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WORLD SCIENTIFIC eBooks

Lecture Notes on Quantum Mechanics

Samuel D Lindenbaum

Published Apr 1, 1999• 57 citations
Publisher

Abstract

Failure of the classical theories wave packets wave functions and probability Schrodinger's Equation quantum mechanical operators Schrodinger's Equation with forces continuous spectrum Eigensolutions the harmonic oscillator three-dimensional problems matrices matrices in different representations approximate methods I applications of WKB method alpha decay approximate methods II continuous spectrum perturbation approximate methods III spin and spinors fine structure of the hydrogen atom anomalous Zeeman effect assemblages of identical particles the Dirac Equation non-relativistic limits of the Dirac Equation Dirac operator and wave function transformations negative energy solutions - positron theory relativistic theory of Coulomb scattering Compton scattering.

DOI
10.1142/4072

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Introductory Quantum Mechanics

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Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)

•Jan 1, 1990•15 citations

Quantum Mechanics

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This book covers the entire span of quantum mechanics whose developments have taken place during the early part of the twentieth century up till the present day. We start with the Rutherford-Bohr model of the atom followed by Schrodinger's wave mechanics with its application to the solution of calculating the energy spectrum of a particle in a box, the harmonic oscillator and finally the hydrogen atom. Heisenberg's matrix mechanics and its duality with Schrodinger's wave mechanics, quantum mechanics in the interaction picture. Dirac's relativistic theory of the electron exhibiting the spin of the electron as a relativistic effect when it interacts with an external electromagnetic field. Feynman's path integral approach to non-relativistic quantum mechanics with is a marvellous intuitive interpretation as a sum over paths and how classical mechanics is obtained from its limit as Planck' constant tends to zero, methods for computing the spectra of the Dirac Hamiltonian in a radial potential, quantum field theory as developed by Feynman, Schwinger, Tomonaga and Dyson for describing the interaction between electrons, positrons, and photons via propagators using both the operator theoretic expansions and Feynman's path integral. We also introduce time independent and time dependent perturbation theory in quantum mechanics with applications to quantum gate design for quantum computers forming a major part of the research conducted by the author's research group, Quantum noise introduced into the Schrodinger and Dirac's equation based on the Hudson-Parthasarathy quantum stochastic calculus in Boson Fock space, scattering theory and wave operators with applications to quantum gate design, some aspects of second quantization like the interpretation of Boson Fock space in terms of harmonic oscillator algebras and the BCS theory of superconductivity, Wigner-Mackey-Frobenius theory of induced representations of a group with applications to Wigner's theory of particle classification, Dirac's equation in a gravitational field and Yang-Mills non-Abelian gauge theories with application to the construction of unified quantum field theories and finally, the more recent theory of super-symmetry which is a Boson-Fermion unification theory. We have discussed the statistics of Boson's, Fermions and Maxwell-Boltzmann based on entropy maximization. The book is written in problem-solution format and it would be of use to physicists and engineers interested respectively in developing unified field theories and in the design of quantum gates. Note: T&F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.

