COMPREHENSIV
VE EXAMIN
NATION 2014
Physicss Departmennt
University of
o Nevada, R
Reno
Classiccal Mechaniccs
Aug
gust 2014
Co
omplete any 4 out of 5 prroblems.
Each problem has
h the samee weight.
A ideal, unifform rope off constant len
ngth L is hannging, with tthe ends eveen, over a sm
mooth
1. An
nail of neglig
gible diameteer. One end of the rope iis pulled dow
wnward a veery small
s starting fr
from rest, thee rope begins to slowly sslide
distance y (aand then releeased), and so,
-w
without friction- off the nail.
n
Assum
ming the ropee does not lifft off the naiil,
a) Find an expreession for thee velocity off the rope aft
fter the end oof the rope has moved
th
hrough a (vertical) distan
nce y. Use co
onservation of energy appproach and then a
Lagrangian
L
ap
pproach and
d compare th
he answers.
b) How
H much tim
me will it tak
ke for the ro
ope to slide ccompletely ooff of the naiil?
c) Find an expreession for thee tension, T, in the rope after the endd of the ropee has moved
hrough a (vertical) distan
nce y. Check
k your resultt by using it to verify thaat the magniitude
th
of the tension
n in the rope is zero as th
he rope leavees the nail.
HINT: Place
P
the zerro point of gravitational potential energy at the m
midpoint of tthe rope (as it
exists
e
when the two endss are even).
y
2. Suppose we have a particle moving in a central force potential. Consider the following
vector, here given in units of m  1 ,
  
A  p  l  krˆ ,


where p is the particle’s linear momentum, l its angular momentum with respect to the
origin (force center), and r̂ is the radial unit vector pointing from the origin (force center) to
 
the particle. Note that p  r in units of m  1 . For a central potential, V  V r  , find a


general expression for the first time derivative of A , that is, find A . Use your answer to

show that A is a constant of the motion for the specific central potential V   k r .





  
 

