Download Contemporary Physics - Department of Physics and Astronomy

University of Louisville
College of Arts and Sciences
Department of Physics and Astronomy PhD Qualifying
Examination (Part I)
Fall 2009
Paper E – Contemporary Physics
Time allowed – 80 minutes
Instructions and Information:
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Attempt any 2 of the 6 questions
This is a closed book examination
Start each question on a new sheet of paper – use only one side of each sheet
Write your identification number on the upper right hand corner of each answer sheet
You may use a non programmable calculator
Partial credit will be awarded.
Correct answers without adequate explanations will not receive full credit.
Make sure your work is legible and clear
The points assigned to each part of each question are clearly indicated
1)
A nuclear isotope of Cesium is represented by
141
55
Cs
(a)
How many nucleons are in the Cesium nucleus ?
(5)
(b)
What is the Z (atomic number) of Cesium ?
€
(5)
(c)
How many neutrons are in the Cs nucleus ?
(5)
(d)
Estimate the radius of this Cesium nucleus.
(13)
(e)
What is the approximate binding energy of a proton in the Cs nucleus ?
(5)
(f)
Sketch the form of the binding energy per nucleon versus mass number for stable nuclei. Be sure to
label your axes clearly and indicate the units.
(10)
(g)
Clearly mark on your graph where you expect the nuclei of Cs, Iron and Helium to be located.
(7)
2)
This problem addresses atmospheric vertical structure
Assume a dry, motionless atmosphere with a vertical temperature sounding T(z), a constant molar mass
(molecular weight) µ, and a constant acceleration of gravity, g.
(a)
Write an expression for hydrostatic balance in terms of pressure, p, and density, ρ, and an expression
for the ideal-gas equation of state in terms of p, ρ, and T.
(8)
(b)
Form a first-order ordinary differential equation for dp/dz and write the general solution for p(z). (8)
(c)
Assume an isothermal atmosphere, T(z) = T0, and solve for p(z).
(d)
The following table gives relevant parameters for the tropospheres of Earth and Jupiter. For each,
calculate the pressure scale height in km. Assume the universal gas constant is R* = 8314 J kmol-1 K-1.
(6)
Atmosphere
Earth
Jupiter
Molar Mass [kg kmol-1]
28.97
2.29
Gravity [m s-2 ]
9.8
25
(8)
Temperature [K]
250
135
(e)
Calculate the mass per area of a column of gas in an atmosphere with a constant density scale height Hρ
and surface density ρ0.
(10)
(f)
Explain qualitatively how the addition of water vapor to the air changes the results above.
(10)
3)
The electric field of an electromagnetic wave (in S.I. units) is described by:
(a)
What is the angular frequency, ω?
(5)
(b)
What are the Cartesian components of the k-vector,
(c)
What is the magnitude of the k-vector,
(d)
What is the wavelength, λ?
(e)
What is the phase velocity,
(f)
What is the refractive index of the medium in which this wave is propagating?
(g)
What are the Cartesian components of the magnetic field associated with this electromagnetic wave?
(5)
(h)
In a Cartesian coordinates system, draw the k-vector
?
(5)
?
(5)
(5)
?
associated with this electromagnetic wave.
(5)
, the electric field
(5)
, and the magnetic field
(5)
(i)
If E0 = 1 V/m, what is the power per unit area that one would measure for this electromagnetic wave?
(5)
(j)
Now suppose that this electromagnetic wave starts to propagate in another medium with a different
refractive index. Among the quantities ω, k , λ, and vphase, list those that will have a different numerical
value and those that will remain at the same value?
(5)
4)
The hydrogen atom with one proton and one electron is the simplest and most common atom of ordinary matter.
Consider the following questions in order, and provide the most complete answer you can to each one.
(a)
Neglect the magnetic moments of the electron and the proton. Make a simple assumption about how the
wavefunction of the electron depends on its momentum, and explain why only certain energies of the
system are likely to be seen in a laboratory experiment.
(5)
(b)
Show that those energies are given by En = −RH/n2 where RH is the Rydberg Constant for hydrogen, and
n is an integer from 1 to ∞. Express RH in terms of the fundamental constants of physics.
(5)
(c)
How do the energies of H differ from those of deuterium (D), with one proton and one neutron in its
nucleus ?
(5)
(d)
For the expression you gave for RH, describe briefly how each of the constants you specified are
known.
(5)
(e)
Show that if a hydrogen atom in the n = 3 state emitted a photon by making an electric dipole transition to
n = 2, the emitted light would be red, while one from n = 2 to n = 1 would be in the extreme ultraviolet.
(5)
(f)
An atomic gas at atmospheric pressure and 273 K has 2.7 × 1019 atoms/cm3. How large would n be if the
electron of the excited hydrogen atom were as far from its nucleus as the average neighboring atom in the
gas.
(5)
(g)
Allow for the magnetic moments, and specify the quantum numbers needed to characterize a state of
atomic hydrogen. In terms of those, what states are possible for n = 1, n = 2, and n = 3? (Hint: some of
these states may have the same energy.)
