You are previewing Quantum Mechanics for Scientists and Engineers.
O'Reilly logo
Quantum Mechanics for Scientists and Engineers

Book Description

If you need a book that relates the core principles of quantum mechanics to modern applications in engineering, physics, and nanotechnology, this is it. Students will appreciate the book's applied emphasis, which illustrates theoretical concepts with examples of nanostructured materials, optics, and semiconductor devices. The many worked examples and more than 160 homework problems help students to problem solve and to practise applications of theory. Without assuming a prior knowledge of high-level physics or classical mechanics, the text introduces Schrödinger's equation, operators, and approximation methods. Systems, including the hydrogen atom and crystalline materials, are analyzed in detail. More advanced subjects, such as density matrices, quantum optics, and quantum information, are also covered. Practical applications and algorithms for the computational analysis of simple structures make this an ideal introduction to quantum mechanics for students of engineering, physics, nanotechnology, and other disciplines. Additional resources available from www.cambridge.org/9780521897839.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. How to use this book
  9. Chapter 1: Introduction
    1. 1.1 Quantum mechanics and real life
    2. 1.2 Quantum mechanics as an intellectual achievement
    3. 1.3 Using quantum mechanics
  10. Chapter 2: Waves and quantum mechanics – Schrödinger’s equation
    1. 2.1 Rationalization of Schrödinger’s equation
    2. 2.2 Probability densities
    3. 2.3 Diffraction by two slits
    4. 2.4 Linearity of quantum mechanics: multiplying by a constant
    5. 2.5 Normalization of the wavefunction
    6. 2.6 Particle in an infinitely deep potential well (“particle in a box”)
    7. 2.7 Properties of sets of eigenfunctions
    8. 2.8 Particles and barriers of finite heights
    9. 2.9 Particle in a finite potential well
    10. 2.10 Harmonic oscillator
    11. 2.11 Particle in a linearly varying potential
    12. 2.12 Summary of concepts
  11. Chapter 3: The time-dependent Schrödinger equation
    1. 3.1 Rationalization of the time-dependent Schrödinger equation
    2. 3.2 Relation to the time-independent Schrödinger equation
    3. 3.3 Solutions of the time-dependent Schrödinger equation
    4. 3.4 Linearity of quantum mechanics: linear superposition
    5. 3.5 Time dependence and expansion in the energy eigenstates
    6. 3.6 Time evolution of infinite potential well and harmonic oscillator
    7. 3.7 Time evolution of wavepackets
    8. 3.8 Quantum mechanical measurement and expectation values
    9. 3.9 The Hamiltonian
    10. 3.10 Operators and expectation values
    11. 3.11 Time evolution and the Hamiltonian operator
    12. 3.12 Momentum and position operators
    13. 3.13 Uncertainty principle
    14. 3.14 Particle current
    15. 3.15 Quantum mechanics and Schrödinger’s equation
    16. 3.16 Summary of concepts
  12. Chapter 4: Functions and operators
    1. 4.1 Functions as vectors
    2. 4.2 Vector space
    3. 4.3 Operators
    4. 4.4 Linear operators
    5. 4.5 Evaluating the elements of the matrix associated with an operator
    6. 4.6 Bilinear expansion of linear operators
    7. 4.7 Specific important types of linear operators
    8. 4.8 Identity operator
    9. 4.9 Inverse operator
    10. 4.10 Unitary operators
    11. 4.11 Hermitian operators
    12. 4.12 Matrix form of derivative operators
    13. 4.13 Matrix corresponding to multiplying by a function
    14. 4.14 Summary of concepts
  13. Chapter 5: Operators and quantum mechanics
    1. 5.1 Commutation of operators
    2. 5.2 General form of the uncertainty principle
    3. 5.3 Transitioning from sums to integrals
    4. 5.4 Continuous eigenvalues and delta functions
    5. 5.5 Summary of concepts
  14. Chapter 6: Approximation methods in quantum mechanics
    1. 6.1 Example problem – potential well with an electric field
    2. 6.2 Use of finite matrices
    3. 6.3 Time-independent nondegenerate perturbation theory
    4. 6.4 Degenerate perturbation theory
    5. 6.5 Tight binding model
    6. 6.6 Variational method
    7. 6.7 Summary of concepts
  15. Chapter 7: Time-dependent perturbation theory
    1. 7.1 Time-dependent perturbations
    2. 7.2 Simple oscillating perturbations
    3. 7.3 Refractive index
    4. 7.4 Nonlinear optical coefficients
    5. 7.5 Summary of concepts
  16. Chapter 8: Quantum mechanics in crystalline materials
    1. 8.1 Crystals
    2. 8.2 One electron approximation
    3. 8.3 Bloch theorem
    4. 8.4 Density of states in k-space
    5. 8.5 Band structure
    6. 8.6 Effective mass theory
    7. 8.7 Density of states in energy
    8. 8.8 Densities of states in quantum wells
    9. 8.9 k·p method
    10. 8.10 Use of Fermi’s Golden Rule
    11. 8.11 Summary of concepts
  17. Chapter 9: Angular momentum
    1. 9.1 Angular momentum operators
    2. 9.2 L squared operator
    3. 9.3 Visualization of spherical harmonic functions
    4. 9.4 Comments on notation
    5. 9.5 Visualization of angular momentum
    6. 9.6 Summary of concepts
  18. Chapter 10: The hydrogen atom
    1. 10.1 Multiple-particle wavefunctions
    2. 10.2 Hamiltonian for the hydrogen atom problem
    3. 10.3 Coordinates for the hydrogen atom problem
    4. 10.4 Solving for the internal states of the hydrogen atom
    5. 10.5 Solutions of the hydrogen atom problem
    6. 10.6 Summary of concepts
  19. Chapter 11: Methods for one-dimensional problems
    1. 11.1 Tunneling probabilities
    2. 11.2 Transfer matrix
    3. 11.3 Penetration factor for slowly varying barriers
    4. 11.4 Electron emission with a potential barrier
    5. 11.5 Summary of concepts
  20. Chapter 12: Spin
    1. 12.1 Angular momentum and magnetic moments
    2. 12.2 State vectors for spin angular momentum
    3. 12.3 Operators for spin angular momentum
    4. 12.4 The Bloch sphere
    5. 12.5 Direct product spaces and wavefunctions with spin
    6. 12.6 Pauli equation
    7. 12.7 Where does spin come from?