HINT: The vector formula a  b  c  a  c b  a  b c may be useful.
3.
a) Show that the following transformation,
Q   p,
P  q  Ap 2
(where A is any constant) is a canonical transformation,
i) by evaluating the Poisson bracket Q, PPB
ii) by expressing pdq  PdQ as an exact differential dF1 q, Q  . WARNING: to do this,
you must first use the transformation equations to express p  pq, Q  and
P  Pq, Q  . Hence find the generating function F  F1 q, Q  that generates this
transformation.
b) Write down the Hamiltonian, H q, p  , for a particle moving vertically in a uniform
~
gravitational field. Using the given transformation, find the Hamiltonian H Q , P  .
Show that we can make Q cyclic by choosing an appropriate value for the constant A.
c) With this choice of A, write down and solve Hamilton’s equations for the new canonical
variables, and then use the transformation equations to find equations for qt  and pt  .
Identify all constants produced.
4. Consider
C
a freee fall from the great alttitude: a freee fall of the bbody from itss original
position r > R (R is the radius of the Earth), wheere v = 0.
a) Find the exprression for veelocity v = v(r).
v What iss the velocityy at surface oof the Earth in
he limit r
?
th
b) Find
F
the exprression for tiime t = t(r) to
t reach the E
Earth.
HINT: In
ntroduce thee distance s, at which thee body has faallen from itts original poosition.
D
the n-body
n
prob
blem in classsical mechannics in detail. Illustrate yyour answer w
with
5. Describe
ex
xamples of twot
or threee-body probllems.
Classical Mechanics Supplement – August 2014
Newton’s second law
 dp
F
dt
Lagrangian
L  T V
Hamiltonian
H   pi qi  L
i
Hamilton’s equations
H
p
H
p  
q
q 
Euler- Lagrange equation
L d  L 
  0
q dt  q 
Poisson bracket
Q, PPB  Q P  Q P
q p
p q
Comprehensive Examination EM (2014 summer)
Department of Physics
Electricity and Magnetism
Answer any four problems. Do not turn in solutions for more than four problems.
Each problem has the same weight.
1: Particles of mass M are singly ionized in an ion source Q and accelerated by the
voltage U . They are entering the magnetic field B through a slit S perpendicular to
the plane of paper, as in Fig. 1. (a) What is the velocity of the particles at the slit
S? (b) Where do they hit the photoplate? (c) Where do the particles entering the
magnetic field at an angle α 1 with respect to the axis hit the photoplate? (e)
How can the mass of the particles be determined by this arrangement?
Fig. 1
2: Let a point charge q be at a distance a in front of an infinitely extending conducting
wall. (a) Obtain the electric field normal to the conducting wall. (b) What charge
density will be induced at the wall? (c) What is the magnitude of the total charge of
the plane?
Fig. 1
1
Comprehensive Examination EM (2014 summer)
Department of Physics
3: Consider a moving charge q in X-direction with velocity v in K-frame (lab frame).
There is another observation frame K 0 , which is comoving with the charge, where the
charge is at rest. (a) Obtain the electric field (Ex0 , Ey0 , Ez0 ) and the magnetic field (Bx0 ,
By0 , Bz0 ) in K 0 -frame. (b) Obtain the electric field (Ex , Ey , Ez ) and the magnetic field
(Bx , By , Bz ) in the K-frame by the Lorentz transformation. (c) Use the obtained
electric field, illustrate the electric fields from the moving charge when the velocity v
is v c and when it is close to the speed of light.
4: A transmission line consists of two identical thin strips of metal, of width b and
separated by a distance a. Assume that b a, and neglect edge effects.
a) Is it possible to propagate a TEM mode on this line? Explain why.
b) Work out the electric E and magnetic H fields associated with the TEM mode.
c) Calculate the net flow of power P .
d) Find out the attenuation constant.
e) Find out the impedance of the line.
f) Find out the series resistance per unit of length.
g) Find out the inductance per unit of length.
5: Consider a circular loop antenna of radius a located on the z = 0 plane that carries
an AC current given by the real part of I(t) = I0 eit .
a) Calculate the potential vector in the radiation zone.
b) Find out the electric and magnetic fields in the radiation zone.
c) In the limit λ a show that the potential and fields in the radiation zone
become those of the magnetic dipole moment of the current distribution.
2
Comprehensive Examination EM (2014 summer)
Department of Physics
Supplements of Electricity and Magnetism
Constant parameters
• Electric permitivity of free space 0 = 8.854 × 10−12 (mks) or 1/4π (cgs)
• Magnetic permeability of free space µ0 = 4π × 10−7 (mks) or 4π/c2 (cgs)
• Electron charge e = 1.6 × 10−19 [C] or 4.8 × 10−10 [esu]
• Electron mass m = 0.91 × 10−30 [kg] or 0.91 × 10−27 [g]
Maxwell equations
MKS
∇·D=ρ
cgs
∇ · D = 4πρ
(Coulomb0 s law)
∇×E+
∂B
=0
∂t
∇×E+
1 ∂B
=0
c ∂t
∇×H−
∂D
=J
∂t
∇×H−
1 ∂D
4π
=
J (Ampere − Maxwell0 s law)
c ∂t
c
∇·B=0
(Faraday0 s law)
∇·B=0
(Absence of free magnetic poles)
The time averaged poynting vector
1
S̄ = Ē × H̄
2
The net flow power of the electromagnetic fields
P =
Z
S̄ · da
Biot-Savart’s law
Magnetic field δB from a small current δI is
MKS
cgs
0
δB =
µ0 δI × (x − x )
,
4π |x − x0 |3
δB =
1 δI × (x − x0 )
,
c |x − x0 |3
here x’ is the location of the current and x is the observation point.
The retarded vector potential
A(x, t) =
µ0 Z 3 0 Z 0 J(x0 , t0 ) 0 |x − x0 |
d x dt
δ(t +
− t)
4π
|x − x0 |
c
3
Comprehensive Examination EM (2014 summer)
Department of Physics
A plane wave and the refractive index
A plane wave has the following relation between electric field and magnetic field,
H
=
E
and the refractive index is given as n =
permitivity (magnetic permeability).
√
s
,
µ
∗ µ∗ , here ∗ (µ∗ ) is the ratio of the electric
Lorentz transformation
x0µ = Λµν xν
4-dimensional vectors and tensors
• Lorentz transformation
to the frame K 0 moving in X-direction with
√ matrix
velocity β and γ = 1/ 1 − β 2 :