CRC Press eBooks•Jul 29, 2021•0 citations

Quantum Mechanics: An Accessible Introduction

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Table of Contents 1. The Origins of Quantum Mechanics 1.1 Introduction 1.2 Blackbody Radiation The Problem with Blackbody Radiation 1.3 The Nature of Light The Photoelectric Effect The Compton Effect Is it a Particle or a Wave? 1.4 TheWave Nature of Matter 1.5 The Bohr Atom 1.6 Where do we Stand? 2. Math Interlude A: Complex Numbers and Linear Operators 2.1 Complex Numbers 2.2 Operators Definition of an Operator Eigenfunctions and Eigenvalues 3. The Schrodinger Equation 3.1 Derivation of the Schrodinger Equation 3.2 The Meaning of theWave Function 3.3 The Time-Independent Schrodinger Equation Derivation of the Time-Independent Schrodinger Equation Qualitative Solutions and the Origin of Quantization 4. One-Dimensional Time-Independent Schrodinger Equation 4.1 Unbound States: Scattering and Tunneling Scattering From Step-Function Potentials 4.2 Bound Systems The Infinite SquareWell The Harmonic Oscillator Potential 5. Math Interlude B: Linear Algebra 5.1 Properties of Linear Operators 5.2 Vector Spaces Inner Products Adjoint and Hermitian Operators Basis Sets 6. The Three-Dimensional Time-Independent Schrodinger Equation 6.1 Solution in Rectangular Coordinates 6.2 Angular Momentum 6.3 The Schrodinger Equation in Spherical Coordinates 6.4 The Hydrogen Atom 7. Math Interlude C: Matrices, Dirac Notation, and the Dirac Delta Function 7.1 The Matrix Formulation of Linear Operators 7.2 Dirac Notation 7.3 The Dirac Delta Function 8. Spin Angular Momentum 8.1 Spin Operators 8.2 Evidence for Spin 8.3 Adding Angular Momentum 8.4 The Matrix Representation of Spin 8.5 The Stern-Gerlach Experiment 8.6 Spin Precession 8.7 Spin Systems with Two Particles Noninteracting Spins Interacting Spins 8.8 Measurement Theory Hidden Variables The ManyWorlds Interpretation of Quantum Mechanics 9. Time-Independent Perturbation Theory 9.1 Derivation of Time-Independent Perturbation Theory 9.2 Perturbations to the Atomic Energy Levels Fine Structure The Lamb Shift 9.3 The Atom in External Electric or Magnetic Fields The Atom in an Electric Field: The Stark Effect The Atom in a Magnetic Field: The Zeeman Effect 10. The Variational Principle 10.1 Variational Principle: Theory 10.2 Variational Principle: Application to the Helium Atom 11. Time-Dependent Perturbation Theory 11.1 Derivation of Time-Dependent Perturbation Theory 11.2 Application: Selection Rules for Electromagnetic Radiation 12. Scattering Theory 12.1 Definition of the Cross Section 12.2 The Born Approximation 12.3 PartialWaves 13. The Multiparticle Schrodinger Equation 13.1 Wave Function for Identical Particles 13.2 Multielectron Atoms 14. Some Modern Applications of Quantum Mechanics 14.1 Magnetic Resonance Imaging 14.2 Quantum Computing 15. What Comes Next? Relativistic Quantum Mechanics 15.1 The Klein-Gordon Equation Derivation of the Klein-Gordon Equation Probability Densities and Currents 15.2 The Dirac Equation Answers and Hints for Selected End-of-Chapter Exercises Index

