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PHYSICS Ph.D. Qualifying Examination
September 2005
Analytical Mechanics (Solve only 2 questions out of 3)
1) A bead of mass m slides without friction along a straight wire that is rotating with
constant angular velocity ω about a vertical axis. The wire makes a fixed angle θ 0 with
the rotation axis. There is a uniform downward gravitational force mg on the bead.
a) Construct the Lagrangian for the bead using as generalized coordinate the distance
s measured along the wire from the point of intersection with the rotation axis.
b) Obtain the Lagrange equation of motion for s and use it to show that the condition
for an equilibrium circular orbit is s0 = g cos θ 0 /(ω sin θ 0 ) 2 .
c) Derive the above result for s0 by directly applying Newton's second law of motion
G
G
( F = ma ) in an inertial frame.
d) Disuss the stability of this orbit against small displacements along the wire by
finding an expression for η (t ) = s (t ) − s0 .
e) Find H, the Hamiltonian of the bead, in terms of a suitable coordinate and
momentum.
2) A particle of mass m is constrained to move under the influence of gravity on the
smooth inner surface of the paraboloid of revolution x2+y2=az, where a is a constant.
Obtain the equations of motion. Show that the angular momentum of the particle about
the axis of symmetry of the system is conserved.
3) A homogeneous disk of radius R and mass M rolls without slipping on a horizontal
surface. The disk’s center is attracted to a point a distance d below the plane. If the force
of attraction is proportional to the distance from the disk’s center of mass to the force
center, find the frequency of oscillation around the position of equilibrium.
Freshman Physics (Solve only 1 question out of 2)
1) A small wooden block, of mass M, lies in the middle of a horizontal table of length L
and height h above the floor. The coefficient of friction between this block and the
surface of the table is µ. A bullet, of mass m, is shot with a horizontal velocity into the
block. As a result, the block (with the embedded bullet) starts moving across the table
and finally lands on the floor a horizontal distance D from the edge of the table. Use this
information to find the speed of the bullet.
2) Calculate the mutual inductance between a very long, straight wire and a conducting
circular loop of radius b with its center d far away from the straight wire as shown in the
figure below in terms of the givens (b and d).
Figure: A long, straight wire and a conducting circular loop of radius b with its center d
away from the straight wire.
Modern Physics (Solve only 1 question out of 2)
1) Many particles in nature are bound to a location in space with a harmonic oscillatorlike force. Suppose that a particle of mass m is inside a potential of the form
1
V (r ) = kr 2
2
where k is a constant and r is the distance from the origin.
a) What is the force attracting this particle to the origin?
b) According to the Bohr theory, what are the allowed values of the radius rn and the
total energy En ? (Restrict yourself to circular orbits.)
c) What is the analog of the Rydberg formula? Show that the correspondence principle is
satisfied for all values of the quantum number n.
2) A beam of atoms emerges from an oven that is at a temperature T. The distribution of
the speeds of the atoms in the beam is proportional to v3exp(-mv2/2kT).
(a) Find the distribution of de Broglie wavelengths of the atoms,
(b) the most probable de Broglie wavelength.
Quantum Mechanics (Solve only 3 questions out of 4)
1) Consider a hydrogen atom in a uniform external electric field ε z (the field direction
defines the z-axis). Neglect the spin of the electron and proton so that the Hamiltonian
can be approximated as H = H 0 + H1 where
GG
p 2 e2
−
and
H1 = eε .r
2µ r
Assume that ε z is weak in the sense that H1 H 0 . The eigenstates of H 0 are ψ nlm and
H0 =
eigenenergies are Enlm . You DO NOT NEED to evaluate the matrix elements explicitly
in this problem, but simply put them in the form < ψ nlmϑ | ψ n 'l ' m ' > where ϑ is the
relevant operator.
a) Give the first order expression for the shift in the ground state energy due to the
electric field and then explain why this linear Stark effect vanishes.
b) Derive an expression for the second order shift in energy of the ground state.
c) Consider the n=2 l=0 first excited state. Suppose that ε = 0 . Explain why this state
cannot decay by emitting a single electric dipole photon.
d) Consider the leading order effect of ε on the n=2 l=0 state. First state the n,l,m values
of all states that are admixed into the n=2, l=0 state with nonvanishing amplitudes. Then
restrict yourself to admixed n=2 state and find the eigenstates in the presence of ε z if the
splitting of the unperturbed n=2, l=0 and n=2, l=1 states can be neglected. Finally give
the energy shifts of these levels.
2) Consider an electron moving in a one-dimensional infinite lattice in the tight binding
approximation. The electron is allowed to exist only at certain equally spaced sites on
the lattice which are labelled by integers. The allowed states of the electron are therefore
denoted by
| n >,
n = 0, ±1, ±2,..., ±∞
The matrix elements of the Hamiltonian for the system are given
< n | H | n ' >= E0δ n ,n ' − ∆δ n ,n +1 − ∆δ n ,n −1
In the absence of tunneling, ∆ = 0 and the energy of the electron is E0 for any site on the
lattice. Tunnelling allows the electron to hop from one site to the next. This hopping is
modelled by the terms proportional to ∆ .
