Download The University of Georgia Department of Physics and Astronomy

The University of Georgia
Department of Physics and Astronomy
Graduate Qualifying Exam –– Part I
August 11, 2008
Instructions: Attempt all problems. Start each problem on a new sheet of paper. Print
your name on each sheet of paper. This is a closed-book and closed-notes exam. You
may use a calculator, but only for arithmetic functions (i.e., not for referring to notes
stored in memory, doing symbolic algebra, etc.). For full credit, you must show your
work and/or explain your answers. Part I has six problems, numbered 1–6.
Problem 1: (2 parts)
Consider a particle of mass m in a 1D “semi-infinite square well” of the form
#!
%
V (x) = $ 0
%V
& 0
for x < 0
for 0 " x " L
for x > L
where L and V0 are positive constants.
(a) Find the transcendental equation determining the allowed energies En of the bound
states. [Hint: you may find it helpful to express your answer using the wave number
kn = 2mEn / ! .]
(b) Show that your solution yields the bound state energies of the infinite square well by
taking the appropriate limit.
Problem 2: (1 part)
An object of uniform horizontal cross-sectional area A and uniform mass density ρ floats
in a large vat of liquid of mass density ρ0. At equilibrium, the object displaces a volume
V of liquid. Derive an expression for the period of small vertical oscillations of the object
about its equilibrium position. You may assume (a) that the object is tall enough that it is
never fully submerged, (b) that the vat is large enough compared to the object that the
rise and fall of the liquid surface as the object bobs up and down is negligible, and (c)
that all damping forces are negligible, as well. [Note: Archimedes’ Principle states that
the buoyant force acting on a floating object equals the weight of the displaced fluid.]
Problem 3: (1 part)
An infinite wire is bent into the geometry shown in the figure. If the wire carries a current I, calculate the magnetic field at point A.
Problem 4: (2 parts)
A particle of mass m moves in the following three-dimensional force field:
(
)
(
)
F(x, y, z) = ! " y 2 + 2# xz 2 x̂ ! 2y (" x + $ z ) ŷ ! $ y 2 + 2# x 2 z ẑ ,
where α, β, and γ are known positive constants, and x̂ , ŷ , and ẑ are Cartesian coordinate unit vectors.
(a) Show that F(x, y, z) is a conservative force field, and determine the potential-energy
function V (x, y, z) , where we choose V = 0 at the origin.
(b) Suppose the particle’s trajectory passes through the origin and comes to an instantaneous rest at the point (1, 1, 1). How fast was the particle traveling at the origin?
Problem 5: (4 parts)
(a) State Maxwell’s equations for an uncharged conductor with conductivity σ, permittivity ε, and permeability µ.
(b) Show that these equations admit decaying plane wave solutions, e.g., for complex K,
E(r,t) = E0 ei(Kz ! " t ) ,
(c) Writing K = k1 + ik2 , and using K 2 = k12 ! k22 + 2ik1k2 , find k2 .
(d) Find an expression for the skin depth ! = 1/ k2 for a poor conductor, i.e., when
! ! "# .
Problem 6: (2 parts)
The set of spectral lines for atomic hydrogen that result when an excited atom decays to
the ground state is called the Lyman series, after the spectroscopist who first observed
these lines and measured their frequencies. The ground-state energy of atomic hydrogen
is E1 = −13.61 eV, and the rest energy of atomic hydrogen is mc2 = 940.1×106 eV.
(a) Calculate the wavelength of the shortest and longest wavelength lines in the Lyman
series. What type of electromagnetic radiation does this correspond to (i.e., infrared,
visible, ultraviolet, etc.)?
(b) The frequencies in the Lyman series get closer and closer together as the principal
quantum number, n, of the excited state gets larger and larger. The observed spectral
lines in a gas of atoms are not absolutely sharp, because the individual atoms are in
motion, and therefore radiation emitted from each atom is slightly Doppler-shifted.
Assume the average speed of an atom is given by the kinetic energy 23 kBT, where kB
is the Boltzmann constant. At what energy level does the Doppler shift of the Lyman
series become comparable to the spacing between the lines, if the gas is at room temperature (T = 300 K)? What is the answer if T = 900 K?
The University of Georgia
Department of Physics and Astronomy
Graduate Qualifying Exam –– Part II
August 12, 2008
Instructions: Attempt all problems. Start each problem on a new sheet of paper. Print
your name on each sheet of paper. This is a closed-book and closed-notes exam. You
may use a calculator, but only for arithmetic functions (i.e., not for referring to notes
stored in memory, doing symbolic algebra, etc.). For full credit, you must show your
work and/or explain your answers. Part I has six problems, numbered 7–12.
Problem 7: (2 parts)
A beam of mixed red and violet light is incident normally on a diffraction grating ruled at
300 lines per millimeter. It is observed that red and violet diffraction spots overlap on the
detector at an angle of 24° with respect to the grating normal.
(a) Find the wavelengths of the red and violet light. (Hint: First estimate wavelengths for
the red and violet light, e.g., for red light ~700nm and for violet light ~400 nm).
(b) Find the angle where red and violet diffraction spots will next overlap again.
Problem 8: (1 part)
A particle at rest at the origin with mass m and charge q is subject to uniform electric and
magnetic fields, E = E ẑ and B = B ẑ . Calculate the trajectory of the particle.
Problem 9: (2 parts)
A billiard ball of mass m and radius b on a billiard table is initially spinning about a horizontal axis through its center at angular speed ! 0 , but has no initial center-of-mass motion. The coefficient of kinetic friction between the ball and the billiard table is µ .
[Note: The moment of inertia about the center of the ball is 25 mb 2 .]