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Preface. 1. Origins of Quantum Physics. 1.1 Historical Note. 1.2 Particle Aspect of Radiation. 1.3 Wave Aspect of Particles. 1.4 Particles versus Waves. 1.5 Indeterministic Nature of the Microphysical World. 1.6 Atomic Transitions and Spectroscopy. 1.7 Quantization Rules. 1.8 Wave Packets. 1.9 Concluding Remarks. 1.10 Solved Problems. Exercises. 2. Mathematical Tools of Quantum Mechanics. 2.1 Introduction. 2.2 The Hilbert Space and Wave Functions. 2.3 Dirac Notation. 2.4 Operators. 2.5 Representation in Discrete Bases. 2.6 Representation in Continuous Bases. 2.7 Matrix and Wave Mechanics. 2.8 Concluding Remarks. 2.9 Solved Problems. Exercises. 3. Postulates of Quantum Mechanics. 3.1 Introduction. 3.2 The Basic Postulates of Quantum Mechanics. 3.3 The State of a System. 3.4 Observables and Operators. 3.5 Measurement in Quantum Mechanics. 3.6 Time Evolution of the System's State. 3.7 Symmetries and Conservation Laws. 3.8 Connecting Quantum to Classical Mechanics. 3.9 Solved Problems. Exercises. 4. One-Dimensional Problems. 4.1 Introduction. 4.2 Properties of One-Dimensional Motion. 4.3 The Free Particle: Continuous States. 4.4 The Potential Step. 4.5 The Potential Barrier and Well. 4.6 The Infinite Square Well Potential. 4.7 The Finite Square Well Potential. 4.8 The Harmonic Oscillator. 4.9 Numerical Solution of the Schrodinger Equation. 4.10 Solved Problems. Exercises. 5. Angular Momentum. 5.1 Introduction. 5.2 Orbital Angular Momentum. 5.3 General Formalism of Angular Momentum. 5.4 Matrix Representation of Angular Momentum. 5.5 Geometrical Representation of Angular Momentum. 5.6 Spin Angular Momentum. 5.7 Eigen functions of Orbital Angular Momentum. 5.8 Solved Problems. Exercises. 6. Three-Dimensional Problems. 6.1 Introduction. 6.2 3D Problems in Cartesian Coordinates. 6.3 3D Problems in Spherical Coordinates. 6.4 Concluding Remarks. 6.5 Solved Problems. Exercises. 7. Rotations and Addition of Angular Momenta. 7.1 Rotations in Classical Physics. 7.2 Rotations in Quantum Mechanics. 7.3 Addition of Angular Momenta. 7.4 Scalar, Vector and Tensor Operators. 7.5 Solved Problems. Exercises. 8. Identical Particles. 8.1 Many-Particle Systems. 8.2 Systems of Identical Particles. 8.3 The Pauli Exclusion Principle. 8.4 The Exclusion Principle and the Periodic Table. 8.5 Solved Problems. Exercises. 9. Approximation Methods for Stationary States. 9.1 Introduction. 9.2 Time-Independent Perturbation Theory. 9.3 The Variational Method. 9.4 The Wentzel Kramers Brillou in Method. 9.5 Concluding Remarks. 9.6 Solved Problems. Exercises. 10. Time-Dependent Perturbation Theory. 10.1 Introduction. 10.2 The Pictures of Quantum Mechanics. 10.3 Time-Dependent Perturbation Theory. 10.4 Adiabatic and Sudden Approximations. 10.5 Interaction of Atoms with Radiation. 10.6 Solved Problems. Exercises. 11. Scattering Theory. 11.1 Scattering and Cross Section. 11.2 Scattering Amplitude of Spinless Particles. 11.3 The Born Approximation. 11.4 Partial Wave Analysis. 11.5 Scattering of Identical Particles. 11.6 Solved Problems. Exercises. A. The Delta Function. A.1 One-Dimensional Delta Function. A.2 Three-Dimensional Delta Function. B. Angular Momentum in Spherical Coordinates. B.1 Derivation of Some General. B.2 Gradient and Laplacianin Spherical Coordinates. B.3 Angular Momentum in Spherical Coordinates. C. Computer Code for Solving the Schrodinger Equation. Index.

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Quantum mechanics : for engineering, materials science, and applied physics

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1. Wave-Particle Duality and Schroedinger Equation. 2. Introduction to Bound States. 3. Rotationally Invariant Potentials: Hydrogen Atom and Beyond. 4. Wave Packets and Uncertainty Relations. 5. Scattering by Simple Barriers. 6. WKB Approximations. 7. Expectation Values and Operators. 8. Electrons in a Magnetic Field. 9. Beyond Hermitian Operators. 10. Harmonic Oscillator: Full Operator Treatment. 11. Composite Systems. 12. Variational Principle. 13. Expansion Principle and Matrix Formulation. 14. Perturbation Theory, I: Degenerate Perturbation Theory. 15. Perturbation Theory, II: Non-Degenerate Perturbation Theory. 16. Symmetry. 17. Electrons in Periodic Crystal Potentials. 18. Rotational Invariance and Angular Momentum. 19. Time-Dependent Perturbation Theory. 20. Elements of Field Quantization. 21. Electron Spin. 22. Indistinguishable Particles: Fermions and Bosons. Appendices: Dirac ...d-Function. Poisson-Distributed Events. Spherical Harmonics. Hydrogen Radial Eigenfunctions. Fourier Integral. Construction of Two Group Character Tables. Selected General References. Fundamental Constants. Index.

