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• Cover
• Foreword
• Acknowledgments
• Chapter I: Waves and particles. Introduction to the fundamental ideas of quantum mechanics
• A. Electromagnetic waves and photons
• B. Material particles and matter waves
• C. Quantum description of a particle. Wave packets
• D. Particle in a time-independent scalar potential
• COMPLEMENTS OF CHAPTER I, READER’S GUIDE
• Complement AI Order of magnitude of the wavelengths associated with material particles
• Complement BI Constraints imposed by the uncertainty relations
• 1. Macroscopic system
• 2. Microscopic system
• Complement CI Heisenberg relation and atomic parameters
• Complement DI An experiment illustrating the Heisenberg relations
• Complement EI A simple treatment of a two-dimensional wave packet
• 1. Introduction
• 2. Angular dispersion and lateral dimensions
• 3. Discussion
• Complement FI The relationship between one- and three-dimensional problems
• 1. Three-dimensional wave packet
• 2. Justification of one-dimensional models
• Complement GI One-dimensional Gaussian wave packet: spreading of the wave packet
• 1. Definition of a Gaussian wave packet
• 2. Calculation of ∆x and ∆p; uncertainty relation
• 3. Evolution of the wave packet
• Complement HI Stationary states of a particle in one-dimensional square potentials
• 1. Behavior of a stationary wave function φ(x)
• 2. Some simple cases
• Complement JI Behavior of a wave packet at a potential step
• 1. Total reflection: E < V0
• 2. Partial reflection: E > V0
• Complement KI
• Exercises
• 2. Bound state of a particle in a “delta function potential”
• 3. Transmission of a “delta function” potential barrier
• 4. Return to exercise 2, using this time the Fourier transform.
• 5. Well consisting of two delta functions
• Chapter II: The mathematical tools of quantum mechanics
• A. Space of the one-particle wave function
• B. State space. Dirac notation
• C. Representations in state space
• D. Eigenvalue equations. Observables
• E. Two important examples of representations and observables
• F. Tensor product of state spaces11
• COMPLEMENTS OF CHAPTER II, , READER’S GUIDE
• Complement AII The Schwarz inequality
• Complement BII Review of some useful properties of linear operators
• 1. Trace of an operator
• 2. Commutator algebra
• 3. Restriction of an operator to a subspace
• 4. Functions of operators
• 5. Derivative of an operator
• Complement CII Unitary operators
• 1. General properties of unitary operators
• 2. Unitary transformations of operators
• 3. The infinitesimal unitary operator
• Complement DII A more detailed study of the { |r〉 }and { |P〉 } representations
• 1. The { |r〉 } representation
• 2. The { |P〉 } representation
• Complement EII Some general properties of two observables, Q and P, whose commutator is equal to 
• 1. The operator S(λ): definition, properties
• 2. Eigenvalues and eigenvectors of Q
• 3. The q representation
• 4. The representation. The symmetric nature of the P and Q observables
• Complement FII The parity operator
• 1. The parity operator
• 2. Even and odd operators
• 4. Application to an important special case
• Complement GII An application of the properties of the tensor product: the two-dimensional infinite well
• 1. Definition; eigenstates
• 2. Study of the energy levels
• Complement HII Exercises
• Dirac notation. Commutators. Eigenvectors and eigenvalues
• Complete sets of commuting observables, C.S.C.O.
