Quantum Mechanics

Cover page
Title page
Copyright page
Preface
Contents
Physical Constants
1. Quantum Phenomena
1.1 The Franck and Hertz Experiment
1.2 Interference of Matter Waves
1.2.1 The Young Double-Slit Experiment
1.2.2 Interference of Atoms in a Double-Slit Experiment
1.2.3 Probabilistic Aspect of Quantum Interference
1.3 The Experiment of Davisson and Germer
1.3.1 Diffraction of X Rays by a Crystal
1.3.2 Electron Diffraction
1.4 Summary of a Few Important Ideas
Further Reading
Exercises
2. The Wave Function and the Schr¨odinger Equation
2.1 The Wave Function
2.1.1 Description of the State of a Particle
2.1.2 Position Measurement of the Particle
2.2 Interference and the Superposition Principle
2.2.1 De Broglie Waves
2.2.2 The Superposition Principle
2.2.3 The Wave Equation in Vacuum
2.3 Free Wave Packets
2.3.1 Definition of a Wave Packet
2.3.2 Fourier Transformation
2.3.3 Structure of the Wave Packet
2.3.4 Propagation of a Wave Packet: the Group Velocity
2.3.5 Propagation of a Wave Packet
2.4 Momentum Measurements and Uncertainty Relations
2.4.1 The Momentum Probability Distribution
2.4.2 Heisenberg Uncertainty Relations
2.5 The Schr¨odinger Equation
2.5.1 Equation of Motion
2.5.2 Particle in a Potential: Uncertainty Relations
2.5.3 Stability of Matter
2.6 Momentum Measurement in a Time-of-Flight Experiment
Further Reading
Exercises
3. Physical Quantities and Measurements
3.1 Measurements in Quantum Mechanics
3.1.1 The Measurement Procedure
3.1.2 Experimental Facts
3.1.3 Reinterpretation of Position and Momentum Measurements
3.2 Physical Quantities and Observables
3.2.1 Expectation Value of a Physical Quantity
3.2.2 Position and Momentum Observables
3.2.3 Other Observables: the Correspondence Principle
3.2.4 Commutation of Observables
3.3 Possible Results of a Measurement
3.3.1 Eigenfunctions and Eigenvalues of an Observable
3.3.2 Results of a Measurement and Reduction of the Wave Packet
3.3.3 Individual Versus Multiple Measurements
3.3.4 Relation to Heisenberg Uncertainty Relations
3.3.5 Measurement and Coherence of Quantum Mechanics
3.4 Energy Eigenfunctions and Stationary States
3.4.1 Isolated Systems: Stationary States
3.4.2 Energy Eigenstates and Time Evolution
3.5 The Probability Current
3.6 Crossing Potential Barriers
3.6.1 The Eigenstates of the Hamiltonian
3.6.2 Boundary Conditions at the Discontinuities of the Potential
3.6.3 Reflection and Transmission on a Potential Step
3.6.4 Potential Barrier and Tunnel Effect
3.7 Summary of Chapters 2 and 3
Further Reading
Exercises
4. Quantization of Energy in Simple Systems
4.1 Bound States and Scattering States
4.1.1 Stationary States of the Schrodinger Equation
4.1.2 Bound States
4.1.3 Scattering States
4.2 The One Dimensional Harmonic Oscillator
4.2.1 Definition and Classical Motion
4.2.2 The Quantum Harmonic Oscillator
4.2.3 Examples
4.3 Square-Well Potentials
4.3.1 Relevance of Square Potentials
4.3.2 Bound States in a One-Dimensional Square-Well Potential
4.3.4 Particle in a Three-Dimensional Box
4.4 Periodic Boundary Conditions
4.4.1 A One-Dimensional Example
4.4.2 Extension to Three Dimensions
4.4.3 Introduction of Phase Space
4.5 The Double Well Problem and the Ammonia Molecule
4.5.1 Model of the NH3 Molecule
4.5.2 Wave Functions
4.5.3 Energy Levels
4.5.4 The Tunnel Effect and the Inversion Phenomenon