(5)
(h)
A so-called “s” electron has no orbital angular momentum. Contrast its behavior in a hydrogen atom in
comparison to an electron with the maximum possible angular momentum of the same n. In what way, if
any, might this affect the energy of atom?
(5)
(i)
Elements with 3, 11, 19, 37, 55, and 87 electrons are chemically similar to hydrogen and to each other.
Their spectra also are similar, with characteristic series of lines following structured patterns. Explain
why sodium, element number 11, has this hydrogenic behavior.
(5)
(j)
When a photon is emitted by a sodium atom undergoing an electric dipole transition, what
“conservation” laws apply? How would these affect the spectrum that would be observed far from the
atom?
(5)
5)
Consider the following questions in order, and provide the most complete answer you can to each one.
(a)
From Kentucky at north latitude 380 what portion of the celestial sphere, if any, can never be seen ? (5)
(b)
The celestial equator is the apparent intersection of the plane of the Earth’s equator with the sky. What is
the altitude of the celestial equator above the southern horizon seen from Kentucky, and what is the
azimuth of the equator where it disappears below the horizon ? (By convention, azimuth is measured from
the north around to the east, so that due east is +900 ).
(5)
(c)
If a telescope with a 10 field of view is directed toward the meridian at the equator, how long would it
take for a star to pass across the field ?
(5)
(d)
Describe how a telescope would be constructed so that rotation around only one axis could compensate
for the Earth’s rotation. At what rate would the telescope have to rotate on this axis to follow a star? (5)
(e)
If, instead, a telescope is constructed to follow a star by rotating about both horizontal and vertical axes,
the field of view will appear to rotate. At what rate does the field rotate, and how would you compensate
for this rotation?
(5)
(f)
The stellar magnitude scale is a logarithmic relationship such that a factor of 100 decrease in light flux
corresponds to an increase of 5 magnitudes. The fainter a star, the larger its magnitude. Therefore, on this
scale a 1st magnitude star is 100 times brighter than a 6th magnitude star, and mathematically, the
relationship is
m2 − m1 = (5 2)log( 1  2 )
Our nearest stellar neighbor, Proxima Centauri, is a red star with an apparent magnitude 11.05, and it is
4.2 light years from us. Its “absolute” magnitude is how bright it would appear to be if it were 10 parsecs
away, where a parsec is 3.26 light years. What is the absolute magnitude of Proxima Centauri?
(5)
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(g)
The Sun’s apparent magnitude is -26.73, but if it were also at 10 parsecs, what would its apparent
magnitude be? If stars emit light as if they are black body sources, what factors would determine the
apparent magnitude of a star as we see it? Is Proxima Centauri smaller in diameter than the Sun, or larger?
(5)
(h)
The Andromeda galaxy, a companion to the Milky Way and similar to it in size, is nearby on a cosmic
scale at only 2.5 million light years distant. Suppose you had a telescope with superb optics that could
detect stars as faint as 25th magnitude. Would it be able to detect a star such as the Sun in the Andromeda
galaxy?
(5)
(i)
A supernova of type Ia, an explosion of a white dwarf star that occurs when it gathers sufficient mass
from a nearby companion to go over a limiting mass for stability, is one of the best distance indicators we
have. These stars reach a maximum luminosity with an absolute magnitude of about -19. What would be
the most distant galaxy in which we could see such a supernova if the limiting apparent magnitude of our
telescope is 25?
(5)
(j)
Suppose that you were given the task of determining the three dimensional distribution of stars in our
Milky Way, and galaxies in space beyond the Milky Way. Describe how you would do this. Keep in
mind that you are observing from a fixed point on a rotating Earth, that except for the nearest stars
distance can be measured only by inference from the magnitude and the spectrum, and that it takes time
for light to reach us from the object we are observing.
(5)
6)
Free electron gas model
(You are advised to first read through the entire problem before attacking individual parts.)
(a)
In the free-electron-gas model, what is the interaction between the electrons?
(5)
(b)
Among the crystals of (a) Na, (b) Fe, (c) GaAs, and (d) Diamond, which is best described by the freeelectron-gas model?
(5)
(c)
Briefly state the main difference between the classical model and the quantum-mechanical model for the
free electron gas?
(5)
(d)
For a three-dimensional (3D) free electron gas confined to a cube of edge L, calculate the density of
states, g(E).
Note: g(E) is defined such that g(E)dE is the number of quantum states for the electron between E and
E+dE.
Note: you only need to obtain the correct dependence on E.
(8.5)
(e)
For a 3D free electron gas of N electrons confined to a cube of edge L, calculate the Fermi energy (EF ) at
0 K.
(8.5)
(f)
Using the quantum-mechanical model, calculate the total kinetic energy for a 3D free electron gas at 0 K.
(6)
(g)
Using the quantum-mechanical model, express the total kinetic energy for the 3D free electron gas at
temperature T with g(E) and the Fermi-Dirac distribution function.
Note: you are required to obtain the correct expression with integral, but not required to finish the
integration that is analytically impossible.
(6)
(h)
For a 3D free electron gas at very low temperature, discuss the difference of the temperature dependence
between the quantum-mechanical heat capacity and the classical heat capacity.
(6)