• Solution of exercise 11
• Solution of exercise 12
• Chapter III: The postulates of quantum mechanics
• A. Introduction
• B. Statement of the postulates
• C. The physical interpretation of the postulates concerning observables and their measurement
• D. The physical implications of the Schrödinger equation
• E. The superposition principle and physical predictions
• COMPLEMENTS OF CHAPTER III, READER’S GUIDE
• Complement AIII Particle in an infinite potential well
• 1. Distribution of the momentum values in a stationary state
• 2. Evolution of the particle’s wave function
• 3. Perturbation created by a position measurement
• Complement BIII Study of the probability current in some special cases
• 1. Expression for the current in constant potential regions
• 2. Application to potential step problems
• 3. Probability current of incident and evanescent waves, in the case of reflection from a two-dimensional potential step
• Complement CIII Root mean square deviations of two conjugate observables
• 1. The Heisenberg relation for P and Q
• 2. The “minimum” wave packet
• Complement DIII Measurements bearing on only one part of a physical system
• 1. Calculation of the physical predictions
• 2. Physical meaning of a tensor product state
• 3. Physical meaning of a state that is not a tensor product
• Complement EIII The density operator
• 1. Outline of the problem
• 2. The concept of a statistical mixture of states
• 3. The pure case. Introduction of the density operator
• 4. A statistical mixture of states (non-pure case)
• 5. Use of the density operator: some applications
• Complement FIII The evolution operator
• 1. General properties
• 2. Case of conservative systems
• Complement GIII The Schrödinger and Heisenberg pictures
• Complement HIII Gauge invariance
• 1. Outline of the problem: scalar and vector potentials associated with an electromagnetic field; concept of a gauge
• 2. Gauge invariance in classical mechanics
• 3. Gauge invariance in quantum mechanics
• Complement JIII Propagator for the Schrödinger equation
• 1. Introduction
• 2. Existence and properties of a propagator K(2, 1)
• 3. Lagrangian formulation of quantum mechanics
• Complement KIII Unstable states. Lifetime
• 1. Introduction
• 2. Definition of the lifetime
• 3. Phenomenological description of the instability of a state
• Complement LIII Exercises
• Complement MIII Bound states in a “potential well” of arbitrary shape
• 1. Quantization of the bound state energies
• 2. Minimum value of the ground state energy
• Complement NIII Unbound states of a particle in the presence of a potential well or barrier of arbitrary shape
• 1. Transmission matrix M(k)
• 2. Transmission and reflection coefficients
• 3. Example
• Complement OIII Quantum properties of a particle in a one-dimensional periodic structure
• 1. Passage through several successive identical potential barriers
• 2. Discussion: the concept of an allowed or forbidden energy band
• 3. Quantization of energy levels in a periodic potential; effect of boundary conditions
• Chapter IV: Application of the postulates to simple cases: spin 1/2 and two-level systems
• A. Spin 1/2 particle: quantization of the angular momentum
• B. Illustration of the postulates in the case of a spin 1/2
• C. General study of two-level systems
• COMPLEMENTS OF CHAPTER IV, READER’S GUIDE
• Complement AIV The Pauli matrices
• 1. Definition; eigenvalues and eigenvectors
• 2. Simple properties
• 3. A convenient basis of the 2 2 matrix space
• Complement BIV Diagonalization of a 2 2 Hermitian matrix
• 1. Introduction
• 2. Changing the eigenvalue origin
• 3. Calculation of the eigenvalues and eigenvectors
• Complement CIV Fictitious spin 1/2 associated with a two-level system
• 1. Introduction
• 2. Interpretation of the Hamiltonian in terms of fictitious spin
• 3. Geometrical interpretation of the various effects discussed in § C of Chapter IV
• Complement DIV System of two spin 1/2 particles
• 1. Quantum mechanical description
• 2. Prediction of the measurement results
• Complement EIV Spin 1 2 density matrix
• 1. Introduction
• 2. Density matrix of a perfectly polarized spin (pure case)
• 3. Example of a statistical mixture: unpolarized spin
• 4. Spin 1/2 at thermodynamic equilibrium in a static field
• 5. Expansion of the density matrix in terms of the Pauli matrices
• Complement FIV Spin 1/2 particle in a static and a rotating magnetic fields: magnetic resonance
• 1. Classical treatment; rotating reference frame
• 2. Quantum mechanical treatment
• 3. Relation between the classical treatment and the quantum mechanical treatment: evolution of M
• 4. Bloch equations
• Complement GIV A simple model of the ammonia molecule
• 1. Description of the model
• 2. Eigenfunctions and eigenvalues of the Hamiltonian
• 3. The ammonia molecule considered as a two-level system
• Complement HIV Effects of a coupling between a stable state and an unstable state
• 1. Introduction. Notation
• 2. Influence of a weak coupling on states of different energies
• 3. Influence of an arbitrary coupling on states of the same energy
• Chapter V: The one-dimensional harmonic oscillator
• A. Introduction
• B. Eigenvalues of the Hamiltonian
• C. Eigenstates of the Hamiltonian
• D. Discussion
• Complement AV Some examples of harmonic oscillators