4.6 Other Applications of the Double Well
Further Reading
Exercises
5. Principles of Quantum Mechanics
5.1 Hilbert Space
5.1.1 The State Vector
5.1.2 Scalar Products and the Dirac Notations
5.1.3 Examples
5.1.4 Bras and Kets, Brackets
5.2 Operators in Hilbert Space
5.2.1 Matrix Elements of an Operator
5.2.2 Adjoint Operators and Hermitian Operators
5.2.3 Eigenvectors and Eigenvalues
5.2.4 Summary: Syntax Rules in Dirac’s Formalism
5.3 The Spectral Theorem
5.3.1 Hilbertian Bases
5.3.2 Projectors and Closure Relation
5.3.3 The Spectral Decomposition of an Operator
5.3.4 Matrix Representations
5.4 Measurement of Physical Quantities
5.5 The Principles of Quantum Mechanics
5.6 Structure of Hilbert Space
5.6.1 Tensor Products of Spaces
5.6.2 The Appropriate Hilbert Space
5.6.3 Properties of Tensor Products
5.6.4 Operators in a Tensor Product Space
5.6.5 Simple Examples
5.7 Reversible Evolution and the Measurement Process
Further Reading
Exercises
6. Two-State Systems, Principle of the Maser
6.1 Two-Dimensional Hilbert Space
6.2 A Familiar Example: the Polarization of Light
6.2.1 Polarization States of a Photon
6.2.2 Measurement of Photon Polarizations
6.2.3 Successive Measurements and “Quantum Logic”
6.3 The Model of the Ammonia Molecule
6.3.1 Restriction to a Two-Dimensional Hilbert Space
6.3.2 The Basis {|S
, |A
}
6.3.3 The Basis {|R
, |L
}
6.4 The Ammonia Molecule in an Electric Field
6.4.1 The Coupling of NH3 to an Electric Field
6.4.2 Energy Levels in a Fixed Electric Field
6.4.3 Force Exerted on the Molecule by an Inhomogeneous Field
6.5 Oscillating Fields and Stimulated Emission
6.6 Principle and Applications of Masers
6.6.1 Amplifier
6.6.2 Oscillator
6.6.3 Atomic Clocks
Further Reading
Exercises
7. Commutation of Observables
7.1 Commutation Relations
7.2 Uncertainty Relations
7.3 Ehrenfest’s Theorem
7.3.1 Evolution of the Expectation Value of an Observable
7.3.2 Particle in a Potential V (r)
7.3.3 Constants of Motion
7.4 Commuting Observables
7.4.1 Existence of a Common Eigenbasis for Commuting Observables
7.4.2 Complete Set of Commuting Observables (CSCO)
7.4.3 Completely Prepared Quantum State
7.4.4 Symmetries of the Hamiltonian and Search of Its Eigenstates
7.5 Algebraic Solution of the Harmonic-Oscillator Problem
7.5.1 Reduced Variables
7.5.2 Annihilation and Creation Operators ˆa and ˆa†
7.5.3 Eigenvalues of the Number Operator ˆ N
7.5.4 Eigenstates
Further Reading
Exercises
8. The Stern–Gerlach Experiment
8.1 Principle of the Experiment
8.1.1 Classical Analysis
8.1.2 Experimental Results
8.2 The Quantum Description of the Problem
8.3 The Observables ˆµx and ˆµy
8.4 Discussion
8.4.1 Incompatibility of Measurements Along Di?erent Axes
8.4.2 Classical Versus Quantum Analysis
8.4.3 Measurement Along an Arbitrary Axis
8.5 Complete Description of the Atom
8.5.1 Hilbert Space
8.5.2 Representation of States and Observables
8.5.3 Energy of the Atom in a Magnetic Field
8.6 Evolution of the Atom in a Magnetic Field
8.6.1 Schr¨odinger Equation
8.6.2 Evolution in a Uniform Magnetic Field
8.6.3 Explanation of the Stern–Gerlach Experiment
8.7 Conclusion
Further Reading
Exercises
9. Approximation Methods
9.1 Perturbation Theory
9.1.1 Definition of the Problem
9.1.2 Power Expansion of Energies and Eigenstates
9.1.3 First-Order Perturbation in the Nondegenerate Case
