The University
of Georgia
Department of Physics and Astronomy
Graduate Qualifying Exam –– Part II
Instructions: Attempt all problems. Start each problem on a new sheet of paper and use one side only. Print your name on each piece of paper that you submit. This is a closed-book and closed-notes exam. You may use a calculator, but only for arithmetic functions (i.e., not for to referring notes stored in memory, doing symbolic algebra, etc.). For full credit, you must show your work and/or explain your answers. Part II has six problems, numbered 7–12.
Problem 7: (3 parts)
A double pendulum consists of two simple pendulums, with one pendulum suspended from the bob of the other. The lengths of the two pendulums are l1 and l2, respectively, and the masses of their bobs are m1 and m2, respectively. The pendulums are confined to swing in the same fixed vertical plane.
(a) Let θ1 and θ2 be the angles that the two pendulums make relative to vertical, respectively. Determine the Lagrangian of the system in terms of θ1 and θ2.
(b) Determine the equations of motion for the system.
(c) Write the equations of motion for the case where m1 = m2, l1 = l2, and the oscillations are assumed to be small.
Problem 8: (2 parts)
Suppose two spin-1/2 particles obey the following Hamiltonian:
,
where denotes the spin operator of the ith particle (i = 1, 2),
is the z-component
of
,
and A and B are positive constants.
(a) Write down the singlet and triplet states of this system. Determine whether these states are stationary states.
(b) Find the ground- and excited-state energies and eigenstates of this system.
Problem 9: (1 part)
The triangular glass prism shown below has an apex angle of 60.0° and an index of refraction of 1.50. The prism is surrounded by air, which has an index of refraction of 1.00. A ray of light coming from the left hits the left face of the prism. What is the smallest angle of incidence, θ1, for which the ray of light can emerge from the right face of the prism?
Problem 10: (4 parts)
A metal bar of mass m slides without friction on two parallel conducting rails a distance l apart, as indicated in the figure below. A resistor with resistance R is connected across the rail, and a uniform magnetic field B pointing perpendicularly out of the page fills the entire region. Ignore gravity.
(a) If the bar moves to the right at speed v, determine the current (magnitude and direction of flow) in the resistor.
(b) Determine the magnetic force, Fm, on the bar.
(c) If the bar is launched to the right with initial speed v0 at time t = 0, determine its speed at a later time, t.
(d)
Show that, by the time the bar comes to rest at t = ∞, the total energy dissipated by the resistor
is ;
i.e., exactly the initial kinetic
energy of the bar.
Problem 11: (2 parts)
One of the tests of special relativity is to compare decay
lifetimes for particles at rest with similar particles moving at high
speeds. Suppose a meson is used for the experiment. At rest, the
meson has a lifetime of 18.0 nanoseconds. Consider an experiment in which you propel a
meson at a speed of 99.0% of the speed of
light. How long do you expect the
meson to live …
(a) in your reference frame?
(b)
in the reference frame of the meson?
Problem 12: (4 parts)
This problem considers a 4d electron in atomic hydrogen:
(a)
List all total angular momentum states, .
(b)
Determine the values of: L, the magnitude of the orbital angular momentum; S, the magnitude of the spin angular
momentum; and Sz, the z-projection of spin. Give your answers in units of .
(c)
For the j
= 3/2, mj = 1/2 state,
determine the values of J, the
magnitude of total angular momentum, and Jz,
the z-projection of total angular
momentum. Give your answers in units of .
(d) Draw a vector diagram showing approximately all possible directions of J for j = 3/2.