Prentice-Hall International eBooks•Jan 1, 1994•15 citations

Theory of quanta

Iwo Białynicki‐Birula, Marek Cieplak, Jerzy Zdzisław Kamiński +1 more

1: The Birth of Quantum Theory. 1.1: The radiation of an ideal black body. 1.2: Line spectra and Bohr's model of the atom. 1.3: The quantum conditions of Bohr-Sommerfield. 1.4: Waves of matters. Comments. Problems. 2: Classical Physics - Quantum Physics. 2.1: The difficulties of classical theory. 2.2: The successes of quantum theory. 2.3: Problems with interpretation of quantum theory. 2.4: Quantum theory (1930. Problems. II: Schrodinger's Equation and the Interpretation of its Solutions. 3: Schrodinger's Equation. 3.1: The Schrodinger equation with time. 3.2: The time-dependent Schrodinger equation. Comments. Problems. 4: The Probabilistic Interpretation of the Wavefunction. Comments. Problems. 5: The Motion of a Free Particle in Wave Mechanics. 5.1: The spreading of wave packets. 5.2: The initial problem for Schrodinger's free particle equation. 5.3: The Gaussian wave packet. Comments. Problems. 6: The Momentum of a Particle and the Fourier Transformation of the Wavefunction. Comments. Problems. 7: Description of Experiments in Wave Mechanics and the Heinsberg Uncertainty Principle. 7.1: Heisenberg's uncertainty principle. 7.2: The entropic uncertainty principle. Comments. Problems. 8: A Charged Particle in an Electromagnetic Field. Comments. Problems. 9: The Hydrodynamic Formulation of Wave Mechanics. 9.1: The motion of the probability fluid in an electromagnetic field. Comments. Problems. 10: The Spin of the Elctron and Pauli's Equation. 10.1: The magnetic moment vector. 10.2: Pauli's equation. 10.3: The Stern-Gerlach experiments. Comments. Problems. 11: The Hydrodynamic Picture of Wave Mechanics of a Particle with Spin. Comments. Problems. 12: Ehrenfest's Equations. 12.1: A particle with spin in the electromagnetic field. Problems. 13: Descriptiopn of Particle Motion in Phase Space -- Wigner's Function. Comments. Problems. 14: Mixed States. 14.1: An unpolarized beam as a mixed state. Comments. Problems. III: Solutions of Schrodinger's Equation. 15: Classification of Solutions to Schrodinger's Equation. 15.1: Scattering states. 15.2: The completeness of the set of bound and scattering state wavefunctions. 15.3: Orthogonality of the wavefunctions. 15.4: Development of the system in time. 15.5: Bound and scattered states. Problems. 16: Bound States and Energy. 16.1: The three-dimensional spherically symmetric potential well. Comments. Problems. 17: Scattering States and Collision Cross-sections. 17.1: One-dimensional scattering. 17.2: The method of integral equations. 17.3: Three-dimensional scattering. 17.4: The scattering amplitude. 17.5: The S matrix. 17.6: Collision cross-section. 17.7: The optical theorem. Comments. Problems. 18: Partial Waves and Wave Effects in Scattering. 18.1: Phase shifts. 18.2: The spherical potential hump. 18.3: The Ramsauer-Townsend effect. 18.4: The extinction paradox. Comments. Problems. 19: Resonance States. Problems. 20: The Coulomb Potential. 20.1: Bound states. 20.2: Scattering states. Comments. Problems. 21