• 1. Vibration of the nuclei of a diatomic molecule
• 2. Vibration of the nuclei in a crystal
• 3. Torsional oscillations of a molecule: ethylene
• 4. Heavy muonic atoms
• Complement BV Study of the stationary states in the representation. Hermite polynomials
• 1. Hermite polynomials
• 2. The eigenfunctions of the harmonic oscillator Hamiltonian
• Complement CV Solving the eigenvalue equation of the harmonic oscillator by the polynomial method
• 1. Changing the function and the variable
• 2. The polynomial method
• Complement DV Study of the stationary states in the representation
• 1. Wave functions in momentum space
• 2. Discussion
• Complement EV The isotropic three-dimensional harmonic oscillator
• 1. The Hamiltonian operator
• 2. Separation of the variables in Cartesian coordinates
• 3. Degeneracy of the energy levels
• Complement FV A charged harmonic oscillator in a uniform electric field
• 1. Eigenvalue equation of in the representation
• 2. Discussion
• 3. Use of the translation operator
• Complement GV Coherent “quasi-classical” states of the harmonic oscillator
• 1. Quasi-classical states
• 2. Properties of the states
• 3. Time evolution of a quasi-classical state
• 4. Example: quantum mechanical treatment of a macroscopic oscillator
• Complement HV Normal vibrational modes of two coupled harmonic oscillators
• 1. Vibration of the two coupled in classical mechanics
• 2. Vibrational states of the system in quantum mechanics
• Complement JV Vibrational modes of an infinite linear chain of coupled harmonic oscillators; phonons
• 1. Classical treatment
• 2. Quantum mechanical treatment
• 3. Application to the study of crystal vibrations: phonons
• Complement KV Vibrational modes of a continuous physical system. Application to radiation; photons
• 1. Outline of the problem
• 2. Vibrational modes of a continuous mechanical system: example of a vibrating string
• 3. Vibrational modes of radiation: photons
• Complement LV One-dimensional harmonic oscillator in thermodynamic equilibrium at a temperature T
• 1. Mean value of the energy
• 2. Discussion
• 3. Applications
• 4. Probability distribution of the observable X
• Complement MV Exercises
• Chapter VI: General properties of angular momentum in quantum mechanics
• A. Introduction: the importance of angular momentum
• B. Commutation relations characteristic of angular momentum
• C. General theory of angular momentum
• D. Application to orbital angular momentum
• Complement AVI Spherical harmonics
• 1. Calculation of spherical harmonics
• 2. Properties of spherical harmonics
• Complement BVI Angular momentum and rotations
• 1. Introduction
• 2. Brief study of geometrical rotations
• 3. Rotation operators in state space. Example: a spinless particle
• 4. Rotation operators in the state space of an arbitrary system
• 5. Rotation of observables
• 6. Rotation invariance
• Complement CVI Rotation of diatomic molecules
• 1. Introduction
• 2. Rigid rotator. Classical study
• 3. Quantization of the rigid rotator
• 4. Experimental evidence for the rotation of molecules
• Complement DVI Angular momentum of stationary states of a two-dimensional harmonic oscillator
• 1. Introduction
• 2. Classification of the stationary states by the quantum numbers nx and ny
• 3. Classification of the stationary states in terms of their angular momenta
• 4. Quasi-classical states
• Complement EVI A charged particle in a magnetic field: Landau levels
• 1. Review of the classical problem
• 2. General quantum mechanical properties of a particle in a magnetic field
• 3. Case of a uniform magnetic field
• Chapter VII: Particle in a central potential. The hydrogen atom
• A. Stationary states of a particle in a central potential
• B. Motion of the center of mass and relative motion for a system of two interacting particles
• C. The hydrogen atom
• COMPLEMENTS OF CHAPTER VII, READER’S GUIDE
• Complement AVII Hydrogen-like systems
• 1. Hydrogen-like systems with one electron
• 2. Hydrogen-like systems without an electron
• Complement BVII A soluble example of a central potential: the isotropic three-dimensional harmonic oscillator
• 1. Solving the radial equation
• 2. Energy levels and stationary wave functions
• Complement CVII Probability currents associated with the stationary states of the hydrogen atom
• 1. General expression for the probability current
• 2. Application to the stationary states of the hydrogen atom
• Complement DVII The hydrogen atom placed in a uniform magnetic field. Paramagnetism and diamagnetism. The Zeeman effect
• 1. The Hamiltonian of the problem. The paramagnetic term and the diamagnetic term
• 2. The Zeeman effect
• Complement EVII Some atomic orbitals. Hybrid orbitals
• 1. Introduction
• 2. Atomic orbitals associated with real wave functions
• 3. sp hybridization
• 4. sp2 hybridization
• 5. sp3 hybridization
• Complement FVII Vibrational-rotational levels of diatomic molecules
• 1. Introduction
• 2. Approximate solution of the radial equation
• 3. Evaluation of some corrections
• Complement GVII Exercises
• 1. Particle in a cylindrically symmetric potential
• 2. Three-dimensional harmonic oscillator in a uniform magnetic field
• Index [The notation (ex.) refers to an exercise]
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