Lectures on Quantum Mechanics: With Problems, Exercises and Solutions

Preface
Contents
1 Perception and Imagination
1.1 Physics and Language
1.2 The Infinitely Complex
1.3 The Paradoxes and the Second Revolution
1.4 Plan of the Text
1.5 Physical Constants
2 Quantum Phenomena
2.1 Planck's Quanta
2.1.1 Einstein and the Photon
2.1.2 Atomic Spectroscopy and Bohr's Model
2.2 Wave Behavior of Particles
2.2.1 Interferences
2.2.2 Wave Behavior of Matter
2.2.3 Analysis of the Phenomenon
2.3 Probabilistic Nature of Quantum Phenomena
2.3.1 Random Behavior of Particles
2.3.2 A Nonclassical Probabilistic Phenomenon
2.3.3 Conclusions
2.4 Phenomenological Description, de Broglie Waves
2.5 First Discovery, Applications
2.6 Appendix: Notions on Probabilities
2.7 Exercises
3 Wave Function, Schrödinger Equation
3.1 Terminology and Methodology
3.2 Principles of Wave Mechanics
3.2.1 The Wave Function
3.2.2 Schrödinger Equation in Presence of a Potential
3.3 Superposition Principle
3.4 Wave Packets
3.4.1 Free Wave Packets
3.4.2 Fourier Transforms
3.4.3 Shape of Wave Packets
3.5 Historical Landmarks
3.6 Momentum Probability Law
3.6.1 Free Particle
3.6.2 General Case
3.7 Heisenberg Uncertainty Relations
3.8 Controversies and Paradoxes
3.9 Appendix. Dirac δ ``Function'', Distributions
3.10 Appendix: Fourier Transformation
3.11 Exercises
4 Physical Quantities and Measurement
4.1 Statement of the Problem
4.1.1 Physical Quantities
4.1.2 Position and Momentum
4.2 Observables
4.2.1 Position and Momentum Observables
4.2.2 Correspondence Principle
4.2.3 Historical Landmarks
4.3 A Counterexample of Einstein and Its Consequences
4.3.1 What Do We Know After a Measurement?
4.3.2 Eigenstates and Eigenvalues of an Observable
4.3.3 Wave Packet Reduction
4.3.4 Decoherence
4.4 Schrödinger's Cat Paradox
4.5 Exercises
5 Energy, Quantization and Quantum Tunneling
5.1 Energy and Time Dependence
5.1.1 The Hamiltonian
5.1.2 The Schrödinger Equation, Time and Energy
5.1.3 Stationary States
5.1.4 Motion: Interference of Stationary States
5.2 Simple Systems
5.2.1 Bound States and Scattering States
5.2.2 One-Dimensional Problems
5.3 The Harmonic Oscillator
5.4 Square Well Potentials
5.5 Crossing a Potential Barrier, Tunnel Effect
5.6 Applications of the Tunnel Effect
5.6.1 Valence Electrons
5.7 Nanotechnologies
5.8 Exercises
5.9 Problem. The Ramsauer effect
5.9.1 Solution
6 Principles of Quantum Mechanics
6.1 Hilbert Space
6.2 Dirac Formalism
6.2.1 Notations
6.2.2 Operators
6.2.3 Syntax Rules
6.2.4 Projectors; Decomposition of the Identity
6.3 Measurement Results
6.3.1 Eigenvectors and Eigenvalues of an Observable
6.3.2 Results of the Measurement of a Physical Quantity
6.3.3 Probabilities
6.3.4 The Riesz Spectral Theorem
6.3.5 Physical Meaning of Various Representations
6.4 Principles of Quantum Mechanics
6.4.1 The Case of a Continuous Spectrum
6.5 Heisenberg's matrices
6.6 Exercises
7 Two-State Systems, Matrix Mechanics
7.1 Double Well, the Ammonia Molecule
7.1.1 The Model
7.1.2 Stationary States and Tunneling
7.1.3 Energy Levels
7.1.4 Wave Functions
7.1.5 Inversion of the Molecule
7.2 ``Two-State'' System
7.3 Matrix Quantum Mechanics
7.4 NH3 in an Electric Field
7.4.1 Uniform Constant Field
7.4.2 Weak and Strong Field Regimes
7.4.3 Other Two-State Systems
7.5 Motion of the Molecule in an Inhomogeneous Field
7.5.1 Force on the Molecule in an Inhomogeneous Field
7.5.2 Population Inversion
7.6 Reaction to an Oscillating Field, The Maser
7.7 Principle and Applications of the Maser
7.7.1 Amplifiers