9.1.4 First-Order Perturbation in the Degenerate Case
9.1.5 First-Order Perturbation to the Eigenstates
9.1.6 Second-Order Perturbation to the Energy Levels
9.1.7 Examples
9.1.8 Remarks on the Convergence of Perturbation Theory
9.2 The Variational Method
9.2.1 The Ground State
9.2.2 Other Levels
9.2.3 Examples of Applications of the Variational Method
Exercises
10. Angular Momentum
10.1 Orbital Angular Momentum and the Commutation Relations
10.2 Eigenvalues of Angular Momentum
10.2.1 The Observables ˆ J2 and ˆ Jz and the Basis States |j,m

10.2.2 The Operators ˆ J±
10.2.3 Action of ˆ J± on the States |j,m

10.2.4 Quantization of j and m
10.2.5 Measurement of ˆ Jx and ˆ Jy
10.3 Orbital Angular Momentum
10.3.1 The Quantum Numbers m and are Integers
10.3.2 Spherical Coordinates
10.3.3 Eigenfunctions of ˆL2 and ˆLz: the Spherical Harmonics
10.3.4 Examples of Spherical Harmonics
10.3.5 Example: Rotational Energy of a Diatomic Molecule
10.4 Angular Momentum and Magnetic Moment
10.4.1 Orbital Angular Momentum and Magnetic Moment
10.4.2 Generalization to Other Angular Momenta
10.4.3 What Should we Think about Half-Integer Values of j and m ?
Further Reading
Exercises
11. Initial Description of Atoms
11.1 The Two-Body Problem; Relative Motion
11.2 Motion in a Central Potential
11.2.1 Spherical Coordinates
11.2.2 Eigenfunctions Common to ˆ H, ˆL2 and ˆLz
11.3 The Hydrogen Atom
11.3.1 Orders of Magnitude: Appropriate Units in Atomic Physics
11.3.2 The Dimensionless Radial Equation
11.3.3 Spectrum of Hydrogen
11.3.4 Stationary States of the Hydrogen Atom
11.3.5 Dimensions and Orders of Magnitude
11.3.6 Time Evolution of States of Low Energies
11.4 Hydrogen-Like Atoms
11.5 Muonic Atoms
11.6 Spectra of Alkali Atoms
Further Reading
Exercises
12. Spin 1/2 and Magnetic Resonance
12.1 The Hilbert Space of Spin 1/2
12.1.1 Spin Observables
12.1.2 Representation in a Particular Basis
12.1.3 Matrix Representation
12.1.4 Arbitrary Spin State
12.2 Complete Description of a Spin-1/2 Particle
12.2.1 Hilbert Space
12.2.2 Representation of States and Observables
12.3 Spin Magnetic Moment
12.3.1 The Stern–Gerlach Experiment
12.3.2 Anomalous Zeeman Effect
12.3.3 Magnetic Moment of Elementary Particles
12.4 Uncorrelated Space and Spin Variables
12.5 Magnetic Resonance
12.5.1 Larmor Precession in a Fixed Magnetic Field B0
12.5.2 Superposition of a Fixed Field and a Rotating Field
12.5.3 Rabi’s Experiment
12.5.4 Applications of Magnetic Resonance
12.5.5 Rotation of a Spin 1/2 Particle by 2ð
Further Reading
Exercises
13. Addition of Angular Momenta,
Fine and Hyperfine Structure of Atomic Spectra
13.1 Addition of Angular Momenta
13.1.1 The Total-Angular Momentum Operator
13.1.2 Factorized and Coupled Bases
13.1.3 A Simple Case: the Addition of Two Spins of 1/2
13.1.4 Addition of Two Arbitrary Angular Momenta
13.1.5 One-Electron Atoms, Spectroscopic Notations
13.2 Fine Structure of Monovalent Atoms
13.3 Hyper.ne Structure; the 21 cm Line of Hydrogen
13.3.1 Interaction Energy
13.3.2 Perturbation Theory
13.3.3 Diagonalization of ˆ H1
13.3.4 The Effect of an External Magnetic Field
13.3.5 The 21 cm Line in Astrophysics
Further Reading
Exercises
14. Entangled States, EPR Paradox and Bell’s Inequality
14.1 The EPR Paradox and Bell’s Inequality
14.1.1 “God Does not Play Dice”
