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University of the Witwatersrand
Physics IIE (Engineering) : PHYS284 : 2002
Examination : June 2002 : Deferred

Instructions: Answer all questions.
Time: 2 hours = 120 minutes
Total Marks: 120 marks
 1.
 The relativistic expression for kinetic energy is
 (a)
 Show that this expression reduces to the normal classical
expression
at low speeds.
(6)
 (b)
 Given that the Lorentz Transformation is :
derive the formula for the relativistic length contraction
(4)
 (c)
 A space shuttle 2 light years away from a space station approaches the
space station with a speed of from the point of view of observers on the space station.
 i)
 How long do astronauts on the space station think the shuttle will
take to reach them ? (2)
 ii)
 How long do astronauts on the shuttle think they will take to reach the
space station ? (3)
 iii)
 If another space shuttle were approaching the space station from the
opposite direction, but with the same speed, then what would be
the relative velocity of the two shuttles, according to Galilean relativity ?
(2)
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PHYS284
June 2002 Physics IIE (Engineering)
 iv)
 What would be the actual relative velocity of the two shuttles ? (3)
[20]
 2.
 (a)
 A linear quantum dot can be modelled as an electron in a box. It can be formed
by depositing contacts on a slab of GaAs and then applying a bias voltage
in such a way as to create a linear confining potential.
 i)
 Suppose the observed
groundstate electron energy is 0.003 eV. How big is the potential
boundary (``box'') of the quantum dot ?
(Note : for GaAs,
(7)
 ii)
 Show that this result is roughly consistent with
Heisenberg's Uncertainty Principle.
(4)
 (b)
 With regard to the Scanning Tunneling Microscope :
 i)
 Explain the principle of operation by referring to
a particle energy drawing showing clearly the three regions corresponding to the
sample, the tip and the airgap in an idealised barrier penetration problem.
(4)
 ii)
 The transmission coefficient of electrons through the barrier in a tunneling microscope
is given by
where nm.
Suppose the electronic detection system is sensitive to electrical
current changes of 0.1%. What surface height sensitivity does this correspond to.
(5)
[20]
 3.
 (a)
 The probability of finding the electron in the hydrogen atom
ground state, at some distance between and is given by
where is the Bohr radius (0.05292 nm). Make a rough plot of in terms of
and calculate the distance from the nucleus where the
electron is most likely to be found.
(15)
 (b)
 The energy levels of certain kinds of twoelectron atoms/ions of
atomic number may be approximated by
where the ground state of hydrogen is
eV.
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PHYS284
June 2001 Physics IIE (Engineering)
 i)
 Sketch the twoelectron atom/ion under conditions where the above expression
could be expected to hold.
(5)
 ii)
 Using the full expression for quantised energy levels in hydrogenlike
(that is, oneelectron) atoms/ions of atomic number ,
describe how each term in the above expression arises.
(6)
 iii)
 Why does the accuracy increase when increases ?
(4)
[30]
 4.
 (a)
 Show that electronic density of states
may be expressed as
at 0 K.
(7)
 (b)
 Now show that the total energy of the electron gas at 0 K is
by evaluating
and hence arrive at the result
Which is typically a few electron volts in magnitude.
(6)
 (c)
 If electrons behaved classically, what would the temperature of the metal have to be
for a similar average energy to of the electrons to be realised.
(6)
 (d)
 With respect to the FermiDirac distribution, show
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)
[25]
 5.
 (a)
 In an intrinsic semiconductor,
,
where
and
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PHYS284
June 2001 Physics IIE (Engineering)
 Show that the number of electrons in the conduction band is :
(Hint :
.)
(7)
 (b)
 A similar equation can be shown to hold for the number of holes in the valence band :
Considering this, show
at room temperature in an intrinsic semiconductor.
(6)
 (c)
 Also derive the expression
for the number of intrinsic carriers and give the explicit form of .
Note that in the case of a donordoped semiconductor, the number of carriers is
(6)
 (d)
 A semiconductor is characterised by the energy band structure shown
in the following figure.
Determine the resistivity of the section of the semiconductor.
(
eV,
cm, eV
cmVs,
C.)
(6)
[25]
Total Marks [120]
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PHYS284
June 2002 Physics IIE (Engineering)
Figure:
Normalised wave functions of the hydrogen atom for and 3.

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