Download The University of Georgia Department of Physics and Astronomy January 6, 2005

The University of Georgia
Department of Physics and Astronomy
Graduate Qualifying Exam –– Part I
January 6, 2005
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 closedbook 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: (1 part)
In a lab class, your lab instructor gives you a sealed box with a terminal at either end.
You are told that the box contains two unknown circuit elements –– one a resistor and the
other either an uncharged capacitor or an inductor –– that are connected either in series
or in parallel across the terminals. At your lab bench you have a 20-V battery, a 100-Ω
resistor, a switch, a voltmeter, and some wire. To determine the contents of the box, you
build the circuit shown below, close the switch at time t = 0, and take the data shown in
the graph. Based on the data, determine as much information as you can about the two
circuit elements in the sealed box and how they are connected to the terminals. Draw a
diagram of the circuit contained in the sealed box. Assume all circuit elements are ideal.
V
20
?
18
100 Ω
16
20.0 V
0
20
40
Time (s)
60
80
Problem 2: (3 parts)
A harmonic oscillator in one dimension is described by the Hamiltonian
h2 d 2 1 2
Hˆ = −
+ kx ,
2m dx 2 2
where m is the mass and k is the Hooke constant.
€
(a) Given that the ground-state
wave function is symmetric in x and centered on x = 0,
construct this wave function. From this determine the wave function for the first
excited state.
(b) Sketch plots of these wave functions.
(c) What are the energies of these wave functions?
Problem 3: (2 parts)
Imagine two concentric, conducting cylinders of radius a and b, respectively, where
a < b. A current I is sent down the inner cylinder and returns along the outer cylinder.
(a) How much magnetic energy does this nested set of cylinders store per unit length?
(b) What is the self-inductance per unit length of this configuration of conductors?
Problem 4: (4 parts)
Let’s consider the classical limit of quantum mechanics through the application of the
Heisenberg Uncertainty Principle. Consider a “stationary” 1-mg grain of sand which is
found to be at a given location within an uncertainty of 550 nm.
(a) What is the minimum uncertainty in the velocity of the sand grain?
(b) How long will it take the sand grain to travel a distance of 1 µm if it moved at a speed
corresponding to the uncertainty in part (a)? Give your answer in years.
(c) Discuss whether classical mechanics is applicable to this situation.
(d) Suppose that we can reasonably measure the velocity of the sand grain to be zero with
an uncertainty of 1 mm/century. What is its de Broglie wavelength and will it behave
as a wave or a particle?
Problem 5: (3 parts)
(a) Write down the differential form of the Maxwell Equations in vacuum.
(b) Derive the wave equations for both the electric field E and the magnetic field B for
electromagnetic waves propagating in vacuum in the absence of charges and currents.
(c) From the resulting wave equations, obtain a relation for the wave speed in terms of
the permittivity and permeability of free space.
Problem 6: (4 parts)
Suppose a particle of mass m is in a bound state, ψ (x) , of some Hamiltonian in one
dimension, where
ψ (x)
€ =
A
2
x + a2
and a is a characteristic length of the system. Assume the potential V(x) vanishes at
infinity.
€
(a)
(b)
(c)
(d)
Determine A in terms of given constants.
Determine V(x) and sketch V(x) vs. x.
Determine the energy of this bound state.
Does this Hamiltonian have bound states of lower energy or higher energy? Explain.