7.7.2 Oscillators
7.7.3 Atomic Clocks and the GPS
7.7.4 Tests of Relativity
7.8 Exercises
7.9 Problem. Aromatic Molecules
7.9.1 Solution
8 The Photon
8.1 Polarization of Light
8.1.1 Polarization States
8.1.2 Polarizers at an Angle
8.1.3 Polarization States
8.1.4 Circular Polarisation
8.1.5 Quantum ``Logic''
8.2 Nature of Photon, Wave or Particle
8.3 Interference Experiments
8.3.1 Attenuated Light Experiments
8.4 Physics with Individual Photons
8.4.1 Single Photon Sources
8.4.2 Particle Nature of the Photon
8.5 Single Photon Interferences
8.6 Exercises
9 The Algebra of Observables
9.1 Commutation of Observables
9.1.1 Fundamental Commutation Relation
9.1.2 Other Commutation Relations
9.1.3 Dirac in the Summer of 1925
9.2 Uncertainty Relations
9.3 Evolution of Physical Quantities
9.3.1 Evolution of an Expectation Value
9.3.2 Particle in a Potential, Classical Limit
9.3.3 Conservation Laws
9.4 Algebraic Resolution of the Harmonic Oscillator
9.5 Commuting Observables
9.5.1 Theorem
9.5.2 Example
9.5.3 Tensor Structure of Quantum Mechanics
9.5.4 Complete Set of Commuting Observables (CSCO)
9.5.5 Completely Prepared Quantum State
9.6 Sunday September 20, 1925
9.7 Exercices
9.8 Problem. Quasi-Classical States of the Harmonic Oscillator
9.8.1 Solution
10 Approximation Methods
10.1 Perturbation Theory
10.1.1 Definition of the Problem
10.1.2 First Order Perturbation Theory
10.1.3 Second Order Perturbation to the Energy Levels
10.2 The Variational Method
10.2.1 The Ground State
10.2.2 Example
10.2.3 Relation with Perturbation Theory
10.2.4 Other Levels
10.3 Exercises
10.4 Problem. Conductivity of Crystals; Energy Bands
10.4.1 Electrons in a Periodic Potential
10.4.2 Solution
11 Angular Momentum
11.1 Fundamental Commutation Relation
11.1.1 Classical Angular Momentum
11.1.2 Definition of an Angular Momentum Observable
11.1.3 Results of the Quantization
11.2 Proof of the Quantization
11.2.1 Statement of the Problem
11.2.2 Vectors |j,m> and Eigenvalues j and m
11.2.3 Operators pm=xpmiy
11.2.4 Quantization
11.3 Orbital Angular Momenta
11.3.1 Formulae in Spherical Coordinates
11.3.2 Integer Values of m and ell
11.3.3 Spherical Harmonics
11.4 Rotation Energy of a Diatomic Molecule
11.5 Interstellar Molecules, the Origin of Life
11.6 Angular Momentum and Magnetic moment
11.6.1 Classical Model
11.6.2 Quantum Transposition
11.6.3 Experimental Consequences
11.6.4 Larmor Precession
11.6.5 What About Half-Integer Values of j and m?
11.7 Exercises
12 The Hydrogen Atom
12.1 Two-Body Problem; Relative Motion
12.2 Motion in a Central Potential
12.2.1 Separation of the Angular Variables
12.2.2 The Radial Quantum Number n'
12.2.3 The Principal Quantum Number n
12.2.4 Spectroscopic Notation (States s,p,d,f,…)
12.3 The Hydrogen Atom
12.3.1 Atomic Units; Fine Structure Constant
12.3.2 The Dimensionless Radial Equation
12.3.3 Spectrum of Hydrogen
12.3.4 Stationary States of the Hydrogen Atom
12.3.5 Dimensions and Orders of Magnitude
12.3.6 Historical Landmarks
12.4 Muonic Atoms
12.5 Exercises
12.6 Problem. Tritium β Decay and Neutrino Mass
12.6.1 Solution
13 Spin 1/2
13.1 Experimental Results
13.2 Spin 1/2 Formalism
13.3 Complete Description of a Spin 1/2 Particle
13.3.1 Mixed Representation
13.3.2 Two-Component Wave Function
13.3.3 Observables
13.4 Physical Spin Effects
13.5 Spin Magnetic Moment
13.6 The Stern–Gerlach Experiment
13.7 Principle of the Experiment
13.7.1 Semi-classical Analysis
13.7.2 Experimental Results
13.7.3 Explanation of the Stern–Gerlach Experiment
