Quantum mechanics: a simplified approach

263 Pages·2019·4.036 MB·English

by  Siddiqui, Shabnam

Physics S
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Quantum Mechanics
A Simplified Approach
Quantum mechanics is one of the most challenging subjects to learn. It is
challenging because a quantum phenomenon is counterintuitive and the mathe-
matics used to explain it is often abstract and difficult to grasp. This textbook is
an attempt to overcome these challenges. Every chapter presents quantum ideas
in a step- by- step structured way along with comparisons between quantum and
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classical concepts that provide clear distinction between classical and quantum
QQuuaannttuumm MMeecchhaanniiccss
logic. Conceptual questions are given after every important section so that U
readers can check their understanding at every step. Tutorials are included on
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special topics to teach problem solving by actively engaging readers in a thinking
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process about the topic. Simulations are provided to aid in the visualization of the
quantum phenomenon and to gain meaningful understanding of the mathematics T
involved. This approach may lead to the development of "quantum mechanical
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intuition" as well as a learning of mathematical techniques for problem solving.
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Most importantly, the book does not contain an overabundance of topics that can
make readers confused and distracted, but rather the most important topics are

discussed and at a deep level. The story of the development of quantum theory is A S i m p l i f i e d A p p r o a c h
M
one of the most inspiring stories in physics. Readers are introduced to the early
ideas and experiments that lead to the theory of quantum mechanics in the first E
two chapters. C
The key features of this book are: H
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(cid:127) A simplified, structured, and step- by- step introduction to quantum
mechanics N
(cid:127) A systematic arrangement of topics and chapters
I Shabnam Siddiqui
(cid:127) Conceptual questions to test logical understanding C
(cid:127) Tutorials on special topics to actively engage students in a thinking
S
process concerning the differences between quantum mechanics and
classical physics
(cid:127) Simulations to aid in the visualization of the physical phenomenon with
demonstrations of the application of mathematics
ISBN-13: 978-1-138-19726-8
90000
9 781138 197268
Quantum Mechanics
A Simplified Approach

Quantum Mechanics
A Simplified Approach
Shabnam Siddiqui
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To my parents and family