Λµν = 
γ −iγβ 0 0
iγβ
γ
0 0
0
0
1 0
0
0
0 1





• Space and time: xµ = (ict, x, y, z)
• Charge and current: jµ = (icρ, jx , jy , jz )
• Potential: Aµ = (icφ, Ax , Ay , Az )
• Velocity wµ = (icγ, γvx , γvy , γvz )
• Lorentz transformation formula of E and B fields (transferred by Λµν ).
Ex0 = Ex
Bx0 = Bx
0
Ey = γ(Ey − βBz ) By0 = γ(By + βEz )
Ez0 = γ(Ez + βBy ) Bz0 = γ(Bz − βEy )
4
Comprehensive Examination
Physics Department
University of Nevada, Reno
Quantum Theory
August 2014
Complete 4 out of 5 problems. Each problem has the same weight.
1.) Consider a particle bound in a double-well potential. We will approximate the system as a two-level
problem, where we let |L> and |R> represent orthogonal state vectors for the particle being in the left and
right well, respectively. Suppose the Hamiltonian for the system is H   | L  R |  | R  L | where
 is a positive real constant.
a.) Find the energy eigenstates and eigenvalues.
b.) If the initial normalized state of the system is |  (t  0)    | L    | R  , where  and  are
complex numbers, what is the probability for observing the particle on the right side of the
potential at time t?
c.) Suppose instead the Hamiltonian is H   | R  L | . Is this a valid Hamiltonian? Why or why
not?
2.) Consider three identical spin-1/2 particles which are bound in a central potential and interact with
each other weakly. Assume the spatial component of the state vector is completely antisymmetric under
the exchange of any pair of two particles. The spin component of the state can be expressed in the basis
of eigenkets | m1 , m2 , m3  of S1z, S2z, and S3z, where m1,m2, and m3 can be -1/2 or 1/2. Here e.g.
S1z | m1 , m2 , m3   m1 | m1 , m2 , m3  .
a.) Is it possible to construct a normalized spin state of the system for two of the particles having m = 1/2
and one having m = -1/2? If so, construct one, or else explain why this is not possible.
b.) Construct all possible normalized spin states of the system for all three of the particles having the
same m values.
c.) For the states given in (a) and (b), what results are possible for a measurement of the z-component of




the total spin S  S1  S 2  S 3 ?
d.) Write all possible spin states for the system assuming instead that the spatial component of the state
vector is symmetric under the exchange of any pair of two particles.
3). Consider a system of three spin-1/2 particles. Assume the initial state of the system is given by
|  


  

1
|  4i |  | . Let the total spin angular momentum be S  S1  S 2  S3 .
18
a.) If S 3 z is measured, what results are possible and with what probability?
b.) If instead S z is measured, what results are possible and with what probability?
c.) Suppose instead S 3 y is measured, what results are possible and with what probability?
4.) Consider a four-state system with state kets | 1 , | 2 , | 3 , | 4  and unperturbed Hamiltonian
0
0
H0  
0

0
0 0
0 0
0 E
0 0
0
0 
.
0

E
Consider adding a time-independent perturbation to this system of the form
0 
 0
W 
0 

0 0
0


0
0
0
. You may assume |  |, |  |  | E | .
0

0
a.) What is the degeneracy of each unperturbed energy level?
nd
b.) Use perturbation theory to find the energy shifts due to this perturbation up to the 2 order.
c.) Find the “correct” zero-order kets to which the perturbed kets reduce to in the limit that
 ,   0.
5.) Consider a one-dimensional simple harmonic oscillator with potential V ( x) 
1
m 2 xˆ 2 .
2
a.) Let |n> denote the energy eigenstates. What are the energy eigenvalues?
2

b.) Consider an initial state at t=0 given by |    exp[ |  | / 2] exp[ aˆ ]| 0  where  is a complex
number. What is the state of the system at later times t?
 Axˆ , t  0
 , where A is a small positive constant,
 0, t  0 
c.) Now add a time dependent perturbation W (t )  
so that matrix elements of W can be treated as small when compared with matrix elements of V. Assume
the initial state of the system for t < 0 is the ground state of the harmonic oscillator.
Using 1st order time-dependent perturbation theory, what is the probability for the oscillator to remain in
the ground state as a function of time t >0 ?
Quantum Theory Supplement - August 2014
Schrodinger’s Equation:
i
∂ψ
= Hψ
∂t
Hamiltonian:
H =−
2 2
∇ +V
2m
mω
2
i