14.1.2 The EPR Argument
14.1.3 Bell’s Inequality
14.1.4 Experimental Tests
14.2 Quantum Cryptography
14.2.1 The Communication Between Alice and Bob
14.2.2 The Quantum Noncloning Theorem
14.2.3 Present Experimental Setups
14.3 The Quantum Computer
14.3.1 The Quantum Bits, or “Q-Bits”
14.3.2 The Algorithm of Peter Shor
14.3.3 Principle of a Quantum Computer
14.3.4 Decoherence
Further Reading
Exercises
15. The Lagrangian and Hamiltonian Formalisms, Lorentz Force in Quantum Mechanics
15.1 Lagrangian Formalism and the Least-Action Principle
15.1.1 Least Action Principle
15.1.2 Lagrange Equations
15.1.3 Energy
15.2 Canonical Formalism of Hamilton
15.2.1 Conjugate Momenta
15.2.2 Canonical Equations
15.2.3 Poisson Brackets
15.3 Analytical Mechanics and Quantum Mechanics
15.4 Classical Charged Particles in an Electromagnetic Field
15.5 Lorentz Force in Quantum Mechanics
15.5.1 Hamiltonian
15.5.2 Gauge Invariance
15.5.3 The Hydrogen Atom Without Spin in a Uniform Magnetic Field
15.5.4 Spin-1/2 Particle in an Electromagnetic Field
Further Reading
Exercises
16. Identical Particles and the Pauli Principle
16.1 Indistinguishability of Two Identical Particles
16.1.1 Identical Particles in Classical Physics
16.1.2 The Quantum Problem
16.2 Two-Particle Systems; the Exchange Operator
16.2.1 The Hilbert Space for the Two Particle System
16.2.2 The Exchange Operator Between Two Identical Particles
16.2.3 Symmetry of the States
16.3 The Pauli Principle
16.3.1 The Case of Two Particles
16.3.2 Independent Fermions and Exclusion Principle
16.3.3 The Case of N Identical Particles
16.3.4 Time Evolution
16.4 Physical Consequences of the Pauli Principle
16.4.1 Exchange Force Between Two Fermions
16.4.2 The Ground State of N Identical Independent Particles
16.4.3 Behavior of Fermion and Boson Systems at Low Temperature
16.4.4 Stimulated Emission and the Laser Effect
16.4.5 Uncertainty Relations for a System of N Fermions
16.4.6 Complex Atoms and Atomic Shells
Further Reading
Exercises
17. The Evolution of Systems
17.1 Time-Dependent Perturbation Theory
17.1.1 Transition Probabilities
17.1.2 Evolution Equations
17.1.3 Perturbative Solution
17.1.4 First-Order Solution: the Born Approximation
17.1.5 Particular Cases
17.1.6 Perturbative and Exact Solutions
17.2 Interaction of an Atom with an Electromagnetic Wave
17.2.1 The Electric-Dipole Approximation
17.2.2 Justification of the Electric Dipole Interaction
17.2.3 Absorption of Energy by an Atom
17.2.4 Selection Rules
17.2.5 Spontaneous Emission
17.2.6 Control of Atomic Motion by Light
17.3 Decay of a System
17.3.1 The Radioactivity of 57Fe
17.3.2 The Fermi Golden Rule
17.3.3 Orders of Magnitude
17.3.4 Behavior for Long Times
17.4 The Time-Energy Uncertainty Relation
17.4.1 Isolated Systems and Intrinsic Interpretations
17.4.2 Interpretation of Landau and Peierls
17.4.3 The Einstein–Bohr Controversy
Further Reading
Exercises
18. Scattering Processes
18.1 Concept of Cross Section
18.1.1 Definition of Cross Section
18.1.2 Classical Calculation
18.1.3 Examples
18.2 Quantum Calculation in the Born Approximation
18.2.1 Asymptotic States
18.2.2 Transition Probability
18.2.3 Scattering Cross Section
18.2.4 Validity of the Born Approximation
18.2.5 Example: the Yukawa Potential
18.2.6 Range of a Potential in Quantum Mechanics
18.3 Exploration of Composite Systems
