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University of the Witwatersrand
Physics IIE (Engineering) : PHYS284
Examination : June 2005

Instructions: Answer all questions.
Time: 2 hours = 120 minutes
Total Marks: 120 marks (105 marks = 100%)
 Given that the Lorentz Transformation is :
derive the formula for the relativistic time dilation
(6)
 A spacecraft is moving relative to the earth. An observer on earth finds that,
according to her clock, 3601s elapse between 1pm and 2pm on the spacecraft's clock.
What is the spacecraft's speed relative to the earth ?
(6)
 Show that the relativistic velocity transformation is
(5)
 Two spaceships, A and B, are approaching each other from opposite directions.
An observer on earth measures their velocities to be 0.750c and 0.850c respectively.
What is their actual relative velocity, as would be measured by an observer
in either ship.
(5)
[22]
 For a particle trapped in a one dimensional box of width :
 Find the average value of the position,
.
(You may need :
.)
(6)
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PHYS284
June 2005 Physics IIE (Engineering)
 []
 [(b)] Show that the average value of the momentum,
is zero, noting that the momentum operator is
.
(You may need :
.)
(4)
 [(c)] Explain why this is a logical result.
(2)
 [(d)] Treat a quantum dot of dimension 30 nm as an electron in a one dimensional box. The dot
is manufactured by using appropriate doping and deposition of contacts on a slab of GaAs
and then applying a bias voltage in such a way as to create a linear confining potential.
Find the minimum energy of the electron. Work in units of eV for energy and use eV.nm.
(Note : for GaAs,
.
(4)
 [(e)] What is the consequence of this result if you
require to operate the device in its ground state ?
(2)
[18]
 [3.]
 The probability of finding an electron, of a given state
of the hydrogen atom, at some distance between and is given by
How much more likely is a electron to be at a distance of
from the nucleus than a distance of ?
(Use the attached table in figure 1
below of the normalised wave functions for the hydrogen atom.)
(8)
 Unsöld's theorem states that for any value of the orbital quantum number ,
the probability densities summed over all possible values of
are angle independent.
 Verify this for by showing
using the attached table in figure 1
below of the normalised wave functions for the hydrogen atom.
(6)
 What is the physical meaning of this mathematical fact.
(2)
 Which group of the periodic table contains elements demonstrating Unsöld's theorem.
(2)
 How does the mathematical form of the probability density illuminate the properties
of these elements ?
(2)
[20]
 [4.]
 What are the differences between bosons and fermions quantum mechanically.
Give one example of each.
(4)
 Describe the behaviour of a system of identical bosons and fermions
respectively at very low temperatures.
(4)
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PHYS284
June 2005 Physics IIE (Engineering)
 []
 [(b)]The density of metallic zinc is 7.13 g/cm and the atomic mass
of the zinc atom is 65.4 u.
The Fermi energy in zinc metal is 11.0 eV.
Work out the effective mass of a delocalised
electron in Zinc metal. Express your answer in terms of the free electron mass
and say why there is a difference.
(Zinc has 2 valence electrons.)
(6)
 [(c)]Show for the FermiDirac distribution that if the
average occupancy of a state of energy
is at any temperature, then the average occupancy of a state of energy
is .
(6)
 [(d)]What is the implication of this for charge carriers in intrinsic material ?
(2)
[22]
 [5.]
 A semiconductor is characterised by the energy band structure.
Draw appropriate structures for :
 An insulator,
 An intrinsic semiconductor,
 A ptype doped semiconductor,
 A metal.
(4)
 Use the expression for the group velocity associated with an
electron in a solid as well as Newtons third Law to
derive a relationship for the effective mass of the electron
in terms of energy and wavenumber.
(4)
 If the energy dependence of an electron in the valance band
is given by
where is a
positive constant and is the lattice spacing,
 Sketch the graphs of electron energy, , the electron velocity
and the effective mass each as a function of for the
valence band from the region
,
(4)
 Determine the hole effective mass at
in terms of and .
(4)
 What would be the effect of effective mass of the charge carrier on
 The appearance of the tunneling limit to miniaturisation.
 The level of the state in the gap.
 The frequency response of the device.
Back up each answer by specifying a formula which demonstrates the physics.
(6)
[22]
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PHYS284
June 2005 Physics IIE (Engineering)
 [6.]
 Show that the transmission of a Gaussian laser beam
(
) of radius ,
through a circular aperture of radius , can be written as:
where is the fraction of energy transmitted through the
aperture, and is the maximum transmission.
(6)
 The table of data given below was collected for a CO laser beam
passing through a circular aperture. Use the data to estimate
the beam radius assuming that the laser beam
has a Gaussian fluence profile.
Aperture diameter (mm) 
Transmitted energy (mJ) 
1.5 
71 
3 
244 
4.0 
368 
5.0 
428 
6.0 
468 

488 
(3)
 Lasers usually emit in a range of closely spaced wavelengths
satisfying the standing wave condition:
, where is an integer
and is the cavity length. Show that the smallest possible
interval between wavelengths can be approximated as:
(2)
 An ultrashort pulsed laser outputs 10nJ pulses of
FWHM duration 50fs at 780nm wavelength. If the resonator
cavity length is 1m:
 How many longitudinal modes would have to be oscillating?
 What would the gain bandwidth of this laser
have to be to support these modes?
 At what repetition rate would you expect this laser to operate at?
 What would the peak power and average power of the laser be?
(5)
[16]
Total Marks [120]
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PHYS284
June 2005 Physics IIE (Engineering)
Figure 1:
Normalised wave functions of the hydrogen atom for and 3.

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Simon H Connell
20060328