    8. 12.8 Summary of concepts
  21. Chapter 13: Identical particles
    1. 13.1 Scattering of identical particles
    2. 13.2 Pauli exclusion principle
    3. 13.3 States, single-particle states, and modes
    4. 13.4 Exchange energy
    5. 13.5 Extension to more than two identical particles
    6. 13.6 Multiple-particle basis functions
    7. 13.7 Thermal distribution functions
    8. 13.8 Important extreme examples of states of multiple identical particles
    9. 13.9 Quantum mechanical particles reconsidered
    10. 13.10 Distinguishable and indistinguishable particles
    11. 13.11 Summary of concepts
  22. Chapter 14: The density matrix
    1. 14.1 Pure and mixed states
    2. 14.2 Density operator
    3. 14.3 Density matrix and ensemble average values
    4. 14.4 Time evolution of the density matrix
    5. 14.5 Interaction of light with a two-level “atomic” system
    6. 14.6 Density matrix and perturbation theory
    7. 14.7 Summary of concepts
  23. Chapter 15: Harmonic oscillators and photons
    1. 15.1 Harmonic oscillator and raising and lowering operators
    2. 15.2 Hamilton’s equations and generalized position and momentum
    3. 15.3 Quantization of electromagnetic fields
    4. 15.4 Nature of the quantum mechanical states of an electromagnetic mode
    5. 15.5 Field operators
    6. 15.6 Quantum mechanical states of an electromagnetic field mode
    7. 15.7 Generalization to sets of modes
    8. 15.8 Vibrational modes
    9. 15.9 Summary of concepts
  24. Chapter 16: Fermion operators
    1. 16.1 Postulation of fermion annihilation and creation operators
    2. 16.2 Wavefunction operator
    3. 16.3 Fermion Hamiltonians
    4. 16.4 Summary of concepts
  25. Chapter 17: Interaction of different kinds of particles
    1. 17.1 States and commutation relations for different kinds of particles
    2. 17.2 Operators for systems with different kinds of particles
    3. 17.3 Perturbation theory with annihilation and creation operators
    4. 17.4 Stimulated emission, spontaneous emission, and optical absorption
    5. 17.5 Summary of concepts
  26. Chapter 18: Quantum information
    1. 18.1 Quantum mechanical measurements and wavefunction collapse
    2. 18.2 Quantum cryptography
    3. 18.3 Entanglement
    4. 18.4 Quantum computing
    5. 18.5 Quantum teleportation
    6. 18.6 Summary of concepts
  27. Chapter 19: Interpretation of quantum mechanics
    1. 19.1 Hidden variables and Bell’s inequalities
    2. 19.2 The measurement problem
    3. 19.3 Solutions to the measurement problem
    4. 19.4 Epilogue
    5. 19.5 Summary of concepts
  28. Appendix A: Background mathematics
    1. A.1 Geometrical vectors
    2. A.2 Exponential and logarithm notation
    3. A.3 Trigonometric notation
    4. A.4 Complex numbers
    5. A.5 Differential calculus
    6. A.6 Differential equations
    7. A.7 Summation notation
    8. A.8 Integral calculus
    9. A.9 Matrices
    10. A.10 Product notation
    11. A.11 Factorial
  29. Appendix B: Background physics
    1. B.1 Elementary classical mechanics
    2. B.2 Electrostatics
    3. B.3 Frequency units
    4. B.4 Waves and diffraction
  30. Appendix C: Vector calculus
    1. C.1 Vector calculus operators
    2. C.2 Spherical polar coordinates
    3. C.3 Cylindrical coordinates
    4. C.4 Vector calculus identities
  31. Appendix D: Maxwell’s equations and electromagnetism
    1. D.1 Polarization of a material
    2. D.2 Maxwell’s equations
    3. D.3 Maxwell’s equations in free space
    4. D.4 Electromagnetic wave equation in free space
    5. D.5 Electromagnetic plane waves
    6. D.6 Polarization of a wave
    7. D.7 Energy density
    8. D.8 Energy flow
    9. D.9 Modes
  32. Appendix E: Perturbing Hamiltonian for optical absorption
    1. E.1 Justification of the classical Hamiltonian
    2. E.2 Quantum mechanical Hamiltonian
    3. E.3 Choice of gauge
    4. E.4 Approximation to linear system
  33. Appendix F: Early history of quantum mechanics
  34. Appendix G: Some useful mathematical formulae
    1. G.1 Elementary mathematical expressions
    2. G.2 Formulae for sines, cosines, and exponentials
    3. G.3 Special functions
  35. Appendix H: Greek alphabet
  36. Appendix I: Fundamental constants
  37. Bibliography
  38. Memorization list
  39. Index