Oxford University Press eBooks•Jan 1, 1992•80 citations
Failure of the classical theories wave packets wave functions and probability Schrodinger's Equation quantum mechanical operators Schrodinger's Equation with forces continuous spectrum Eigensolutions the harmonic oscillator three-dimensional problems matrices matrices in different representations approximate methods I applications of WKB method alpha decay approximate methods II continuous spectrum perturbation approximate methods III spin and spinors fine structure of the hydrogen atom anomalous Zeeman effect assemblages of identical particles the Dirac Equation non-relativistic limits of the Dirac Equation Dirac operator and wave function transformations negative energy solutions - positron theory relativistic theory of Coulomb scattering Compton scattering.
Part 1: the need for the quantum mechanics quantization of radiation the special theory of relativity wave-particle duality the nature of matter waves the uncertainty principle the Bohr atom. Part 2 Schrodinger's equation: the postulates of quantum mechanics the wave equation the wave equation in three dimensions the time-independent Schrodinger equation stationary states normalisation probability flux physical constraints on the wave-function wave packets Gaussian wave-packets and the uncertainty principle. Part 3 One-dimensional potential wells: potential wells the one-dimensional infinite square well discussion of infinite square-well results eigenvalue equations the finite square well finite square well eigenfunctions the control of the wave-function by the wave equation numerical integration of Schrodinger's equation continuum solutions of the square-well potential interpretation of unbound-particle wave functions the double potential well the quantum bouncer electric fields and the tilted well the harmonic oscillator - introduction the quantum oscillator discussion of harmonic oscillator eigenvalues and eigenfunctions the classical probability density more potential wells. Part 4 Schrodinger's equation in three dimensions: particle in a three dimensional box degeneracy Schrodingers equation in spherical polar coordinates - the hydrogen atom normalisation. Part 5 Formal development of quantum mechanics: eigenvalues equations, operators and measurements the momentum operator, eigenvalues and eigenfunctions the position operator and its eigenfunctions Hermitian operators orthogonality of eigenfunctions the expansion postulate example - the b-decay of tritium expectation values compatibility of observables - commutators non-commuting operators time-dependence of expectation values - Ehrenfest's theorem abstract operators - the parity operator, matrix mechanics. Part 6 Angular momentum: angular momentum operators angular momentum commutators angular momentum eigenvalues and eigenfunctions interpretation of angular momentum eigenvalues interpretation of angular momentum eigenfunctions the Stern-Gerlach apparatus and electron spin spin operators - the Pauli Spin matrices the Zeeman effect measurements in quantum mechanics. Part 7 Spherically symmetric potential wells: the rigid rotator the radial equation the hydrogen atom and hydrogen like ions the isotropic harmonic oscillator molecular vibrations - the Morse potential the spherical square well general properties of spherically symmetric potential wells. (Part contents)
This book covers the entire span of quantum mechanics whose developments have taken place during the early part of the twentieth century up till the present day. We start with the Rutherford-Bohr model of the atom followed by Schrodinger's wave mechanics with its application to the solution of calculating the energy spectrum of a particle in a box, the harmonic oscillator and finally the hydrogen atom. Heisenberg's matrix mechanics and its duality with Schrodinger's wave mechanics, quantum mechanics in the interaction picture. Dirac's relativistic theory of the electron exhibiting the spin of the electron as a relativistic effect when it interacts with an external electromagnetic field. Feynman's path integral approach to