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PHYS 2012 Quantum Physics

创建日期：2024-05-16 15:11:18

所属分类：物理Physical

标签： PHYS 2012

Quantum Physics

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CC0933

PHYS 2012 PHYSICS 2B
SEMESTER 2, 2014 TIME ALLOWED: 3 HOURS
ALL QUESTIONS HAVE THE VALUE SHOWN
INSTRUCTIONS:
This paper consists of 2 sections.
Section A Electromagnetic Properties of Matter 60 marks
Section B Quantum Physics 60 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. Non-
programmable calculators are permitted.
CC0933 SEMESTER 2, 2014 Page 2 of 9
Table of constants
Avogadro’s number NA = 6.022× 1023 mol−1
speed of light c = 2.998× 108 m.s−1
electronic charge e = 1.602× 10−19 C
electron rest mass me = 9.110× 10−31 kg
electron rest energy E = 511 keV
electron volt 1 eV = 1.602× 10−19 J
proton rest mass mp = 1.673× 10−27 kg
neutron rest mass mn = 1.675× 10−27 kg
Planck’s constant h = 6.626× 10−34 J.s
Planck’s constant (reduced) ~ = 1.055× 10−34 J.s
Boltzmann’s constant kB = 1.380× 10−23 J.K−1
Stefan’s constant σ = 5.670× 10−8 W.m−2.K−4
Coulomb constant (4pi0)
−1 = 8.988× 109 N.m2.C−2
permittivity of free space 0 = 8.854× 10−12 C2.N−1.m−2
permeability of free space µ0 = 4pi × 10−7 kg.m.C−2
gravitational constant G = 6.673× 10−11 N.m2.kg−2
atomic mass constant u = 1.660× 10−27 kg
gas constant R = 8.314 J.mol−1.K
CC0933 SEMESTER 2, 2014 Page 3 of 9
SECTION A: ELECTROMAGNETIC PROPERTIES OF MATTER
= r0 µ = µrµ0
c =
1√
0µ0
v =
c
n
n =
√
rµr
F =
1
4pi
q1q2
r2
r̂ E =
1
4pi
q
r2
r̂ V =
1
4pi
q
r
E = −∇V Vb − Va = −
∫ b
a
E · dl
∮
E · dl = 0
FE = qE∮
E · dA = qenclosed
0
=
qf − qb
0
=
qf
r0∮
D · dA = qf D = E = σf
∮
D · dA =
∫
ρfdV
σb = P · n̂ ρb = −∇ ·P E = 1
0
(D−P) P = χe0E r = 1 + χe
C =
Q
V
C =
qf
V
C =
A
d
Cseries =
1
C−11 + C2
−1 + · · · Cparallel = C1 + C2 + · · ·
U =
1
2
Q2
C
=
1
2
CV 2 =
1
2
QV W =
1
2

