Table Of ContentQuantum Mechanics of
Charged Particle Beam Optics
Multidisciplinary and Applied Optics
SeriesEditors:
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L.SrinivasaVaradharajan,UniversityofWaterloo,Ontario,Canada
UnderstandingOpticswithPython
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CRCMULAPPOPT
Quantum Mechanics of
Charged Particle Beam Optics
Understanding Devices from Electron Microscopes to
Particle Accelerators
RamaswamyJagannathanandSameenAhmedKhan
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Dedication
Tothememoryofmyparents
Thirali Aravamudha Ramaswamy Iyengar & Andal Ramaswamy
—–RamaswamyJagannathan
Tothememoryofmyparents
Hamid Ahmed Khan & Nighat Hamid Khan
—–SameenAhmedKhan
Contents
Preface.......................................................................................................................xi
Authors.....................................................................................................................xv
Chapter1 Introduction.....................................................................................1
Chapter2 AnIntroductoryReviewofClassicalMechanics............................9
2.1 SingleParticleDynamics........................................................9
2.1.1 LagrangianFormalism...............................................9
2.1.1.1 BasicTheory..............................................9
2.1.1.2 Example:MotionofaCharged
ParticleinanElectromagneticField........10
2.1.2 HamiltonianFormalism...........................................12
2.1.2.1 BasicTheory............................................12
2.1.2.2 Example:MotionofaCharged
ParticleinanElectromagneticField........14
2.1.3 HamiltonianFormalisminTermsofthePoisson
Brackets....................................................................16
2.1.3.1 BasicTheory............................................16
2.1.3.2 Example:DynamicsofaCharged
ParticleinaConstantMagneticField.....18
2.1.4 ChangingtheIndependentVariable.........................21
2.1.4.1 BasicTheory............................................21
2.1.4.2 Example: Dynamics of a Charged
ParticleinaConstantMagneticField.....22
2.1.5 CanonicalTransformations......................................25
2.1.5.1 BasicTheory............................................25
2.1.5.2 Optical Hamiltonian of a Charged
Particle Moving Through an
ElectromagneticOpticalElementwith
aStraightAxis.........................................29
2.1.6 SymplecticityofCanonicalTransformations..........31
2.1.6.1 Time-Independent Canonical
Transformations.......................................31
2.1.6.2 Time-DependentCanonicalTransfor-
mations:HamiltonianEvolution..............32
2.1.6.3 CanonicalInvariants:PoissonBrackets...35
2.2 DynamicsofaSystemofParticles.......................................36
vii
viii Contents
Chapter3 AnIntroductoryReviewofQuantumMechanics..........................41
3.1 Introduction...........................................................................41
3.2 GeneralFormalismofQuantumMechanics.........................42
3.2.1 Single Particle Quantum Mechanics:
FoundationalPrinciples...........................................42
3.2.1.1 QuantumKinematics...............................42
3.2.1.2 QuantumDynamics.................................51
3.2.1.3 DifferentPicturesofQuantum
Dynamics.................................................64
3.2.1.4 Ehrenfest’sTheorem................................68
3.2.1.5 Spin..........................................................70
3.3 NonrelativisticQuantumMechanics....................................78
3.3.1 NonrelativisticSingleParticleQuantum
Mechanics................................................................78
3.3.1.1 FreeParticle.............................................78
3.3.1.2 LinearHarmonicOscillator.....................90
3.3.1.3 Two-DimensionalIsotropicHarmonic
Oscillator...............................................100
3.3.1.4 ChargedParticleinaConstant
MagneticField.......................................103
3.3.1.5 ScatteringStates....................................107
3.3.1.6 ApproximationMethods,
Time-DependentSystems,andthe
InteractionPicture.................................111
3.3.1.7 Schro¨dinger–Pauli Equation for the
Electron..................................................117
3.3.2 Quantum Mechanics of a System of Identical
Particles..................................................................119
3.3.3 PureandMixedStates:DensityOperator..............126
3.4 RelativisticQuantumMechanics........................................132
3.4.1 Klein–GordonEquation.........................................132
3.4.1.1 Free-ParticleEquationand
DifficultiesinInterpretation..................132
3.4.1.2 Feshbach–VillarsRepresentation..........137
3.4.1.3 Charged Klein–Gordon Particle in a
ConstantMagneticField........................139
3.4.2 DiracEquation.......................................................141
3.4.2.1 Free-ParticleEquation...........................141
3.4.2.2 Zitterbewegung......................................149
3.4.2.3 SpinandHelicityoftheDirac
Particle...................................................150
3.4.2.4 SpinMagneticMomentoftheElectron
andtheDirac–PauliEquation................153
3.4.2.5 ElectroninaConstantMagneticField...154
Contents ix
3.4.3 Foldy–WouthuysenTransformation.......................156
3.4.3.1 Foldy–WouthuysenRepresentationof
theDiracEquation.................................156
3.4.3.2 Foldy–WouthuysenRepresentationof
the Feshbach–Villars form of the
Klein–GordonEquation.........................168
3.5 Appendix: The Magnus Formula for the Exponential
SolutionofaLinearDifferentialEquation.........................170
Chapter4 AnIntroductiontoClassicalChargedParticleBeamOptics......173
4.1 Introduction: Relativistic Classical Charged Particle
BeamOptics........................................................................173
4.2 FreePropagation.................................................................174
4.3 OpticalElementswithStraightOpticAxis........................178
4.3.1 Axially Symmetric Magnetic Lens: Imaging in
ElectronMicroscopy..............................................178
4.3.2 NormalMagneticQuadrupole...............................197
4.3.3 SkewMagneticQuadrupole...................................202
4.3.4 AxiallySymmetricElectrostaticLens...................204
4.3.5 ElectrostaticQuadrupole........................................205
4.4 Bending Magnet: An Optical Element with
aCurvedOpticAxis...........................................................206
4.5 NonrelativisticClassicalChargedParticle
BeamOptics........................................................................210
Chapter5 QuantumChargedParticleBeamOptics:ScalarTheoryfor
Spin-0andSpinlessParticles.......................................................213
5.1 GeneralFormalismofQuantumChargedParticle
BeamOptics........................................................................213
5.2 Relativistic Quantum Charged Particle Beam Optics
BasedontheKlein–GordonEquation................................214
5.2.1 GeneralFormalism.................................................214
5.2.2 FreePropagation:Diffraction................................230
5.2.3 Axially Symmetric Magnetic Lens: Electron
OpticalImaging.....................................................232
5.2.3.1 ParaxialApproximation:Point-to-Point
Imaging..................................................232
5.2.3.2 Going Beyond the Paraxial
Approximation:Aberrations..................250
5.2.3.3 QuantumCorrectionstotheClassical
Results...................................................260
5.2.4 NormalMagneticQuadrupole...............................262
5.2.5 SkewMagneticQuadrupole...................................267
5.2.6 AxiallySymmetricElectrostaticLens...................269
