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Quantum Mechanics Demystified, 2nd Edition

Book Description

If you think projection operators work in the cinema, or learning about spin-1/2 makes your head, well, spin, Quantum Mechanics DeMYSTiFieD will energize your knowledge of this topic’s fundamental concepts and theories, and allow you to learn at your own pace.This thoroughly revised and updated guide eases youinto the subject, beginning with wave mechanics thenintroducing you to the mathematical foundations neededto do modern quantum physics.

Table of Contents

  1. Cover 
  2. Title Page
  3. Copyright Page
  4. About the Author
  5. Contents 
  6. Introduction
  7. Chapter 1: Historical Review
    1. Blackbody Radiation and Planck’s Formula
      1. Planck’s Radical Assumption
    2. The Photoelectric Effect
    3. The Bohr Theory of the Atom
      1. Bohr Makes Two Key Assumptions about the Atom
    4. de Broglie’s Hypothesis
    5. Summary
    6. Quiz
  8. Chapter 2: Basic Developments
    1. The Schrödinger Equation
    2. Solving the Schrödinger Equation
    3. The Probability Interpretation and Normalization
    4. Expansion of the Wavefunction and Finding Coefficients
    5. The Phase of a Wavefunction
    6. Operators in Quantum Mechanics
    7. Momentum and the uncertainty Principle
    8. The Conservation of Probability
    9. Summary
    10. Quiz
  9. Chapter 3: The Time-Independent Schrödinger Equation
    1. The Free Particle
    2. Bound States and 1-D Scattering
    3. Parity
    4. Ehrenfest Theorem
    5. Summary
    6. Quiz
  10. Chapter 4: An Introduction to Hilbert Space
    1. Basic Definitions
    2. Hilbert Space Definitions
      1. The Dirac Delta Function
    3. Summary
    4. Quiz
  11. Chapter 5: The Mathematical Structure of Quantum Mechanics I
    1. Linear Vector Spaces
      1. More Properties of Vector Spaces
      2. The Inner Product
      3. Properties of the Inner Product
    2. Basis Vectors
      1. Dimension of a Space
    3. Expanding a Vector in Terms of a Basis
    4. Orthonormal Sets and the Gram-Schmidt Procedure
    5. Dirac Algebra with Bras and Kets
    6. Finding the Expansion Coefficients in the Representation of Bras and Kets
    7. Quiz
  12. Chapter 6: The Mathematical Structure of Quantum Mechanics II
    1. Linear Operators
    2. The Representation of an Operator
      1. The Action of an Operator on Kets in Matrix Representation
      2. Calculating Expectation Values of Operators
    3. Eigenvalues and Eigenvectors
    4. The Hermitian Conjugate of an Operator
      1. Finding the Hermitian Conjugate of a Matrix
      2. Properties of the Transpose Operation
      3. Hermitian and Unitary Operators
    5. The Commutator
      1. Properties of the Commutator
    6. Quiz
  13. Chapter 7: The Mathematical Structure of Quantum Mechanics III
    1. Change of Basis and Unitary Transformations
    2. Similarity Transformations
    3. Finding a Similarity Transformation for a Matrix C
    4. The Generalized Uncertainty Relation
    5. Projection Operators
    6. Functions of Operators
    7. Generalization to Continuous Spaces
    8. Summary
    9. Quiz
  14. Chapter 8: The Foundations of Quantum Mechanics
    1. The Postulates of Quantum Mechanics
      1. Two-Level System Example: State Vectors
      2. Two-Level System Example: The Pauli Operators
      3. Writing an Operator in Terms of Outer Products
    2. Spectral Decomposition
      1. Two-Level System Example: Spectral Decomposition of Z
    3. Projective Measurements
      1. Two-Level System Example: Projection Operators
    4. The Completeness Relation
      1. Aside on Degeneracy
      2. Two-Level System Example: State of the System after Measurement
    5. Completely Specifying a State with a CSCO
    6. The Heisenberg versus Schrödinger Pictures
    7. Describing Composite Systems in Quantum Mechanics
      1. Dimension of a Vector Space Formed by a Tensor Product
    8. The Matrix Representation of a Tensor Product
    9. The Tensor Product of State Vectors
    10. The Density Operator
      1. Properties of the Density Operator
    11. The Density Operator for a Completely Mixed State
    12. A Brief Introduction to the Bloch Vector
    13. Summary
    14. Quiz
  15. Chapter 9: The Harmonic Oscillator
    1. The Solution of the Harmonic Oscillator in the Position Representation
    2. The Operator Method for the Harmonic Oscillator
    3. Number States of the Harmonic Oscillator
    4. More on the Action of the Raising and Lowering Operators
    5. Summary
    6. Quiz
  16. Chapter 10: Angular Momentum
    1. The Commutation Relations of Angular Momentum
    2. The Uncertainty Relations for Angular Momentum
    3. Generalized Angular Momentum and the Ladder Operators
      1. Ladder Operators for Angular Momentum
      2. Solving the Eigenvalue Problem for Angular Momentum
    4. Matrix Representations of Angular Momentum
      1. Finding the Possible Results of Measurement of Angular Momentum
    5. Coordinate Representation of Orbital Angular Momentum and the Spherical Harmonics
      1. Angular Momentum and Rotations
    6. Summary
    7. Quiz
  17. Chapter 11: Spin-1/2 Systems
    1. The Stern-Gerlach Experiment
    2. The Basis States for Spin-1/2 Systems
    3. Using the Ladder Operators to Construct Sx, Sy
    4. Unitary Transformations for Spin-1/2 Systems
    5. The Outer Product Representation of the Spin Operators
    6. The Pauli Matrices
    7. The Time Evolution of Spin-1/2 States
    8. The Density Operator for Spin-1/2 Systems
    9. Summary
    10. Quiz
  18. Chapter 12: Quantum Mechanics in Three Dimensions
    1. The 2-D Square Well
    2. An Overview of a Particle in a Central Potential
    3. An Overview of the Hydrogen Atom
    4. Summary
    5. Quiz
  19. Chapter 13: Identical Particles
    1. Basic Principles of Systems of Identical Particles
    2. Fermions and Bosons
    3. The Pauli Exclusion Principle
    4. The Periodic Table
    5. More General Cases
    6. Summary
    7. Quiz
  20. Chapter 14: Time-Independent Perturbation Theory
    1. When Is Perturbation Theory Needed?
    2. The Stark Effect
    3. Summary
    4. Quiz
  21. Chapter 15: Applications of Perturbation Theory to the Hydrogen Atom
    1. The Fine Structure of Hydrogen
    2. The Zeeman Effect
    3. Hyperfine Splitting
    4. Summary
    5. Quiz
  22. Chapter 16: Time-Dependent Perturbation Theory
    1. “Pictures” in Quantum Theory
    2. Two-Level Systems
    3. Fermi’s Golden Rule
    4. The Emission and Absorption of Radiation
    5. Summary
    6. Quiz
  23. Chapter 17: Scattering
    1. Introduction
    2. Scattering Amplitudes
    3. The Born Approximation
    4. Summary
    5. Quiz
  24. Final Exam
  25. Answers to Quiz and Exam Questions
  26. Index