pˆ 
 xˆ +
mω 

Raising and lowering operators:
aˆ =
aˆ + =
mω
2
aˆ | n > =
i


pˆ 
 xˆ −
mω 

n | n −1 >
aˆ + | n > = n + 1 | n + 1 >
Angular Momentum:
J ± | j , m > =  j ( j + 1) − m(m ± 1) | j , m ± 1 >
Comprehensive Examination
Physics Department
University of Nevada, Reno
Statistical Mechanics
August 2014
Complete any 4 out of 5 problems.
Each problem has the same weight.
1. Particles in a magnetic field.
When a particle with spin
1
is placed in a magnetic field H, its energy level is split into
2
  H and   H . Suppose a system consisting of N such particles is in a magnetic field H
and is kept at temperature T.
a) Find the partition function and the Helmholtz free energy F with the help of the canonical
distribution.
b) Find the entropy S and internal energy U using the results from (a). What is the entropy in
two limiting cases of T0 and T? Explain your answer.
c) Find the total magnetic momentum M of this system with the help of the canonical
distribution. What is a relation between U and M? Explain your answer.
Hint: M  
F
.
H
d) Find the heat capacity CH and sketch it as a function of (kBT/H). Comment on the graph.
Hint: C H  (
U
)H .
T
2. Formal thermodynamic manipulations.
From the fundamental thermodynamic relation show that
(
C P
 2V
) T , N  T ( 2 ) P , N
P
T
(1)
where V , N , P, and C P are volume, number of particles, pressure, and heat capacity at the
constant pressure.
Hint: The fundamental thermodynamic relation is dU  TdS  PdV   dN , so the
independent variables are V , N , S .To prove the desired expression you need to
define the thermodynamic potential that has independent variables of interest in (1)
and to use a Maxwell relation approach
1
3. Ideal gas.
An ideal gas consisting of N particles of mass m (classical statistics being obeyed) is enclosed in
an infinitely tall cylindrical container of a cross section  placed in a uniform gravitational field,
and is in the thermal equilibrium.
a) Find a classical partition function of this system.
b) Calculate the Helmholtz free energy and mean energy of the system and compare it with
ideal gas results.
c) Calculate the heat capacity at constant volume of this system and compare it with the
result for an ideal gas.
Hint: the translational Hamiltonian per particle is given by
H trans 
p x2  p y2  p z2
2m
 mgz.
4. Chain.
There is a one-dimensional chain consisting of N elements (N>>1), as is seen in the figure.
Let the length of each element be a and the distance between the end points x.
a) Find the entropy of this chain as a function of x.
b) Obtain the relation between the temperature T of the chain and the force (tension) X
which is necessary to maintain the distance x, assuming the joints to turn freely.
c) For x << Na, what would be the tension X?
Hint: in order to specify a possible configuration of the chain, you may consider indicating
successively, starting from the left end, whether each consecutive element is directed to the
right (+) or to the left (-). For example, in the case shown in the figure, we have (+ + - + +
+ - - - + + - + + +).
2
5. Quantum gas.
a) Prove that in the non-relativistic case of 3-D Fermi gas, the Fermi energy is
6 2 n 2 / 3  2
)
F  (
g
2m
(2)
where n is a particle density and g is a weight factor arising from the „internal structure“ (for
example, g=2 for electrons).
Hint: you may consider using the following expression of the total number of particles N
through the density of states a ( ) :
F
 a ( )d  N .
0
b) How will the expression for the Fermi energy shown in (2) change for the case of 2-D
Fermi gas?
c) Obtain the numerical estimates of the Fermi energy (in eV) and the Fermi temperature (in
K) for the electron gas in the interior of white dwarf stars with n=1030 cm-3. Are the
electron energies in the relativistic regime?
d) Obtain the numerical estimates of the Fermi energy (in eV) and the Fermi temperature (in
K) for the conduction electrons in silver with the concentration of atoms 5.76x1022 cm-3.
Compare to the results for the electron gas in the interior of white dwarf stars (c).
3
Statistical Mechanics Supplement – August 2014
Thermodynamic potentials
-TS
U
F
+PV
G
H
Stirling’s approximation for large N
ln( N !)  N ln( N )  N .
Multiplicity, entropy, partition functions, and chemical potential
Z   exp( i /  ),
S  k B ln ,
 X 
X
i
  1 /(k BT )  1 / 
ZN 
1
N!h3N
 U  [
 dq dq
3
1
   exp[  ( N i    i )]
i
3
2
 ln Z
]V , N

r
e   Er
r
Z
F  k B T ln Z
...dq N3  dp13 dp 23 ...dp N3 e  H
 (T ,V , N )  (
F
) T ,V
N
 (T , P, N )  (
G
)T ,P
N
Ideal gas
Z
VN
h2
 h2 1/ 2
1/ 2
;

(
)

(
)

N !3 N
2mk BT
2 m
Quantum gas
 nj 
1
e
 ( j   )
1
4
Fundamental Physical Constants
Name
Symbol
Speed of light
c
Planck constant
h
Planck constant
h
Value
Planck hbar
Planck hbar
Gravitation constant
G
Boltzmann constant
k
Boltzmann constant
k
Molar gas constant
R
Avogadro's number
NA
Charge of electron
e
6.0221 x 1023 mol-1
Permeability of vacuum
Permittivity of vacuum
Coulomb constant
Faraday constant
F
Mass of electron
Mass of electron
Mass of proton
Mass of proton
Mass of neutron
Mass of neutron
Atomic mass unit
u
5
Atomic mass unit
u
Avogadro's number
Stefan-Boltzmann constant
Rydberg constant
Bohr magneton
Bohr magneton
Flux quantum
Bohr radius
Standard atmosphere
Wien displacement constant
atm
b
6