∫
E2dV u =
1
2
E2
p = qd τ = p× E U = −p · E
P = np Jb = −∂P
∂t
p = αE
α = 4pi0a
3 α =
30
n
r − 1
r + 2
P = np
[
coth(pE/kT )− 1
pE/kT
]
CC0933 SEMESTER 2, 2014 Page 4 of 9
F = qv ×B dF = idl×B dB = µ0
4pi
idl× r
r3
Fc =
mν2
r
ΦB =
∫
B · dA
∮
B · dA = 0
emf =
∮
E · dl = −dΦB
dt∮
B · dl = µ0itotal
∮
H · dl = ifree
pm = NiAn̂ τ = pm ×B U = −pm ·B
M = χmH B = µ0(H+M) B = µH = µrµ0H µr = 1 + χm
W = V
∫
HdB
i =
dq
dt
i = nqνdriftA i =
∫
J · dA
R =
V
I
R = ρ
L
A
σ =
1
ρ
J = σE ρ =
m
e2nτ
CC0933 SEMESTER 2, 2014 Page 5 of 9
Please use a separate book for this section.
Answer ALL QUESTIONS in this section.
1. A chargeQ > 0 is placed at the centre of a water-filled sphere with radius a (the dielectric
constant of water is εr = 81).
(a) Draw a diagram of the system, showing the polarization density vectors and indi-
cating the induced polarization charges with + and − signs.
(b) Find the electric field both inside and outside the sphere.
(c) Find the polarization densityP inside the sphere, and useP to determine the polar-
ization surface charge density and the total polarization charge Qpol on the sphere.
(d) Show that Q/εr = Q−Qpol, and explain the physical significance of this relation-
ship.
(15 marks)
2. A long solenoid with lengthL and radius a has n turns per unit length and carries a current
I . Assume that there is a uniform magnetic field inside the solenoid parallel to its axis,
given by B0 = µ0nI .
(a) A cylindrical iron bar of length L and radius a/2 is fully inserted into the solenoid
so that its axis coincides with that of the solenoid. Draw a diagram for a cross
section of the solenoid, showing the direction of the magnetization density and the
direction of currents in the solenoid and the iron bar. Find the total magnetic field
everywhere in the solenoid, and sketch it as a function of the radial distance from
the axis in units of B0 (use µr = 1000 for the iron bar).
(b) Repeat the question in (a) for the case where the iron bar is replaced with an iden-
tical platinum bar (µr = 1.0003).
(c) Next consider the case where the iron bar in (a) is replaced with a cylindrical shell
made of iron, which has length L, outer radius a, and inner radius a/2 (i.e., it has a
hole of radius a/2 in the centre). Repeat the question in (a) for this configuration.
(15 marks)
CC0933 SEMESTER 2, 2014 Page 6 of 9
3. Give brief physical answers to the following questions.
(a) Why does filling a capacitor with a dielectric material which has a large εr increase
its capacity for charge storage and hence storage of potential energy?
(b) State whether the expression σ = Nq2τ/m for the conductivity of a material over-
estimates or underestimates the conductivity of (i) electrons in metals and (ii) ions
in electrolytes, and then explain why. Here N , q, τ and m are the number density,
charge, mean collison time, and mass of the charge carriers, respectively.
(c) Discuss the temperature dependence of magnetization in paramagnetic, diamag-
netic and ferromagnetic materials and give a physical explanation for each be-
haviour.
(15 marks)
4. We consider a slab of doped silicon of thickness d = 10µm, with current I along the
slab, immersed in a uniform mangetic fieldB parallel to the slab and perpendicular to the
direction of the current (see figure below).
Figure 1: Geometry for Question 4(a-c).
(a) Using a schematic of the slab showing the various forces and charge accumulations
that are relevant, explain the Hall effect and its origin.
(b) For I = 0.1A, and B = 0.12T, the voltage is measured to be VB − VA = +500mV.
Are the predominant charge carriers positive or negative? Calculate the density of
carriers N .
(c) Maintaining the same current, would the voltage increase or decrease if the temper-
ature is increased by 20◦C? Justify your answer.
(d) A Hall sensor as as depicted in the figure, with N = 1.5 × 1022m−3 is used as
an electronic compass in a mobile phone. The Earth’s magnetic field in Sydney
is 55µT. Assuming the smallest voltage change that can be reliably detected is
∆V = 100µV, determine the smallest error ∆θ with which the magnetic north
can be determined with this sensor, for I = 0.1A. How could this be improved?
(15 marks)
CC0933 SEMESTER 2, 2014 Page 7 of 9
SECTION B: QUANTUM PHYSICS
There is no formula sheet for the Quantum Physics Section
Please use a separate book for this section.
Answer ALL QUESTIONS in this section.
5. Explain briefly (less than 50 words each) what is meant by each of the following.
(a) The Stern-Gerlach experiment.
(b) The Hamiltonian operator.
(c) A quantum dot.
(12 marks)
6. Consider a standard Stern-Gerlach experiment, with a beam of spin-1/2 particles prepared
in the state
|ψ〉 = |+〉 − 2eipi/3|−〉 .
(a) Normalise this state vector.
(b) What are the possible results of a measurement of the spin component Sz, and with
what probabilities do they occur?
(c) What is the expectation value 〈Sz〉 for this state?
(d) What is the uncertainty ∆Sz for this state?
(e) What are the possible results of a measurement of the spin component Sy, and with
what probabilities do they occur? Hint: the eigenstates for Sy, expressed in terms
of the eigenstates of Sz, are:
|+〉y = 1√
2
(|+〉+ i|−〉) , |−〉y = 1√
2
(|+〉 − i|−〉) ,
(f) Suppose that the Sy measurement was performed, with the result Sy = −~/2.
Subsequent to that result, a second measurement is performed to measure the spin
component Sz. What are the possible results of that measurement, and with what
probabilities do they occur?

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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