If a proton (m =
1.67  10-27 kg) moves as a particle in this box, calculate (using h = 6.63  10-34 J.s):
a) the ground state energy of the proton in MeV
b) the energies of the first and second excited states
c) What are the degeneracies of these states?
Exercise #2 (Problem 9)
If an electron has orbital angular momentum of |L| = 4.714  10-34 J.s, what is the orbital quantum number for this state
of the electron?
Exercise #3 (Problem 16)
Calculate the possible values of Lz, the z-component of angular momentum for an electron in d subshell.
Exercise #4 (Problem 17)
Calculate the magnitude of angular momentum L for an electron in:
a) the 4d state of hydrogen
6f state of hydrogen
b) the
Exercise #5 (Problem 21)
Suppose that a hydrogen atom is in the 2s state. Taking r = a0, calculate values for:
a) 2s (a0)
(a0)|2
b) |2s
c) P2s(a0)
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 9
Atomic Structure
Exercise #1 (Similar to Problem 4)
List the possible sets of quantum numbers for an electron in the:
a) 4p subshell
b) 4f subshell
Exercise #2 (Problem 13)
An electron is in the 4F5/2 state.
a) Find the values of the quantum numbers n, , and j.
What is |J|, the magnitude of the electron’s total angular momentum?
What are the possible values for Jz, the z-component of the electron’s total angular momentum?
b)
c)
Exercise #3 (similar to Problem 17)
Six identical, non-interacting particles are placed in a cubical box of sides L = 0.500 nm. Find the lowest energy of the
system (in eV) and list the quantum numbers of all occupied states if:
a) the particles are electrons
particles are identical to electrons, except do not obey the Pauli exclusion principle
Exercise #4 (similar to Problem 21)
Write out:
a) the electronic configuration for nitrogen (Z=7)
b) the values for the set of quantum numbers n, , m and ms for each electron in nitrogen.
b) the
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 10
Statistical Physics
Exercise #1 (Similar to Problem 1)
A system of six indistinguishable particles has total energy 8E. Calculate the probability of finding the particle with
energy (a) 1E and (b) 8E.
Exercise #2 (similar to Problem 6)
An unspecified element possesses an energy difference of 1.92 eV between n = 1 (ground) state and n = 2 (first excited)
state, assumed to have equal statistical weights. If a sample of 1020 atoms of this substance is in an enclosed chamber
in thermal equilibrium at T = 1000 K, calculate the approximate number of atoms in the ground state (n1) and in the first
excited state (n2). Assume n2 << n1.
Exercise #3 (Problem 15)
The Fermi energy of aluminum (atomic mass 26.98, density 2.70 g/cm3) is 11.63 eV.
a) Assuming that the free electron model applies to aluminum, calculate n = N/V, the number of free electrons per unit
volume at low temperatures.
b) Determine the valence of aluminum by dividing the answer found in part (a) by the number of aluminum atoms per
unit volume, as calculated from the density & atomic mass.
Exercise #4 (Problem 20)
Find the probability that a conduction electron in a metal has an energy equal to the Fermi energy EF at the temperature
300 K .
Exercise #5 (Problem 22)
The Fermi energy for silver is 5.48 eV at T = 800 K. Also, the probability of finding the electron in that state is 0.95.
Note the Boltzmann constant is kB = 8.617  10-5 eV/K.
a) Calculate the energy E of a conduction electron in silver at T = 800 K.
Calculate the Fermi speed vF for electrons in silver.
b)
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 12
The Solid State
Exercise #1 (Problem 7)
The crystal structure of KCl is the same as NaCl. Calculate the:
a) ionic cohesive energy for KCl. Take r0 = 0.314 nm and m = 9.
b) atomic cohesive energy of KCl by using the fact that ionization energy of potassium is 4.34 eV
eV  K+ + e) and that electron affinity of chlorine is 3.61 eV (Cl - + 3.61 eV  Cl + e)
(K + 4.34
Exercise #2 (Similar to Problem 10)
Sodium (Na) is a monovalent metal (each Na atom contributes one e-) having a density 0.971 g/cm3, an atomic weight
23.0 g/mol, and resistivity 4.20  10-8 .m at 300 K. Assume the classical free-electron model (MB distribution).
Calculate the:
a) average density n = N/V of free electrons
b) average time  between collisions of the electrons
thermal velocity vrms of an electron
path L
d in nm, and compare with L
c)
d) mean free
e) typical lattice spacing
Exercise #3 (Problem 14)
Sodium (Na) is a monovalent metal (each Na atom contributes one e-) having a density 0.971 g/cm3, an atomic weight
23.0 g/mol, and resistivity 4.20  10-8 .m at 300 K. Assuming the Fermi-Dirac gas, use this information to calculate:
a) the density of free electrons n = N/V
b) the
Fermi energy EF at 0 K
c) the Fermi
velocity vF of an electron
d) the average time 
between electronic collisions
e) the mean free path L of
the electrons (assuming EF at 300 K is the same as at 0 K) and compare with the typical lattice spacing d (from Exercise
#2)
f) the thermal conductivity K of Na
Exercise #4 (similar to Problem 16)
From the optical absorption spectrum of a certain semiconductor, one finds that the longest wavelength of radiation
absorbed is 2.09 m. Calculate the energy gap for this semiconductor.
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 12 Addendum
Superconductivity
Exercise #1
An iron-core toroid is wrapped with 250 turns of wire per meter of its length. The current in the winding is 8.00 A.
