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THE UNIVERSITY OF SYDNEY
FACULTY OF SCIENCE
INTERMEDIATE PHYSICS
PHYS 2902 PHYSICS 2B (ADVANCED)
NOVEMBER 2003 TIME ALLOWED: 3 HOURS
ALL QUESTIONS HAVE THE VALUE SHOWN
INSTRUCTIONS:
This paper consists of 2 sections.
Section A Quantum Physics 75 marks Section B Electromagnetic Properties of Matter 75 marks
Candidates should attempt all questions.
USE A SEPARATE ANSWER BOOK FOR EACH SECTION.
In answering the questions in this paper, it is particularly important to give rea- sons for your answer. Only partial marks will be awarded for correct answers with inadequate reasons.
No written material of any kind may be taken into the examination room. Calcu- lators are permitted.
Table of constants
Avogadro’s number =
speed of light =
electronic charge = C
electron rest mass = kg
electron rest energy energy = 511 keV
electron volt 1 eV = J
protron rest mass = kg
neutron rest mass = kg
Planck’s constant = J.s
Planck’s constant (reduced) = J.s
Boltzmann’s constant =
Stefan’s constant =
Coulomb constant =
permittivity of free space =
permeability of free space =
gravitational constant =
atomic mass constant =
Please use a separate book for this section.
Answer ALL QUESTIONS in this section.
- In the Schr¨odinger formulation of quantum mechanics, any observable quan- tity ( ) is associated with an operator ( ). The observable quantity momentum ( ) enters quantum mechanics through the de Broglie relation
and the operator corresponding to the observable (considered to be one dimensional only) is
(a) Why is it necessary to use a differential operator for , for example when calculating the expectation value of?
(b) What does it mean to say that a function is an eigenfunction of some operator?
(c) Which of the following functions are eigenfunctions of the momentum operator? And what are the eigenvalues, where appropriate?
(i) ,
(ii).
(iii) ,
(iv) ,
(d) Why should the expectation value of momentum be zero for a par- ticle in a bound energy eigenstate? (13 marks)
- A proton with total energy attempts to tunnel through a rectangular potential barrier of height and width.
(a) Using a model for this system in which a steady beam of protons is incident upon the left-hand side of the barrier, draw a carefully labelled diagram to show what the absolute value of the wave function for the proton beam should look like.
(b) Explain briefly how you might estimate the reflection coefficient from your diagram.
(c) A deuteron is a particle with the same charge as a proton, but twice the mass. If a beam of deuterons, each with energy , tried to tunnel through the same barrier (with height ), would the reflection co- efficient be less than, greater than or the same as that for the beam of protons? Explain. (12 marks)
- (a) Describe briefly three of the important properties of superconductors.
(b) A simple demonstration of the Meissner effect shows a small perma- nent magnet being levitated above a disk shaped ceramic superconduc- tor, which is cooled by liquid nitrogen. Draw a diagram of the magnetic field involved, and label it to explain why the small magnet levitates.
(c) In the BCS theory of superconductivity, one of the most important ideas is that the carriers of the supercurrent are pairs of electrons (called Cooper pairs). Why, in order to explain the observations, is it important that the carriers are pairs, rather than single electrons? (13 marks)
- (a) State the Pauli Exclusion Principle, and discuss its relevance to the sta- tistical behaviour of a gas of particles which obey Fermi-Dirac statis- tics.
(b) For a gas of particles which obey Fermi-Dirac statistics, describe how the low temperature distribution of particles among the energy levels of the system differs from that of a gas of classical particles which obey Maxwell-Boltzmann statistics.
(c) The constant which appears in the standard formula for the Fermi- Dirac distribution function is known as the Fermi energy. The meaning of this quantity is often expressed by the following state- ment:
In a gas of fermions, the energy states corresponding to have a 50% probability of being occupied, at any temperature.
Demonstrate, from the formula for the Fermi-Dirac distribution func- tion, that this statement is true. (12 marks)
- (a) Draw schematically the energy level diagram for a one-dimensional “lattice” of length , consisting of a large number of simple poten- tial wells. Indicate on your diagram what is meant by the terms “energy band” and “energy gap”.
(b) Draw the shapes of the two lowest energy eigenfunctions ( ) of this system. Indicate on each what its quantum number is. (The diagrams need not be accurate nor to scale, but they should show the main features of the shape. In particular they must show clearly how features of the wave function correspond to the positions of the lattice ions.)
(c) In theoretical expositions of band theory, the wave functions are as- sumed to be Bloch waves. If the two eigenfunctions you drew in part (b) are interpreted as Bloch waves, what are the magnitudes of the wave numbers associated with each eigenfunction?
(d) In discussions of the physical properties of solids, it is said that some of the energy bands are “full”. What does this mean? Under what conditions do energy bands become “filled up”? (13 marks)
SECTION B
ELECTROMAGNETIC PROPERTIES OF MATTER
FORMULAS
- (a) Sketch a graph showing hysteresis in a ferromagnetic material. Label the axes. On the graph, indicate remanence and coercivity.
(b) Briefly define remanence and coercivity.
(c) The sketch below shows an electromagnet. The length of the iron core is and the air gap is.
(i) Assuming that the permeability of the iron core is large, use Am- pere’s Law to obtain an expression for the field in the gap in terms of the number of turns and the current in each turn.
(ii) Copy the diagram and indicate the closed loop you used when ap- plying Ampere’s Law.
(iii) If you have made any further assumptions, say what they are. (15 marks)
- Answer both of the following items. You may include equations if you wish, but do not derive them. You may answer in point form.
(a) Briefly describe the difference between diamagnetism and paramag- netism.
(b) By considering a test charge in the vicinity of a current carrying wire, and with reference to relativity, explain briefly why one observer may ascribe a force on a charged object to the Coulomb force, while another ascribes it to a magnetic force. (8 marks)
- A conductor of width and thickness is placed in a magnetic field of strength. The field is directed parallel to the axis and into the page, as indicated in the figure below. When a current is passed through the conductor in the direction, a potential difference known as the Hall voltage appears across the conductor, which can be measured between the electrodes shown on either side of the specimen.
(a) Starting from formulas given on the formula sheet, show that the mag- nitude of the Hall voltage developed across the conductor is
where is the density of charge carriers in the conductor and their individual charges. Carefully explain your working.
(b) Suppose that the magnitude of the Hall voltage for the conductor is measured to be. If the conductor is thick and mm wide, the magnetic field strength is and the current is , calculate the carrier density, assuming that each charge carrier carries the same magnitude of charge as an electron.
(c) Copy the above diagram twice, and indicate the direction of the electric field associated with the Hall voltage for the case of (i) negative charge carriers and (ii) positive charge carriers. (15 marks)
- (a) For static conditions, Ampere’s Law is given by
Maxwell found it necessary to modify this Law by adding an extra term to the equation. By considering a charging parallel plate capacitor, derive Maxwell’s modified equation which applies even in the presence of time-varying electric fields.