Consider the operator T(i) which is defined by
T (i ) | n >=| n + i >
where i is an integer.
a) Show that T(i) is unitary.
b) Show that [T(i),H] = 0.
c) What is the physical interpretation of the operator T(i), i.e. what symmetry of the
Hamiltonian does it correspond to?
d) Find the energy eigenvalues and eigenfunctions of H. Is the spectrum continuous or
discrete?
3) In 3 dimensions, a particle of mass m is bound in a potential
V (r ) =
−e 2
x2 + z 2
a) The definite energy states will, of course, be eigenfunctions of H. What other operators
can they be eigenfunctions of?
b) Simplify the three dimensional Schrodinger equation by using these operators.
G
4) Let Si , i = 1, 2 denote the spin vectors of two spin-1/2 particles. The interaction
is given by
G G
H = V0 ( S1.S2 − 3S1z S2 z )
Find the energy eigenstates and eigenvalues.
Statistical Mechanics (Solve only 2 questions out of 3)
1) Consider a system of N classical harmonic oscillators of the same frequency ν .
a) Find the number of states
b) Find a relation between the energy and the temperature of the system.
Hint: Volume of a n-dimensional hypersphere of radius R is
Vn =
π n/2
n
Γ( + 1)
2
Rn
2) Particles of mass m, and charge q, are emitted from an oven of temperature T. These
particles enter a mass spectrometer, where they experience a magnetic field B and hit the
florescent screen as shown in the figure. The brightness of a point on the screen is
directly proportional to the number of particles per unit time hitting that point.
Find: 1) The distance between the brightest point on the screen and the entry point.
2) The brightness distribution as a function distance from the entry point.
B
T
d
3) A particular quantum system has two different energy levels, the ground energy
E0 = 0 and the excited energy E1 . The ground state is unique (i.e. has degeneracy of one),
and the excited state has degeneracy n (i.e. there are n distinguishable states of energy
E1 ). At a temperature of T:
a) Give the free energy of the system.
b) Give the probability that the excited state will be occupied.
c) Give the average energy of the system.
d) Give the entropy of the system. Calculate or state the values as T → 0 and T → ∞ .
Explain why these results are reasonable.
Experimental Physics (Solve only 1 question out of 2)
1) State the techniques of temperature measurement from µKelvins to thousands of
Kelvins and their range of applicability.
2) The classical method for measuring the polarization of a particle such as a proton or
neutron is to scatter it from a suitable analyzing target and to measure the asymmetry in
the scattered particle distribution. One can, for example, count the number of particles
scattered to the left of the beam at certain angle and to the right of the beam at the same
corresponding angle. If R is the number scattered to the right and L is the number
scattered to the left, the asymmetry is given by ε=(R-L)/(R+L). Calculate the error on ε as
a function of the counts R and L. [Hint: Since the measurements are independent, the
covariance is zero, the counts R and L obey the Poisson distribution.]
Mathematical Physics (Solve only 2 questions out of 3)
1) Calculate the following series
1+
1 1 1
+ + + ...
32 52 7 2
by using the Parseval identity
∞
∫
f 2 ( x)dx =
−∞
π
2
∞
∑| f
n =1
n
|2
− kx
2) Expand e in terms of Laguerre polynomials Ln ( x) and find the expression for all
expansion coefficients.
Laguerre Polynomials  Lνn=0 ( x) ≡ Ln ( x)  :
∞
−x
∫ e Ln ( x) Lm ( x)dx = δ nm ,
Lm ( x) =
0
∞
Other information:
∫x
0
m − ax
e
dx =
ex d m m − x
 x e 
m ! dx m 
n!
a n +1
3) State the type of the following integral equation and solve it using the Neumann series
(i.e. iterative) approach. Also, determine the closed form expression for your solution
(that is to say, sum the infinite series).
x
φ ( x) = 1 − 2 ∫ dt t φ (t ).
0
Electromagnetic Theory (Solve only 2 questions out of 3)
1) A metal ring has radius r and total resistance R. A magnetic dipole µ which points
along the symmetry axis of the ring is moved with constant velocity v from - ∞ to ∞,
passing through the center of the ring. Find the total work done to move the magnetic
dipole. Hint: The magnetic field generated by a magnetic dipole is given (in CGS) as
G 3( µG.rˆ)rˆ − µG
B=
r3
(4u 2 − 1) 15
also given
du
∫ (1 + u 2 )5 = 128 π
R
r
v
µ
2) Show that in cylindrical coordinates ρ , φ , z Maxwell’s equations for empty space
have a solution
∂ 2ψ
H ρ = 0,
Hφ =
,
Hz = 0
∂ρ∂t
Eρ = −
∂ 2ψ
,
∂ρ∂z
Eφ =
1 ∂  ∂ψ
ρ
ρ ∂ρ  ∂ρ
and find the differential equation satisfied byψ .

,

Ez = 0
G
3) Consider a uniform charge distribution ρ (r ) of net charge Q confined to a loop of
radius a in the x-y plane with negligible wire thickness. In sperical coordinates the charge
density is
Q
G
ρ (r ) =
δ (cos θ )δ (r − a)
2π a 2
G
Determine the first two non-vanishing terms of the expansion of the potential Φ (r ) in
Legendre polynomials Pl (cos θ ) and powers of r l for short distance r a .
Hint: First determine the potential along the z-axis and then match; the leading Legendre
polynomials are
P0 ( x) = 1
P1 ( x) = x
P2 ( x) = (3 x 2 − 1) / 2