13.7.4 Successive Stern–Gerlach Setups
13.7.5 Measurement Along an Arbitrary Axis
13.8 The Discovery of Spin
13.8.1 The Hidden Sides of the Stern–Gerlach Experiment
13.8.2 Einstein and Ehrenfest's Objections
13.8.3 Anomalous Zeeman Effect
13.8.4 Bohr's Challenge to Pauli
13.8.5 The Spin Hypothesis
13.8.6 The Fine Structure of Atomic Lines
13.9 Magnetism, Magnetic Resonance
13.9.1 Spin Effects, Larmor Precession
13.9.2 Larmor Precession in a Fixed Magnetic Field
13.9.3 Rabi's Calculation and Experiment
13.9.4 Nuclear Magnetic Resonance
13.9.5 Magnetic Moments of Particles
13.10 Entertainment: Rotation by 2π of a Spin 1/2
13.11 Exercises
13.12 Problem. Lorentz Force in Quantum Mechanics
13.12.1 Solution
14 Fine and Hyperfine Structure of Spectral Lines
14.1 Addition of Angular Momenta
14.1.1 A Simple Case: The Addition of Two Spins 1/2
14.1.2 Addition of Two Arbitrary Angular Momenta
14.2 One-Electron Atoms, Spectroscopic Notations
14.2.1 Fine Structure of Monovalent Atoms
14.3 Hyperfine Structure; The 21 cm Line of Hydrogen
14.3.1 Remarks
14.4 Radioastronomy
14.5 The 21-cm Line of Hydrogen
14.6 The Intergalactic Medium; Star Wars
14.7 Exercises
15 Identical Particles, The Pauli Principle
15.1 Indistinguishability of Two Identical Particles
15.1.1 Identical Particles in Classical Physics
15.1.2 The Quantum Problem
15.1.3 Example of Ambiguities
15.2 Two-Particle System; The Exchange Operator
15.2.1 The Hilbert Space for the Two-Particle System
15.2.2 The Exchange Operator Between Two Identical Particles
15.2.3 Symmetry of the States
15.3 The Pauli Principle
15.3.1 The Case of Two Particles
15.3.2 Independent Fermions and Exclusion Principle
15.3.3 The Case of N Identical Particles
15.4 Physical Consequences of the Pauli Principle
15.4.1 Exchange Force Between Two Fermions
15.4.2 The Ground State of N Identical Independent Particles
15.4.3 Behavior of Fermion and Boson Systems at Low Temperatures
15.4.4 Stimulated Emission and the Laser Effect
15.4.5 Heisenberg Uncertainty Relations for N Fermions
15.5 Exercises
15.6 Problem. Pauli Principle and the Aging of Stars
15.6.1 White Dwarfs
15.6.2 Neutron Stars
15.6.3 Mini-Boson Stars
15.6.4 Solution
16 The Evolution of Systems
16.1 Time-Dependent Perturbation Theory
16.1.1 Example: A Collision Process
16.1.2 Constant Perturbation
16.1.3 Sinusoidal Perturbation
16.2 Interaction of an Atom with an Electromagnetic Wave
16.2.1 The Electric Dipole Approximation
16.2.2 Justification of the Electric Dipole Interaction
16.2.3 Absorption of Energy by an Atom
16.2.4 Selection Rules
16.2.5 Spontaneous Emission
16.2.6 Control of an Atomic Motion by Light
16.3 Decay of a System
16.3.1 The Radioactivity of 57Fe
16.3.2 The Fermi Golden Rule
16.3.3 Orders of Magnitude
16.3.4 Behavior for Long Times
16.4 The Time-Energy Uncertainty Relation
16.4.1 Isolated Systems and Intrinsic Interpretations
16.4.2 Interpretation of Landau and Peierls
16.4.3 The Einstein–Bohr controversy
16.5 Exercises
16.6 Problem. Molecular Lasers
16.6.1 Solution
17 Entangled States. The Way of Paradoxes
17.1 The EPR Paradox
17.2 The Version of David Bohm
17.2.1 Bell's Inequality
17.2.2 Experimental Tests
17.3 The GHZ Experiment
17.3.1 Quantum Situation
17.3.2 Local Realistic Situations
18 Quantum Information and the Fruits of Entanglement
18.1 Quantum Information: How to Take Advantage of an Embarrassment
18.2 Quantum Teleportation
18.2.1 Bell States
18.3 Quantum Cryptography
18.4 The Quantum Computer
18.5 The Quantum Professions
19 Solutions to the Exercises
Index
Index