Problems and Solutions in Quantum Physics

194 Pages·2016·1.305 MB·English

by  Ficek, Zbigniew

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Problems and
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o
Readers studying the abstract field of quantum physics need to
b
solve plenty of practical, especially quantitative, problems. This
l Solutions in
e
book contains tutorial problems with solutions for the textbook
m
Quantum Physics for Beginners. It places emphasis on basic problems
s
of quantum physics together with some instructive, simulating, and Quantum Physics
a
useful applications. A considerable range of complexity is presented n
d
by these problems, and not too many of them can be solved using

formulas alone. S
o
l
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Zbigniew Ficek
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Zbigniew Ficek is professor of quantum optics and i
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quantum information at the National Centre for Applied n
s
Physics, King Abdulaziz City for Science and Technology

i
(KACST), Saudi Arabia. He received his PhD from Adam n

Mickiewicz University, Poland, in 1985. Before KACST, he has held Q
various positions at Adam Mickiewicz University; University of u
a
Queensland, Australia; and Queen’s University of Belfast, UK. He
n
has also been an honorary adjunct professor in the Department of t
u
Physics, York University, Canada. He has authored or coauthored
m
over 140 scientific papers and 2 research books and been an invited

speaker at more than 25 conferences and talks. He is particularly P
h
well known for his contributions to the fields of multi-atom
y
effects, spectroscopy with squeezed light, quantum interference, s
i
multichromatic spectroscopy, and entanglement. c
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F
i
c
e
k
V493
ISBN 978-981-4669-36-8
Problems and
Solutions in
Quantum Physics
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Problems and
Solutions in
Quantum Physics
Zbigniew Ficek
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20160406
International Standard Book Number-13: 978-981-4669-37-5 (eBook - PDF)
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March18,2016 13:47 PSPBook-9inx6in Zbigniew-Ficek-tutsol
Preface
This book contains problems with solutions of a majority of
the tutorial problems given in the textbook Quantum Physics for
Beginners. Not presented are solutions to only those problems
whose solutions the reader can find in the textbook. You should
readthetextofachapterbeforetryingthetutorialproblemsinthe
chapter.Solutionstotheproblemsgivethereaderaself-checkand
reassuranceontheprogressoflearning.
ZbigniewFicek
TheNationalCentreforAppliedPhysics
KingAbdulazizCityforScienceandTechnology
Riyadh,SaudiArabia
Spring2016
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
March18,2016 13:47 PSPBook-9inx6in Zbigniew-Ficek-tutsol
Chapter 1
Radiation (Light) is a Wave
Problem1.2
UsingEq.(1.13)ofthetextbook,showthat
E(cid:2) =−cκ(cid:2)×B(cid:2) , (1.1)
k k
which is the same relation one can obtain from the Maxwell
Eq.(1.4).
