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PHYSICS 447 SUPPLEMENTAL PROBLEMS
1-1A. (Probability Distributions)
The roll of a six-sided weighted die is described by the discrete probability distribution P(x) = A[10-(x-4)2]
where x is the number on the top side of the die (x = 1, 2, 3, 4, 5, or 6).
(a) Find the normalization constant A.
(b) What number is most likely to come up when the die is rolled? What number is least likely?
(c) What is the probability that the most likely number will come up on a roll? How does this compare to the
probability that this number would be rolled with a straight die?
(d) Find the average value of x. (This would be the average of many rolls.)
(e) Find the probability that a roll will yield a 5 or a 6.
1-1B. (Probability Distributions)
Consider the Gaussian probability distribution P( x)  Ae ( x  xo ) / 2 where x can have any value. Show the
following:
(a) A  1  2 .
(b) the average value of x is xo.
(c) the probability of x being in between xo- and xo+ is 68%. ( is the standard deviation of the mean)
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1-2A. (Boltzman Distribution)
Consider the preliminary form of the Boltzman probability distribution, P( E )  Ce  E / Eo , for a system where
an entity can have any energy between zero and infinity.
(a) Normalize the distribution and show that C = 1/Eo.
(b) Show that the average energy of an entity is Eo.
1-2B. (Boltzman Distribution)
Using the Maxwell-Boltzman distribution for the x-component of an atom’s velocity in a gas, show:
(a) the average value of vx is zero.
(b) the average value of vx2 is kT/m. Note that this is consistent with the equipartition of energy theorem since
the average value of (1/2)mvx2 is (1/2)kT.
1-2C. (Boltzman Distribution)
Using the Maxwell-Boltzman distribution for the speed of an atom in a gas, do the following.
(a) Show that the most probable speed is (2kT / m)1/2.
(b) Show that the average speed is (8kT / m)1/2. Why is the average speed larger than the most probable speed?
(c) Evaluate these two speeds for a gas of hydrogen at room temperature. At this temperature, the hydrogen will
be in the form of diatomic molecules (H2).
1-2D. (Boltzman Distribution)
Using the Maxwell-Boltzman distribution for the kinetic energy of an atom in a gas, show:
(a) The normalization constant of the distribution is indeed 2  (kT ) 3 / 2 .
(b) The average kinetic energy is (3/2)kT, consistent with the equipartition theorem. (You can perform an
integration by parts on the integral used in (a) or use Maple.)
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1-3A.(Thermal Radiation)
(a) It is useful to know the form of Planck's blackbody spectrum in terms of wavelength since spectrometers
usually measure light intensity as a function of wavelength. Beginning with the energy density frequency
distribution function, (), show that the energy density wavelength distribution function is
() = (8hc/5)/(ehc/kT - 1)
(1)
(Follow Ex. 1-5 in the text.)
(b) Derive the Wien displacement law for maximum wavelength by maximizing the energy density wavelength
distribution function with respect to wavelength. That is, show that
maxT = 0.2014 hc/k = 2.9 x 10-3 m-K
(2)
by solving ()/ = 0. (Hint: Set hc/kT = x and show that you obtain the equation e-x + x/5 = 1. Then show
that x = 4.965 is a solution, and finally that Wien's law as expressed in (2) follows from this value of x.)
(c) Derive the Wien displacement law for maximum frequency by maximizing the energy density frequency
distribution function with respect to frequency. That is, show that
max/T = 2.821 k/h = 5.88 x 1010 Hz/K
(3)
by solving ()/ = 0. (Hint: Set h/kT = x and show that you obtain the equation e-x + x/3 = 1. Then show
that x = 2.821 is a solution, and finally that Wien's law as expressed in (3) follows from this value of x.)
(d) Show that maxmax = 0.568c. But wait! Since  = c for electromagnetic radiation, then why doesn't
maxmax = c ? This is a subtle point. (Hint: Recall that the energy density wavelength distribution function is
for a constant uniform wavelength interval d, while the energy density frequency distribution function is for a
constant uniform frequency interval d.)
1-8A.(Bohr Model)
Consider a Hydrogen atom in the second excited state (n=3). Use the Bohr model to answer the following:
(a) The maximum number of photons that can be emitted by this atom is two. Explain.
(b) Find the energy (in eV), wavelength (in nm), and momentum (in eV/c) of each of the photons in (a).
Suppose instead that the excited atom makes one transition to the ground state.
(c) Find the energy (in eV), wavelength (in nm), and momentum (in eV/c) of the single emitted photon.
(d) Using conservation of momentum, find the recoil speed of the atom. (A H atom has a rest mass of
1.007825 u. You can use the nonrelativistic expression p=Mov since its kinetic energy is much less than its rest
energy.)
(e) Find the kinetic energy of the recoiling atom in eV. (You can use the nonrelativistic expression K= p2/2Mo =
(1/2)Mov2.)
(f) Compare this recoil energy of the atom to the photon energy found in (c). Are we justified in neglecting the
recoil energy of the atom in calculating the energies and wavelengths of emitted photons?
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1-8B.(Extending the Bohr Model)
Let's treat the motion of a satellite orbiting the Earth quantum mechanically borrowing from Bohr's model of the
atom. The mass of the Earth is 6x1024 kg and let's assume the mass of the satellite is 200 kg.
(a) Show that the allowed radii of the satellite’s orbit measured from the center of the Earth obey the
relationship
rn = n2Ao (1)
where Ao = 7x10-88 m. Do this by noting that the gravitational force exerted by the Earth on the satellite is
providing a centripetal acceleration and by assuming Bohr's angular momentum quantization:
L = mvr = n
(2)
(b) The satellite can't be orbiting inside the Earth which has a radius of 6380 km. So what is the approximate
minimum allowed value of n for the satellite?
(c) What is the approximate value of n if the satellite is 35900 km above the Earth's surface (near geostationary
orbit)?
(d) Find the distance (dr) between allowed orbits near this altitude by differentiating (1) and using dn=1.
(e) Are we justified in treating the satellite motion classically and assuming the satellite can have any altitude?
2-1A.(deBroglie’s Postulates)
An electron and a photon each have a wavelength of 5 Å. Find the following quantitites for each:
(a) momentum, in units of eV/c
(b) energy (not including rest energy), in units of eV
(c) speed, expressed as a number times c
2-2A.(HUP)
(a) Show that you can write the uncertainty condition for position and momentum as x  2 / 4  .
(b) Suppose the uncertainty in the wavelengths of the electron and photon in Problem 2-1A is 1% when you
perform the measurements. To within what distance could you have simultaneously located the electron? To
within what distance could you have located the photon? [It is interesting to note that even though the electron
travels much slower than the photon, you cannot measure the location of the electron with any more certainty
than the photon!]
2-4A.(Infinite Square Well: Normalization of Eigenfunctions)
Show that the amplitudes of all of the normalized eigenfunctions for an infinite square well of width a are
identical and equal to (2/a)1/2.
Do this by performing one normalization for the eigenfunction n(x)=Ancosknx where n is odd and showing that
An=(2/a)1/2. Then perform one more normalization for the eigenfunction n(x)=Bnsinknx where n is even and
show that Bn=(2/a)1/2.
*Do the integrals with the variable n. Don't substitute a number in for n. You don't have to! You should find
that there is no dependence of A or B on n.
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2-4B.(Infinite Square Well: Orthonormal Eigenfunctions)
Show that the eigenfunctions for an infinite square well are orthonormal. That is, show that
0 for m  n
*


