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Principles of Quantum Mechanics

Book Description

This book comprehensively covers all relevant topics to meet the requirements of both undergraduate and postgraduate students of physics. The initial chapters of the book introduce the basic fundamentals of the subject to help the first-time learners and the later chapters cover aspects that prepare to apply quantum mechanics to understand the various physical phenomena. The book includes a detailed discussion on why classical mechanics, which is applicable at macroscopic level, cannot be applicable at microscopic level.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. About the Author
  5. Dedication
  6. Foreword
  7. Preface
  8. Chapter 1: Introduction
    1. 1.1 Motivation
    2. 1.2 The usual (classical) mechanics
    3. 1.3 The process of observation
    4. 1.4 The new mechanics
  9. Chapter 2: Wave-particle Duality
    1. 2.1 Introduction
    2. 2.2 Young’s two-slit experiment
    3. 2.3 Bragg’s x-ray diffraction
    4. 2.4 Photoelectric effect
    5. 2.5 Compton effect
    6. 2.6 Wave-particle nature of electromagnetic radiations
    7. 2.7 Electron/neutron diffraction
    8. 2.8 Davisson and Germer electron diffraction experiment
    9. 2.9 Wave-particle nature of matter
    10. 2.10 What is the real nature of matter and radiations?
    11. Exercises
    12. Solutions
    13. References
  10. Chapter 3: Wave Packets and Uncertainty Principle
    1. 3.1 Introduction
    2. 3.2 Superposition of waves
    3. 3.3 Phase velocity and group velocity
    4. 3.4 de Broglie relation
    5. 3.5 Measurement and uncertainty principle
    6. 3.6 Uncertainty principle: Thought experiments
    7. 3.7 Conclusions
    8. Exercises
    9. Solutions
    10. References
  11. Chapter 4: Operators, Eigenstates, Eigenvalues and Schrodinger Equation
    1. 4.1 Introduction
    2. 4.2 Measurement process as operator operating on the state function/wave function of a particle having definite linear momentum
    3. 4.3 Physical interpretation of wave function ψ(r, t)
    4. 4.4 Schrodinger equation for a free particle
    5. 4.5 Schrodinger equation for a free wave packet
    6. 4.6 Schrodinger equation for a particle in a potential
    7. 4.7 Expectation value and operators
    8. 4.8 Probability current density: Equation of continuity
    9. 4.9 Gaussian wave packet and its spread with time
    10. 4.10 Wave function in momentum space
    11. 4.11 The Ehrenfest theorem
    12. 4.12 The uncertainty relations (revisited)
    13. 4.13 The (resulting) quantum logic
    14. Exercises
    15. Solutions
    16. References
  12. Chapter 5: One-dimensional Problems
    1. 5.1 Introduction
    2. 5.2 Time-independent schrodinger equation and stationary states
    3. 5.3 Some characteristics of wave functions
    4. 5.4 Particle in a one-dimensional potential box
    5. 5.5 Potential box with periodic boundary conditions
    6. 5.6 The potential step
    7. 5.7 Rectangular potential barrier
    8. 5.8 Potential well of finite depth
    9. 5.9 Kronig–Penney model
    10. Exercises
    11. Solutions
    12. References
  13. Chapter 6: The Linear Harmonic Oscillator
    1. 6.1 Introduction
    2. 6.2 Classical harmonic oscillator
    3. 6.3 Quantum harmonic oscillator
    4. 6.4 The Normalized Wave Functions
    5. 6.5 The Hermite polynomials
    6. 6.6 Parity
    7. 6.7 Conclusions
    8. Exercises
    9. Solutions
    10. References
  14. Chapter 7: The Linear Vector Space
    1. 7.1 Introduction
    2. 7.2 Some characteristics of eigenstates of Hermitian operators
    3. 7.3 Dirac bra and ket notations
    4. 7.4 More about bra, ket vectors and linear vector space
    5. 7.5 Matrix representation of state vectors and operators
    6. 7.6 Some special matrices/operators
    7. 7.7 Change of basis: Unitary transformation
    8. 7.8 Tensor product or direct product of vector spaces
    9. 7.9 Outer product operators
    10. Exercises
    11. Solutions
    12. References
  15. Chapter 8: Linear Harmonic Oscillator—Revisited
    1. 8.1 Introduction
    2. 8.2 The creation and annihilation operators
    3. 8.3 Energy eigenstates
    4. 8.4 Matrix representation of various operators
    5. 8.5 Expectation values of various operators
    6. 8.6 The coherent states
    7. 8.7 Time evolution of the coherent state and its comparison with classical oscillator
    8. 8.8 The Schrodinger and Heisenberg pictures
    9. Exercises
    10. Solutions
    11. References
  16. Chapter 9: Angular Momentum
    1. 9.1 Introduction
    2. 9.2 Orbital angular momentum operator
    3. 9.3 Commutation relations
    4. 9.4 Angular momentum operator in spherical polar coordinates
    5. 9.5 The eigenvalues and eigenfunctons of L2 and Lz
    6. 9.6 Measurement of angular momentum components and the uncertainty relations
    7. 9.7 Orbital angular momentum and spatial rotation
    8. Exercises
    9. Solutions
    10. References
  17. Chapter 10: Three-dimensional Systems
    1. 10.1 Introduction
    2. 10.2 A particle in a cubic potential box
    3. 10.3 Cubic box with periodic boundary conditions
    4. 10.4 Density of states of free particles (free electron gas in metals)
    5. 10.5 Spherically symmetric potentials
    6. 10.6 The free particle in spherical polar coordinates