Taking the magnetic permeability of iron to be m = 50000, calculate the:
a) magnetic field strength H (in A/m)
b) magnetic flux density B (in T)
Exercise #2
A wire made of Nb3 Al model has radius 2.0 mm and is maintained at 4.2 K. Using data for T c and Bc2(0) provided in
Table 12.5, calculate:
a) the upper
critical field Bc(T) for this wire at this temperature
b) the maximum
current Imax that can pass through the wire before its superconductivity is destroyed
c) the magnetic field B
at 6.0 mm from the wire surface (8.0 mm from center) when the current has its maximum value.
Exercise #3
The penetration depth for lead at 0.0 K is 39 nm. Using the critical temperature T c = 7.193 K for lead, find the
penetration depth in lead at: a) 1.0 K b) 4.2 K c) 7.0 K
Exercise #4
Persistence currents. In an experiment carried out by S.C. Collins between 1955 and 1958, a current was maintained
in a superconducting lead ring for 2.5 years with no observed loss. If the inductance in the ring was 3.14  10-8 H, and
the sensitivity of the experiment was 1 part in 109, determine the maximum resistance of the ring. Hint: Treat this as a
decaying current in an RL circuit, and use the fact that
e-x  1 - x
for small x
Exercise #5
Calculate energy gaps Eg as predicted by BCS theory for (a) type I superconductor Zn and (b) type II superconductor
Nb3Sn. Tc is listed in Table 12.9. Compare the values for types I and II superconductors.
Exercise #6
The radius of a ring that would fit on a finger is ~ 8.0 mm. Calculate:
a) the magnetic flux through the ring due to Earth’s magnetic field (B = 5.8  10-5 T)
b) the number of fluxons that the ring would enclose.
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 13
Nuclear Structure
Exercise #1 (Problem 4)
The Larmor precessional frequency is
f = E/h = 2B/h
Calculate the radio-wave frequency at which resonance absorption occurs for:
a) free neutrons in a magnetic field B = 1.0 T
b) free protons in a magnetic field B = 1.0 T
c) free protons in the Earth's magnetic field at a location where B = 5.0  10-5 T
Exercise #2 (Problem 7)
Consider a H atom with the electron in the 2p state. The magnetic field at the nucleus produced by the orbiting electron
has a value B = 12.5 T. The proton can have its magnetic moment aligned in either of two directions perpendicular to
the plane of the electron's orbit. Because of the interaction of the proton's magnetic moment with the electron's
magnetic field, there will be a difference in energy between the states with the two different orientations of the proton's
magnetic moment. Find that energy difference E in eV.
Exercise #3 (Problem 11)
In Example 13.3, the binding energy of the deuteron was calculated to be 2.224 MeV. This corresponds to a value of
1.112 MeV/nucleon. What is the binding energy per nucleon for the heaviest isotope of hydrogen, 3H (called tritium),
for which m(3H) = 3.01605 u.
Exercise #4 (Problem 17)
Using the graph in Fig. 13.10, read off the values of binding energy per mass number Eb/A at A1 = 100 and A2 = 200 to
estimate how much energy is released when a nucleus of mass number 200 is split into two nuclei, each of mass number
100.
Exercise #5 (Problem 24)
Tritium has a half-life of 12.33 yr. What percentage of the 3H nuclei in a tritium sample will decay during a period of
5 yr?
Exercise #6 (Problem 26)
How many radioactive atoms are present in a sample that has an activity of 0.2 Ci and a half-life of 8.1 days?
Exercise #7 (Problem 32)
A by-product of some fission reactors is the isotope 239Pu94, which is an alpha emitter with a half-life of 24,000 yr:
239
Pu94 
235
U92 + 4He2
Consider a sample of 1 kg of pure 239Pu94 at t = 0. Calculate:
a) the number N0 of 239Pu94 nuclei present at t = 0
b) the initial activity R0 in the sample
c) the time t required for the activity to decrease to 1 decay/s.
Exercise #8 (Problem 41)
Find the energy released in the alpha decay 238U92 shown below. Use the mass values in Table 13.6.
238
U92 
234
Th90 + 4He2
Widener University
Spring 2004
PHYS 261 Modern Physics
Prof. Augensen
Name _______________________
Worksheet for Chap. 14
Nuclear Physics Applications
Exercise #1 (Problem 3)
The following reaction, first observed in 1930, led to the discovery of the neutron by Chadwick:
9
Be4 (, n) 12C6
Calculate the Q value of this reaction.
Exercise #2 (Problem 11)
The density of liquid hydrogen target in a bubble chamber is 70 kg/m3. If 20% of a beam of slow neutrons incident on
the bubble chamber has reacted with the hydrogen by the time the beam has traveled 2 m through the hydrogen, what is
the cross section, in barns, for the reaction of these slow neutrons with hydrogen atoms?
Exercise #3 (Problem 20)
A particle cannot generally be localized to distances much smaller than its de Broglie wavelength. This means that a
slow neutron appears to be larger to a target particle than does a fast neutron, in the sense that the slow neutron will
probably be found over a large volume of space. For a thermal neutron at room temperature (300 K), find:
a) the linear momentum p
b) the de Broglie wavelength B
c) Compare this effective neutron size in b) with both nuclear & atomic dimensions.
Exercise #4 (Problem 30)
The Sun radiates energy at the rate 4  1026 W. Assuming that the reaction
4(1H1)  4He2 + 2e+ + 2 + 
accounts for all the energy released, calculate:
a) the energy Q released every time this reaction occurs
b) the number of protons fused per second
c) the mass transformed into energy per second.