dx




mn
 m n
1 for m  n
To show this, you must perform 5 integrations:
(1) m=n = even integer
(2) m=n = odd integer
(3) mn both integers even
(4) mn both integers odd
(5) mn one integer even, one integer odd
*Again, do not substitute numbers in for m and n. Perform the integrations with m and n and then make
observations about whether combinations of m and n are odd or even to evaluate the definite integrals.
2-4C. (Finite Square Well)
By running the “Finite/Infinite Well Program” (447well.exe), answer the following questions. (You can
download a copy of the program from my home page by going to ‘My Courses” and selecting ‘Physics 447’.)
For a=100 Å, V=100 meV, m=mo:
1. How many energy levels does the finite well have? ...the infinite well?
2. What are the ground state energies of the two wells?
3. Is there better agreement between the energies for small or large n?
4. Is there more penetration of the finite well walls for smaller or larger energies?
By trying different parameters with both models:
5. Do the finite & infinite well energies agree better for wide or narrow wells?
6. Do the finite & infinite well energies agree better for large or small well heights?
7. Find the largest well width (to the nearest Å) for V=100 meV, m=mo where the infinite square model
predicts a ground state energy that is just above the finite well height of 100 meV.
8. Does the finite well model predict a confined ground state energy for this width? What is the energy?
9. Does a finite well always have a ground state energy, no matter the well width, height, or particle mass?
By trying different parameters with the finite well model:
10. How do a, V, and m each affect the size of the ground state energy?
11. How do a, V, and m each affect the separation of the confined energies of a well?
12. For the range of widths, heights, and masses allowed in the program, what is the largest ground state
energy that you can obtain for the finite well? Write down the values of a, V, and m for this energy.
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3-1A.(Spin-Orbit Interaction)
(a) Find one exact expression for the average spin-orbit energies for the n=2, =1 state of the Hydrogen atom.
(b) Find the values of the two energies (corresponding to the two allowed values of j).
(c) Find the total splitting.
Comments:
Use the expression derived in class for the average spin-orbit potential energy. Recall that you must find the


average value of (1/r3) using (1 / r 3 ) =  (1 / r 3 ) P21 (r)dr
P
0
21
(r)dr .
0
You should find that E = (1.5x10-5 eV)[j(j+1)-11/4] and that the total splitting is 4.5x10-5 eV. In class we
estimated the total splitting to be on the order of 10-4 eV. (This is also done in Ex. 8-3 of the book.)
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Some useful integrals for the Supplemental Problems


0


0


0


0
1
e
u 2
du   / 2

2
e u / 2 du  0.3413 2
0

u e u du  1 / 2
2

u e u du   / 2
0
2 u 2
u e

du   / 4
u 3 e  u du  1 / 2
2
n!
 u n e - au du= n+1
a
0
for a>0 and n = 1,2,3,...