    7. 10.7 Schrodinger equation for a two-body system
    8. 10.8 The hydrogenic atom
    9. Exercises
    10. Solutions
    11. References
  18. Chapter 11: Angular Momentum—Revisited
    1. 11.1 Introduction
    2. 11.2 Raising and lowering operators (the ladder operators)
    3. 11.3 Eigenvalues and eigenstates of orbital angular momentum operators: Second construction of spherical harmonics
    4. 11.4 The constants C+ and C–
    5. 11.5 Matrix representation of angular momentum operator corresponding to j = 1
    6. 11.6 Matrix representation of angular momentum operator corresponding to j = 1/2
    7. Exercises
    8. Solutions
    9. References
  19. Chapter 12: The Spin
    1. 12.1 Introduction
    2. 12.2 Orbital angular momentum and magnetic moment
    3. 12.3 The electron spin: Spin operators and spin eigenstates
    4. 12.4 Total wave function of an electron
    5. 12.5 The Stern–Gerlach experiment
    6. 12.6 Spin and rotation (spinor transformation)
    7. 12.7 A magnetic moment in a uniform magnetic field: The Larmor precession
    8. 12.8 Electron spin resonance
    9. Exercises
    10. Solutions
    11. References
  20. Chapter 13: Addition of Angular Momenta
    1. 13.1 Introduction
    2. 13.2 Addition of two angular momenta
    3. 13.3 Recursion relations for the C–G coefficients
    4. 13.4 The possible values of j
    5. 13.5 Addition of two spin 1/2 angular momenta
    6. 13.6 Addition of j = 1 and j = 1/2 angular momenta
    7. Exercises
    8. Solutions
    9. References
  21. Chapter 14: WKB Approximation and Electron Tunnelling
    1. 14.1 Introduction
    2. 14.2 The essential idea of WKB method
    3. 14.3 Development of WKB approximation
    4. 14.4 Validity of WKB approximation
    5. 14.5 The connection formulae
    6. 14.6 Application of WKB technique to barrier penetration
    7. 14.7 Cold emission of electrons from metals
    8. 14.8 Alpha-decay of nuclei
    9. Exercises
    10. Solutions
    11. References
  22. Chapter 15: Time-independent Perturbation Theory
    1. 15.1 Introduction
    2. 15.2 Non-degenerate perturbation theory
    3. 15.3 Harmonic oscillator subject to perturbing potential
    4. 15.4 Degenerate perturbation theory
    5. 15.5 The Stark effect
    6. 15.6 The fine structure of hydrogen
    7. 15.7 The Zeeman effect
    8. Exercises
    9. Solutions
    10. References
  23. Chapter 16: Time-dependent Perturbation Theory
    1. 16.1 Introduction
    2. 16.2 Time development of states and transition probability
    3. 16.3 Constant perturbation
    4. 16.4 The adiabatic approximation
    5. Exercises
    6. Solutions
    7. References
  24. Chapter 17: Semi-classical Theory of Radiations
    1. 17.1 Introduction
    2. 17.2 Interaction of one-electron atom with electromagnetic field
    3. 17.3 Harmonic perturbation theory
    4. 17.4 Spontaneous emission: Einstein A and B coefficients
    5. 17.5 Selection rules for electric dipole transitions
    6. 17.6 Lifetime and line-width
    7. Exercises
    8. Solutions
    9. References
  25. Chapter 18: Theory of Scattering
    1. 18.1 Introduction
    2. 18.2 Scattering experiments and scattering cross-section
    3. 18.3 Classical theory of scattering: Rutherford scattering
    4. 18.4 Quantum theory of scattering
    5. 18.5 Solution of Schrodinger equation for scattering problem: Green’s function
    6. 18.6 The Born approximation
    7. 18.7 Method of partial waves and phase shifts
    8. Exercises
    9. Solutions
    10. References
  26. Chapter 19: Theory of Measurement in Quantum Mechanics
    1. 19.1 Introduction
    2. 19.2 Process of measurement: A simple treatment
    3. 19.3 Measurement of spin of an atom
    4. 19.4 The EPR paradox
    5. 19.5 The hidden variables and Bell’s theorem
    6. 19.6 Time–evolution of a system: Quantum zeno paradox
    7. Exercises
    8. Solutions
    9. References
  27. Chapter 20: Introduction to Quantum Computing
    1. 20.1 Introduction
    2. 20.2 Binary number systems
    3. 20.3 Classical logic gates
    4. 20.4 Turing machine
    5. 20.5 Qubits
    6. 20.6 Entanglement
    7. 20.7 Quantum logic gates
    8. 20.8 Quantum computation
    9. Exercises
    10. Solutions
    11. References
  28. Appendices
    1. Appendix A: Early Quantum Mechanics
      1. A.1 Planck’s formula of black body radiations
      2. A.2 Atomic spectra and Bohr’s model of hydrogen atom
      3. A.3 Bohr’s correspondence principle
      4. A.4 The Franck–Hertz experiment
    2. Appendix B: Some Supplementary Topics
      1. B.1 Fourier transform
      2. B.2 Dirac delta function
      3. B.3 Bloch theorem
      4. B.4 The variational method
      5. Exercises
      6. Solutions
    3. Appendix C: Some Mathematical Relations
      1. C.1 Some algebraic relations
      2. C.2 Some trigonometric relations
      3. C.3 Coordinate systems
      4. C.4 Some vector relations
      5. C.5 Some calculus relations
      6. C.6 Some definite integrals
      7. C.7 An important integral
    4. Appendix D: Various Tables
      1. D.1 Table of fundamental physical constants
      2. D.2 Table of conversion factors between units used to express magnitude of energy
      3. D.3 Table of spectrum of electromagnetic radiations
      4. D.4 Range of radiations
      5. D.5 Symbols and values of pre-factors
      6. D.6 The Greek alphabets
  29. Copyright
  30. Back Cover