non-relativistic quantum mechanics with is a marvellous intuitive interpretation as a sum over paths and how classical mechanics is obtained from its limit as Planck' constant tends to zero, methods for computing the spectra of the Dirac Hamiltonian in a radial potential, quantum field theory as developed by Feynman, Schwinger, Tomonaga and Dyson for describing the interaction between electrons, positrons, and photons via propagators using both the operator theoretic expansions and Feynman's path integral. We also introduce time independent and time dependent perturbation theory in quantum mechanics with applications to quantum gate design for quantum computers forming a major part of the research conducted by the author's research group, Quantum noise introduced into the Schrodinger and Dirac's equation based on the Hudson-Parthasarathy quantum stochastic calculus in Boson Fock space, scattering theory and wave operators with applications to quantum gate design, some aspects of second quantization like the interpretation of Boson Fock space in terms of harmonic oscillator algebras and the BCS theory of superconductivity, Wigner-Mackey-Frobenius theory of induced representations of a group with applications to Wigner's theory of particle classification, Dirac's equation in a gravitational field and Yang-Mills non-Abelian gauge theories with application to the construction of unified quantum field theories and finally, the more recent theory of super-symmetry which is a Boson-Fermion unification theory. We have discussed the statistics of Boson's, Fermions and Maxwell-Boltzmann based on entropy maximization. The book is written in problem-solution format and it would be of use to physicists and engineers interested respectively in developing unified field theories and in the design of quantum gates. Note: T&F does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka.
Table of Contents 1. The Origins of Quantum Mechanics 1.1 Introduction 1.2 Blackbody Radiation The Problem with Blackbody Radiation 1.3 The Nature of Light The Photoelectric Effect The Compton Effect Is it a Particle or a Wave? 1.4 TheWave Nature of Matter 1.5 The Bohr Atom 1.6 Where do we Stand? 2. Math Interlude A: Complex Numbers and Linear Operators 2.1 Complex Numbers 2.2 Operators Definition of an Operator Eigenfunctions and Eigenvalues 3. The Schrodinger Equation 3.1 Derivation of the Schrodinger Equation 3.2 The Meaning of theWave Function 3.3 The Time-Independent Schrodinger Equation Derivation of the Time-Independent Schrodinger Equation Qualitative Solutions and the Origin of Quantization 4. One-Dimensional Time-Independent Schrodinger Equation 4.1 Unbound States: Scattering and Tunneling Scattering From Step-Function Potentials 4.2 Bound Systems The Infinite SquareWell The Harmonic Oscillator Potential 5. Math Interlude B: Linear Algebra 5.1 Properties of Linear Operators 5.2 Vector Spaces Inner Products Adjoint and Hermitian Operators Basis Sets 6. The Three-Dimensional Time-Independent Schrodinger Equation 6.1 Solution in Rectangular Coordinates 6.2 Angular Momentum 6.3 The Schrodinger Equation in Spherical Coordinates 6.4 The Hydrogen Atom 7. Math Interlude C: Matrices, Dirac Notation, and the Dirac Delta Function 7.1 The Matrix Formulation of Linear Operators 7.2 Dirac Notation 7.3 The Dirac Delta Function 8. Spin Angular Momentum 8.1 Spin Operators 8.2 Evidence for Spin 8.3 Adding Angular Momentum 8.4 The Matrix Representation of Spin 8.5 The Stern-Gerlach Experiment 8.6 Spin Precession 8.7 Spin Systems with Two Particles Noninteracting Spins Interacting Spins 8.8 Measurement Theory Hidden Variables The ManyWorlds Interpretation of Quantum Mechanics 9. Time-Independent Perturbation Theory 9.1 Derivation of Time-Independent Perturbation Theory 9.2 Perturbations to the Atomic Energy Levels Fine Structure The Lamb Shift 9.3 The Atom in External Electric or Magnetic Fields The Atom in an Electric Field: The Stark Effect The Atom in a Magnetic Field: The Zeeman Effect 10. The Variational Principle 10.1 Variational Principle: Theory 10.2 Variational Principle: Application