(a) Find the distance the ball has traveled at the instant slipping ceases to occur.
(b) Determine the work done by friction in this interval.
Problem 10: (3 parts)
Imagine a quantum mechanical system in which there are just three linearly independent
states: 1 = (1 0 0 ) , 2 = ( 0 1 0 ) , and 3 = ( 0 0 1) . The most general state
T
T
T
2
2
2
is a normalized linear combination: s = a 1 + b 2 + c 3 , with a + b + c = 1 . In this
basis, the Hamiltonian is represented by the matrix:
& 2'
$
H =$ g
$ 0
%
g
0#
!
2' 0 ! , where g ! 0
0 3' !"
(a) What are the possible results when the energy of the system is measured? Under
what conditions can there be degenerate energy levels? What is the maximum possible degeneracy in this system?
(b) A particle is initially in the state, ! (0) =
1
(i "i i )T , find ! (t ) .
3
(c) Suppose that the energy of this system was measured and a value of E = 2" ! g was
found. Subsequently a measurement is made of an observable A described in the
same basis ( 1 , 2 , 3 ) by:
&5 0 0#
$
!
A = $0 2 i ! .
$ 0 ' i 2!
%
"
What is the expectation value of A?
Problem 11: (2 parts)
The tension f of a rubber band with an unstretched length of L0 is found experimentally
to have the following behavior under changes in temperature and length:
2
bL " " L0 % %
" !f %
=
1
(
$#
'
$ '
!T & L L0 $# # L & '&
bT
" !f %
and $ ' =
# !L & T L0
" " L0 % 2 %
$ 1 + $# L '& ' ,
#
&
where b is a positive constant.
(a) Show that df is an exact differential, and find the equation of state, f (T , L) .
(b) Compute ( !L / !T ) f , and explain its physical meaning for an elongated rubber band
under tension.
Problem 12: (2 parts)
A particle of mass m is constrained to move on a circle of radius r that lies in the x-y
plane. The external potential is zero.
(a) Use matter-wave arguments to find the energy eigenvalues of the system.
(b) Now solve the Schrödinger equation to find the energy eigenvalues.
Note Sheet for Qualifying Exam
Vector Identities: (a, b, c and d are vector fields and ψ is a scalar field)
a ! (b " c) = b ! (c " a) = c ! (a " b)
a ! (b ! c) = b(a " c) # c(a " b)
(a ! b) " (c ! d) = (a " c)(b " d) # (a " d)(b " d)
! " !# = 0
! " (! # a) = 0
2
! " (#a) = a " !# + #! " a
! " (! " a) = !(! # a) $ ! a
! " (#a) = !# " a + #! " a
! " (a # b) = b " (! # a) $ a " (! # b)
!(a " b) = (a " !)b + (b " !)a + a # (! # b) + b # (! # a)
! " (a " b) = a(! # b) $ b(! # a) + (b # !)a $ (a # !)b
Vector Differential Operators in Non-Cartesian Coordinates:
Cylindrical Coordinates: ( !, ",z)
#"
1 #"
#"
â $ +
â% +
â z
#$
$ #%
#z
1 $
1 $A% $Az
!"A =
( #A# ) +
+
# $#
# $%
$z
$ A# *
' 1 $ Az $ A% *
' $ A# $ Az *
1' $
!"A=)
&
â # + )
&
â% + ) ( # A% ) &
â z
,
,
$z +
$# +
# ( $#
$% ,+
( # $%
( $z
1 $ % $" ( 1 $ 2" $ 2"
! 2" =
+
'# * +
# $# & $# ) # 2 $+ 2 $z 2
!" =
Spherical Coordinates: (r,!, " )
#"
1 #"
1 #"
â r +
â$ +
â%
#r
r #$
r sin $ #%
1 #
1 #
1 #A%
! " A = 2 (r 2 Ar ) +
(A$ sin $ ) +
r #r
r sin $ #$
r sin $ #%
' 1 $ Ar 1 $
*
1 ' $
$A *
1' $
$A *
!"A=
(A% sin # ) & # , â r + )
&
(rA% ), â# + ) (rA# ) & r , â%
)
( r sin # $% r $ r
+
r sin # ( $#
$% +
r ( $r
$# +
(
1 #2
1
# % #"
1
# 2"
! 2" =
r
"
+
sin
$
+
'
*
(
)
) r 2 sin 2 $ #+ 2
r #r 2
r 2 sin $ #$ & #$
!" =
Solar-System Physical Data:
Earth Mass: 5.98×1024 kg
Earth Radius: 6.38×106 m
Mean Earth-Sun Distance: 1.50×1011 m
Solar Mass: 1.99×1030 kg
Moon Mass: 7.35×1022 kg
Moon Radius: 1.74×106 m
Mean Moon-Earth Distance: 3.85×108 m
Solar Radius: 6.96×108 m
Fundamental Constants:
Gravitational constant
Coulomb constant (k = 1/4πε0)
Planck’s constant (! = h /2! )
Boltzmann constant
Speed of light in vacuum
Proton charge
Proton rest mass
Electron rest mass
Avogadro’s number
G = 6.673×10–11 N·m2/kg2
k = 8.9876×109 N·m2/C2
h = 6.6262×10–34 J·s
kB = 1.3086×10–23 J/K
c = 2.9979×108 m/s
e = 1.6022×10–19 C
mp = 1.6726×10–27 kg
me = 9.1096×10–31 kg
NA = 6.0222×1023
Pauli Spin Matrices:
"0 1%
!x = $
'
#1 0&
#0 "i&
!y = %
(
$i 0'
#1 0 &
!z = %
(
$0 "1'