Contents
Preface ................................................................................... xiii
Author....................................................................................xvii
AcronymsandAbbreviations .............................................................. xix
Chapter1 AnIntroductoryHistoryofQuantumMechanics-I............................... 1
1.1 ClassicalViewofanElectromagneticWave.............................. 1
1.1.1 WhatIsanElectromagneticWave?..............................2
1.1.2 WhatArethePropertiesofanElectromagneticWave?............3
1.1.3 HowAretheElectromagneticWavesGenerated?.................3
1.2 TheBlack-BodyRadiationParadox...................................... 4
1.2.1 TheExperimentalObservations..................................5
1.2.2 TheMathematicalFormoftheRadiationFunction................7
1.3 TheBeginningofQuantumMechanicalView............................ 9
Problems..................................................................... 10
1.4 ThePhotoelectricEffect................................................ 11
Problems..................................................................... 14
1.5 TheComptonEffect................................................... 14
Problems..................................................................... 16
1.6 Tutorial ............................................................... 17
1.6.1 PhotoelectricEffect............................................17
Chapter2 AnIntroductoryHistoryofQuantumMechanics-II............................. 21
2.1 DeBroglieMatterWaves.............................................. 21
Problems..................................................................... 25
2.2 ElectronDiffraction.................................................... 25
Problems..................................................................... 27
2.3 Double-SlitExperiment................................................ 27
Problems..................................................................... 32
2.4 UncertaintyPrinciple .................................................. 32
2.4.1 TheUncertaintyPrinciple ......................................33
2.4.2 ASimpleProofoftheUncertaintyPrinciple.....................35
2.5 Measurement.......................................................... 36
Chapter3 Formalism ................................................................... 39
3.1 Introduction........................................................... 39
3.2 DescribingaQuantumMechanicalSystem:BasicPostulatesof
theModel............................................................. 39
3.2.1 Postulate1:DefininganObservable ............................39
3.2.2 IndeterminacyAssociatedwiththeMeasurementofan
Observable ....................................................40
vii
viii Contents
3.2.3 Postulate2:DefiningtheStatesAssociatedwith
anObservable .................................................40
3.2.4 Postulate3:MeasurementofanObservable.....................41
3.3 MathematicalFoundation .............................................. 41
3.3.1 StateVector ...................................................41
3.3.2 KetandBraSpace.............................................42
3.3.3 MultiplicationRulesforKetandBra............................43
3.3.3.1 InnerProduct....................................... 43
3.3.3.2 OuterProduct....................................... 43
3.3.3.3 Orthogonality....................................... 43
3.3.3.4 Normalization....................................... 43
3.3.4 Operators......................................................45
3.3.5 BaseKets .....................................................45
3.3.6 Completeness..................................................45
3.3.6.1 DiscreteSpectrum................................... 48
3.3.6.2 ContinuousSpectrum ............................... 50
Problems..................................................................... 50
3.4 WaveFunction........................................................ 51
3.4.1 FunctionSpaceandHilbertSpace ..............................51
3.4.2 Eigenfunctions.................................................53
3.4.2.1 Orthogonal ......................................... 53
3.4.2.2 Normal............................................. 54
3.4.2.3 Completeness....................................... 54
3.4.3 ProbabilityDensity ............................................54
3.4.4 ExpectationValueofanObservable ............................56
3.4.4.1 Position ............................................ 56
3.4.4.2 Momentum......................................... 57
3.4.4.3 KineticEnergy...................................... 57
Problems..................................................................... 58
3.5 DynamicalPropertiesoftheSystem .................................... 59
3.5.1 Time-DependentSchrodingerEquation .........................59
3.5.2 TimeEvolutionOperator.......................................60
3.5.3 SeparationofVariables:Time-Independent
SchrodingerEquation..........................................61
3.5.4 StationaryStates...............................................62
Problems..................................................................... 63
3.6 ConservationofProbability ............................................ 64
Problem ..................................................................... 65
3.7 HeisenbergUncertaintyPrinciple....................................... 65
3.7.1 ProofoftheUncertaintyPrinciple ..............................67
3.7.1.1 TheSchwarzInequality ............................. 67
3.7.2 ApplicationsofthePrinciple ...................................68
3.7.2.1 Heisenberg’sMicroscope............................ 68
3.7.2.2 DefiningOrbitsinAtoms............................ 69
Problems..................................................................... 70
3.8 MatrixMechanics ..................................................... 71
3.9 Tutorials .............................................................. 73
3.9.1 Spin-1/2 ......................................................73
3.9.2 WaveFunction ................................................75
Contents ix
Chapter4 ApplicationsoftheFormalism-I............................................... 81
4.1 Introduction........................................................... 81
4.2 TheFreeParticle ...................................................... 81
Problems..................................................................... 86
4.3 TheInfiniteSquareWell............................................... 86
Problems..................................................................... 92
4.4 StepPotential.......................................................... 92
Problems..................................................................... 98
4.5 PotentialBarrierPenetration(Tunneling)................................ 98
Problems.................................................................... 100
4.6 TheFiniteSquareWell ............................................... 100
Problems.................................................................... 104
4.7 Tutorials ............................................................. 105
4.7.1 InfiniteSquareWell.......................................... 105
4.7.2 Tunneling ................................................... 110
4.7.3 QuantumWavePacket....................................... 119
Chapter5 ApplicationsoftheFormalism-II............................................. 127
5.1 TheHarmonicOscillator.............................................. 127
5.1.1 AnalyticalMethod........................................... 127
5.1.1.1 NormalizationConstant ............................ 131
Problems.................................................................... 133
5.1.2 AlgebraicMethod............................................ 134
5.1.2.1 NormalizationConstant ............................ 137
Problems.................................................................... 138
5.2 TheSchrodingerEquationinThreeDimensions........................ 138
5.2.1 TheSchrodingerEquationinCartesianCoordinates ........... 138
5.2.1.1 WhatIstheNewQuantumPropertyofthe
3-DimensionalInfiniteSquareWell?................ 140
Problems.................................................................... 141
5.2.2 TheSchrodingerEquationinSphericalCoordinates ........... 141
5.2.3 TheAngularEquation........................................ 142
5.2.4 TheRadialEquation ......................................... 145
5.3 TheHydrogenAtom.................................................. 146
5.3.1 TheRadialEquationfortheHydrogenAtom.................. 147
5.3.1.1 AsymptoticBehavior............................... 147
5.3.1.2 PowerSeriesSolution.............................. 148
5.3.1.3 TheLaguerrePolynomialsandtheAssociated
LaguerrePolynomials.............................. 151
5.3.1.4 DegeneracyofHydrogenAtom..................... 153
5.3.2 HydrogenAtomSpectrum.................................... 153
Problems.................................................................... 156
5.4 TheAngularMomentum.............................................. 156
5.4.1 WhatAretheEigenfunctionsandEigenvalues
ofTheseOperators?.......................................... 159
5.4.2 IsAngularMomentumConservedinaQuantum
MechanicalSystem? ......................................... 159