to the Helium Atom 11. Time-Dependent Perturbation Theory 11.1 Derivation of Time-Dependent Perturbation Theory 11.2 Application: Selection Rules for Electromagnetic Radiation 12. Scattering Theory 12.1 Definition of the Cross Section 12.2 The Born Approximation 12.3 PartialWaves 13. The Multiparticle Schrodinger Equation 13.1 Wave Function for Identical Particles 13.2 Multielectron Atoms 14. Some Modern Applications of Quantum Mechanics 14.1 Magnetic Resonance Imaging 14.2 Quantum Computing 15. What Comes Next? Relativistic Quantum Mechanics 15.1 The Klein-Gordon Equation Derivation of the Klein-Gordon Equation Probability Densities and Currents 15.2 The Dirac Equation Answers and Hints for Selected End-of-Chapter Exercises Index
Preface. 1. Origins of Quantum Physics. 1.1 Historical Note. 1.2 Particle Aspect of Radiation. 1.3 Wave Aspect of Particles. 1.4 Particles versus Waves. 1.5 Indeterministic Nature of the Microphysical World. 1.6 Atomic Transitions and Spectroscopy. 1.7 Quantization Rules. 1.8 Wave Packets. 1.9 Concluding Remarks. 1.10 Solved Problems. Exercises. 2. Mathematical Tools of Quantum Mechanics. 2.1 Introduction. 2.2 The Hilbert Space and Wave Functions. 2.3 Dirac Notation. 2.4 Operators. 2.5 Representation in Discrete Bases. 2.6 Representation in Continuous Bases. 2.7 Matrix and Wave Mechanics. 2.8 Concluding Remarks. 2.9 Solved Problems. Exercises. 3. Postulates of Quantum Mechanics. 3.1 Introduction. 3.2 The Basic Postulates of Quantum Mechanics. 3.3 The State of a System. 3.4 Observables and Operators. 3.5 Measurement in Quantum Mechanics. 3.6 Time Evolution of the System's State. 3.7 Symmetries and Conservation Laws. 3.8 Connecting Quantum to Classical Mechanics. 3.9 Solved Problems. Exercises. 4. One-Dimensional Problems. 4.1 Introduction. 4.2 Properties of One-Dimensional Motion. 4.3 The Free Particle: Continuous States. 4.4 The Potential Step. 4.5 The Potential Barrier and Well. 4.6 The Infinite Square Well Potential. 4.7 The Finite Square Well Potential. 4.8 The Harmonic Oscillator. 4.9 Numerical Solution of the Schrodinger Equation. 4.10 Solved Problems. Exercises. 5. Angular Momentum. 5.1 Introduction. 5.2 Orbital Angular Momentum. 5.3 General Formalism of Angular Momentum. 5.4 Matrix Representation of Angular Momentum. 5.5 Geometrical Representation of Angular Momentum. 5.6 Spin Angular Momentum. 5.7 Eigen functions of Orbital Angular Momentum. 5.8 Solved Problems. Exercises. 6. Three-Dimensional Problems. 6.1 Introduction. 6.2 3D Problems in Cartesian Coordinates. 6.3 3D Problems in Spherical Coordinates. 6.4 Concluding Remarks. 6.5 Solved Problems. Exercises. 7. Rotations and Addition of Angular Momenta. 7.1 Rotations in Classical Physics. 7.2 Rotations in Quantum Mechanics. 7.3 Addition of Angular Momenta. 7.4 Scalar, Vector and Tensor Operators. 7.5 Solved Problems. Exercises. 8. Identical Particles. 8.1 Many-Particle Systems. 8.2 Systems of Identical Particles. 8.3 The Pauli Exclusion Principle. 8.4 The Exclusion Principle and the Periodic Table. 8.5 Solved Problems. Exercises. 9. Approximation Methods for Stationary States. 9.1 Introduction. 9.2 Time-Independent Perturbation Theory. 9.3 The Variational Method. 9.4 The Wentzel Kramers Brillou in Method. 9.5 Concluding Remarks. 9.6 Solved Problems. Exercises. 10. Time-Dependent Perturbation Theory. 10.1 Introduction. 10.2 The Pictures of Quantum Mechanics. 10.3 Time-Dependent Perturbation Theory. 10.4 Adiabatic and Sudden Approximations. 10.5 Interaction of Atoms with Radiation. 10.6 Solved Problems. Exercises. 11. Scattering Theory. 11.1 Scattering and Cross Section. 11.2 Scattering Amplitude of Spinless Particles. 11.3 The Born Approximation. 11.4 Partial Wave Analysis. 11.5 Scattering of Identical Particles. 11.6 Solved Problems. Exercises. A. The Delta Function. A.1 One-Dimensional Delta Function. A.2 Three-Dimensional Delta Function. B. Angular Momentum in Spherical Coordinates. B.1 Derivation of Some General. B.2 Gradient and Laplacianin Spherical Coordinates. B.3 Angular Momentum in Spherical Coordinates. C. Computer Code for Solving the Schrodinger Equation. Index.