(Hint:Usethevectoridentity A(cid:2) ×(B(cid:2) ×C(cid:2))= B(cid:2)(A(cid:2) ·C(cid:2))−C(cid:2)(A(cid:2) ·B(cid:2)).)
Solution
Equation (1.13) of the textbook shows the relation between the
directionsoftheelectricandmagneticfieldsoftheelectromagnetic
wave
B(cid:2) = 1κ(cid:2)×E(cid:2) , (1.2)
k k
c
whereκ(cid:2)istheunitvectorinthedirectionofpropagationofthewave.
By taking a cross product of both sides from the left with the
vectorκ(cid:2),weget
κ(cid:2)×B(cid:2) = 1κ(cid:2)×(κ(cid:2)×E(cid:2) ). (1.3)
k k
c
ProblemsandSolutionsinQuantumPhysics
ZbigniewFicek
Copyright(cid:2)c 2016PanStanfordPublishingPte.Ltd.
ISBN978-981-4669-36-8(Hardcover),978-981-4669-37-5(eBook)
www.panstanford.com
March18,2016 13:47 PSPBook-9inx6in Zbigniew-Ficek-tutsol
2 Radiation(Light)isaWave
Next,usingthevectoridentity A(cid:2) ×(B(cid:2) ×C(cid:2)) = B(cid:2)(A(cid:2) ·C(cid:2))−C(cid:2)(A(cid:2) · B(cid:2)),
wecanwritetheright-handsideoftheaboveequationas
(cid:2) (cid:3)
1κ(cid:2)×(κ(cid:2)×E(cid:2) )= 1 κ(cid:2)(κ(cid:2)·E(cid:2) )−E(cid:2) (κ(cid:2)·κ(cid:2)) . (1.4)
k k k
c c
Sinceκ(cid:2) ·κ(cid:2) = 1andtheelectricandmagneticfieldsaretransverse
fields(κ(cid:2)·E(cid:2) =0),wearriveat
k
E(cid:2) =−cκ(cid:2)×B(cid:2) . (1.5)
k k
(cid:2) (cid:2) (cid:2)
Thisresultfor E andthatfor B ,Eq.(1.2),showthatboth B and
k k k
(cid:2)
E ofanelectromagneticwaveareperpendiculartothedirectionof
k
propagationofthewave.
Problem1.3
Show,usingthedivergenceMaxwellequations,thattheelectromag-
neticwavesinvacuumaretransversewaves.
Solution
Consider an electromagnetic wave propagating in the z direction.
The wave is represented by the electric and magnetic fields of the
form
E(cid:2) = E(cid:2) ei(ωt−kz),
0
B(cid:2) = B(cid:2) ei(ωt−kz). (1.6)
0
Thepropagationofthewaveischaracterizedbythefrequencyωand
thewavenumberk.
Whencalculatingdivergences∇·E(cid:2) and∇·B(cid:2),weget
∇·E(cid:2) = ∂Ex + ∂Ey + ∂Ez =0+0+ ∂Ez,
∂x ∂y ∂z ∂z
∇·B(cid:2) = ∂Bx + ∂By + ∂Bz =0+0+ ∂Bz. (1.7)
∂x ∂y ∂z ∂z
Sinceinvacuum∇·E(cid:2) =0and∇·B(cid:2) =0alwaysinelectromagnetism,
wehave
∂E ∂B
z =0 and z =0. (1.8)
∂z ∂z
March18,2016 13:47 PSPBook-9inx6in Zbigniew-Ficek-tutsol
Radiation(Light)isaWave 3
However,fortheelectricandmagneticfieldsofaplanewave,
∂E ∂B
z =−ikE and z =−ikB . (1.9)
∂z z ∂z z
Hence,theright-handsidesmustbezero,whichmeansthateither
k = 0or E = 0and B = 0,thatboth E(cid:2) and B(cid:2) aretransverseto
z z
thedirectionofpropagation.Sincek(cid:4)=0forapropagatingwave,the
(cid:2) (cid:2)
waveistransverseinbothE andB.
Problem1.4
Calculatetheenergyofanelectromagneticwavepropagatinginone
dimension.
Solution
Consider a plane electromagnetic wave propagating in the z
direction in a vacuum with the electric field polarized in the x
direction:
E(cid:2) = E sin(ωt−kz)iˆ, (1.10)
0
whereiˆistheunitvectorinthex direction.
(cid:2)
Having E,wecancalculatethemagneticfieldofthewaveusing
theMaxwellequation
∂B(cid:2) =−∇×E(cid:2), (1.11)
∂t
andget
∂B(cid:2) =−∇×E(cid:2) =kE cos(ωt−kz)ˆj. (1.12)
∂t 0
Integratingthisequation,wefind
(cid:4)
B(cid:2) =kE dtcos(ωt−kz)ˆj = kE0 sin(ωt−kz)ˆj. (1.13)
0 ω
Sincek/ω=1/c,wefinallyobtain
B(cid:2) = B sin(ωt−kz)ˆj, (1.14)
0
whereB = E /c.
0 0