18.3.1 Scattering Off a Bound State and the Form Factor
18.3.2 Scattering by a Charge Distribution
18.4 General Scattering Theory
18.4.1 Scattering States
18.4.2 The Scattering Amplitude
18.4.3 The Integral Equation for Scattering
18.5 Scattering at Low Energy
18.5.1 The Scattering Length
18.5.2 Explicit Calculation of a Scattering Length
18.5.3 The Case of Identical Particles
Further Reading
Exercises
19. Qualitative Physics on a Macroscopic Scale
19.1 Con.ned Particles and Ground State Energy
19.1.1 The Quantum Pressure
19.1.2 Hydrogen Atom
19.1.3 N-Fermion Systems and Complex Atoms
19.1.4 Molecules, Liquids and Solids
19.1.5 Hardness of a Solid
19.2 Gravitational Versus Electrostatic Forces
19.2.1 Screening of Electrostatic Interactions
19.2.2 Additivity of Gravitational Interactions
19.2.3 Ground State of a Gravity-Dominated Object
19.2.4 Liquefaction of a Solid and the Height of Mountains
19.3 White Dwarfs, Neutron Stars and the Gravitational Catastrophe
19.3.1 White Dwarfs and the Chandrasekhar Mass
19.3.2 Neutron Stars
Further Reading
20. Early History of Quantum Mechanics
20.1 The Origin of Quantum Concepts
20.1.1 Planck’s Radiation Law
20.1.2 Photons
20.2 The Atomic Spectrum
20.2.1 Empirical Regularities of Atomic Spectra
20.2.2 The Structure of Atoms
20.2.3 The Bohr Atom
20.2.4 The Old Theory of Quanta
20.3 Spin
20.4 Heisenberg’s Matrices
20.5 Wave Mechanics
20.6 The Mathematical Formalization
20.7 Some Important Steps in More Recent Years
Further Reading
Appendix A. Concepts of Probability Theory
1 Fundamental Concepts
2 Examples of Probability Laws
2.1 Discrete Laws
2.2 Continuous Probability Laws in One or Several Variables
3 Random Variables
3.1 Definition
3.2 Conditional Probabilities
3.3 Independent Random Variables
3.4 Binomial Law and the Gaussian Approximation
4 Moments of Probability Distributions
4.1 Mean Value or Expectation Value
4.2 Variance and Mean Square Deviation
4.3 Bienaym´e–Tchebyche. Inequality
4.4 Experimental Verification of a Probability Law
Exercises
Appendix B. Dirac Distribution, Fourier Transformation
1 Dirac Distribution, or ä “Function”
1.1 Definition of d(x)
1.2 Examples of Functions Which Tend to ä(x)
1.3 Properties of ä(x)
2 Distributions
2.1 The Space S
2.2 Linear Functionals
2.3 Derivative of a Distribution
2.4 Convolution Product
3 Fourier Transformation
3.1 Definition
3.2 Fourier Transform of a Gaussian
3.3 Inversion of the Fourier Transformation
3.4 Parseval–Plancherel Theorem
3.5 Fourier Transform of a Distribution
3.6 Uncertainty Relation
Exercises
Appendix C. Operators in Infinite-Dimensional Spaces
1 Matrix Elements of an Operator
2 Continuous Bases
Appendix D. The Density Operator
1 Pure States
1.1 A Mathematical Tool: the Trace of an Operator
1.2 The Density Operator of Pure States
1.3 Alternative Formulation of Quantum Mechanics for Pure States
2 Statistical Mixtures
2.1 A Particular Case: an Unpolarized Spin-1/2 System
2.2 The Density Operator for Statistical Mixtures
3 Examples of Density Operators
3.1 The Micro-Canonical and Canonical Ensembles
3.2 The Wigner Distribution of a Spinless Point Particle
4 Entangled Systems
4.1 Reduced Density Operator
4.2 Evolution of a Reduced Density Operator
4.3 Entanglement and Measurement
Further Reading
Exercises
Solutions to the Exercises
Index