1. Wave-Particle Duality and Schroedinger Equation. 2. Introduction to Bound States. 3. Rotationally Invariant Potentials: Hydrogen Atom and Beyond. 4. Wave Packets and Uncertainty Relations. 5. Scattering by Simple Barriers. 6. WKB Approximations. 7. Expectation Values and Operators. 8. Electrons in a Magnetic Field. 9. Beyond Hermitian Operators. 10. Harmonic Oscillator: Full Operator Treatment. 11. Composite Systems. 12. Variational Principle. 13. Expansion Principle and Matrix Formulation. 14. Perturbation Theory, I: Degenerate Perturbation Theory. 15. Perturbation Theory, II: Non-Degenerate Perturbation Theory. 16. Symmetry. 17. Electrons in Periodic Crystal Potentials. 18. Rotational Invariance and Angular Momentum. 19. Time-Dependent Perturbation Theory. 20. Elements of Field Quantization. 21. Electron Spin. 22. Indistinguishable Particles: Fermions and Bosons. Appendices: Dirac ...d-Function. Poisson-Distributed Events. Spherical Harmonics. Hydrogen Radial Eigenfunctions. Fourier Integral. Construction of Two Group Character Tables. Selected General References. Fundamental Constants. Index.
1: The Birth of Quantum Theory. 1.1: The radiation of an ideal black body. 1.2: Line spectra and Bohr's model of the atom. 1.3: The quantum conditions of Bohr-Sommerfield. 1.4: Waves of matters. Comments. Problems. 2: Classical Physics - Quantum Physics. 2.1: The difficulties of classical theory. 2.2: The successes of quantum theory. 2.3: Problems with interpretation of quantum theory. 2.4: Quantum theory (1930. Problems. II: Schrodinger's Equation and the Interpretation of its Solutions. 3: Schrodinger's Equation. 3.1: The Schrodinger equation with time. 3.2: The time-dependent Schrodinger equation. Comments. Problems. 4: The Probabilistic Interpretation of the Wavefunction. Comments. Problems. 5: The Motion of a Free Particle in Wave Mechanics. 5.1: The spreading of wave packets. 5.2: The initial problem for Schrodinger's free particle equation. 5.3: The Gaussian wave packet. Comments. Problems. 6: The Momentum of a Particle and the Fourier Transformation of the Wavefunction. Comments. Problems. 7: Description of Experiments in Wave Mechanics and the Heinsberg Uncertainty Principle. 7.1: Heisenberg's uncertainty principle. 7.2: The entropic uncertainty principle. Comments. Problems. 8: A Charged Particle in an Electromagnetic Field. Comments. Problems. 9: The Hydrodynamic Formulation of Wave Mechanics. 9.1: The motion of the probability fluid in an electromagnetic field. Comments. Problems. 10: The Spin of the Elctron and Pauli's Equation. 10.1: The magnetic moment vector. 10.2: Pauli's equation. 10.3: The Stern-Gerlach experiments. Comments. Problems. 11: The Hydrodynamic Picture of Wave Mechanics of a Particle with Spin. Comments. Problems. 12: Ehrenfest's Equations. 12.1: A particle with spin in the electromagnetic field. Problems. 13: Descriptiopn of Particle Motion in Phase Space -- Wigner's Function. Comments. Problems. 14: Mixed States. 14.1: An unpolarized beam as a mixed state. Comments. Problems. III: Solutions of Schrodinger's Equation. 15: Classification of Solutions to Schrodinger's Equation. 15.1: Scattering states. 15.2: The completeness of the set of bound and scattering state wavefunctions. 15.3: Orthogonality of the wavefunctions. 15.4: Development of the system in time. 15.5: Bound and scattered states. Problems. 16: Bound States and Energy. 16.1: The three-dimensional spherically symmetric potential well. Comments. Problems. 17: Scattering States and Collision Cross-sections. 17.1: One-dimensional scattering. 17.2: The method of integral equations. 17.3: Three-dimensional scattering. 17.4: The scattering amplitude. 17.5: The S matrix. 17.6: Collision cross-section. 17.7: The optical theorem. Comments. Problems. 18: Partial Waves and Wave Effects in Scattering. 18.1: Phase shifts. 18.2: The spherical potential hump. 18.3: The Ramsauer-Townsend effect. 18.4: The extinction paradox. Comments. Problems. 19: Resonance States. Problems. 20: The Coulomb Potential. 20.1: Bound states. 20.2: Scattering states. Comments. Problems. 21