CHEM 110A: Introduction to Quantum Mechanics

by Delmar Larsen

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This text focuses on introducing students to the postulates and general principles of quantum mechanics. Approximations based on variational method and time independent perturbation theory. Application to harmonic oscillator, rigid rotor, one-electron and many-electron atoms, and homo-and hetero-nuclear diatomic molecules.

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Publisher
University of California, Davis
Subjects
Chemistry
Licence
CC BY-NC-SA 4.0, as LibreTexts states it. Non-commercial use only.
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Tweens profiles and older, judged from its title and subjects

Contents

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  1. AGENDA
  2. Exam 2 Review Questions
  3. Text
  4. Front Matter
  5. TitlePage
  6. InfoPage
  7. Table of Contents
  8. Licensing
  9. 1: The Dawn of the Quantum Theory
  10. 1.1: Blackbody Radiation Cannot Be Explained Classically
  11. 1.2: Quantum Hypothesis used for Blackbody Radiation Law
  12. 1.3: Photoelectric Effect Explained with Quantum Hypothesis
  13. 1.4: The Hydrogen Atomic Spectrum
  14. 1.5: The Rydberg Formula and the Hydrogen Atomic Spectrum
  15. 1.6: Matter Has Wavelike Properties
  16. 1.7: de Broglie Waves can be Experimentally Observed
  17. 1.8: The Bohr Theory of the Hydrogen Atom
  18. 1.9: The Heisenberg Uncertainty Principle
  19. 1.E: The Dawn of the Quantum Theory (Exercises)
  20. 2: The Classical Wave Equation
  21. 2.1: The One-Dimensional Wave Equation
  22. 2.2: The Method of Separation of Variables
  23. 2.3: Oscillatory Solutions to Differential Equations
  24. 2.4: The General Solution is a Superposition of Normal Modes
  25. 2.5: A Vibrating Membrane
  26. 2.E: The Classical Wave Equation (Exercises)
  27. 3: The Schrödinger Equation and the Particle-in-a-Box Model
  28. 3.1: The Schrödinger Equation
  29. 3.2: Linear Operators in Quantum Mechanics
  30. 3.3: The Schrödinger Equation is an Eigenvalue Problem
  31. 3.4: The Quantum Mechanical Free Particle
  32. 3.5: Wavefunctions Have a Probabilistic Interpretation
  33. 3.6: The Energy of a Particle in a Box is Quantized
  34. 3.7: Wavefunctions Must Be Normalized
  35. 3.8: The Average Momentum of a Particle in a Box is Zero
  36. 3.9: The Uncertainty Principle Redux - Estimating Uncertainties from Wavefunctions
  37. 3.10: Particle in a Finite Box and Tunneling (Optional)
  38. 3.11: A Particle in a Two-Dimensional Box
  39. 3.12: A Particle in a Three-Dimensional Box
  40. 3.E: The Schrödinger Equation and a Particle in a Box (Exercises)
  41. 3.I: Interactive Worksheets
  42. 3.I.1: Free Particle (Python Notebook)
  43. 3.I.2: Particle in an Infinite Potential Box (Python Notebook)
  44. 3.I.3: Particle in a Finite Potential Box (Python Notebook)
  45. 4: Postulates and Principles of Quantum Mechanics
  46. 4.1: The Wavefunction Specifies the State of a System
  47. 4.2: Quantum Operators Represent Classical Variables
  48. 4.3: Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators
  49. 4.4: The Time-Dependent Schrödinger Equation
  50. 4.5: The Eigenfunctions of Operators are Orthogonal
  51. 4.6: Heisenburg Uncertainy Principle III - Commuting Operators
  52. 4.E: Postulates and Principles of Quantum Mechanics (Exercises)
  53. 5: The Harmonic Oscillator and the Rigid Rotor
  54. 5.1: A Harmonic Oscillator Obeys Hooke's Law
  55. 5.2: The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule
  56. 5.3: The Harmonic Oscillator is an Approximation
  57. 5.4: The Harmonic Oscillator Energy Levels
  58. 5.5: The Harmonic Oscillator and Infrared Spectra
  59. 5.6: The Harmonic-Oscillator Wavefunctions Involve Hermite Polynomials
  60. 5.7: Hermite Polynomials are either Even or Odd Functions
  61. 5.8: The Energy Levels of a Rigid Rotor
  62. 5.9: The Rigid Rotator is a Model for a Rotating Diatomic Molecule
  63. 5.E: The Harmonic Oscillator and the Rigid Rotor (Exercises)
  64. 5.I: Interactive Worksheets
  65. 5.2I: Spherical Harmonics Visualization
  66. 5.I1: Simple Harmonic Oscillator - Plotting Eigenstates
  67. 6: The Hydrogen Atom
  68. 6.1: The Schrödinger Equation for the Hydrogen Atom Can Be Solved Exactly
  69. 6.2: The Wavefunctions of a Rigid Rotator are Called Spherical Harmonics
  70. 6.3: The Three Components of Angular Momentum Cannot be Measured Simultaneously with Arbitrary Precision
  71. 6.4: Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers
  72. 6.5: s Orbitals are Spherically Symmetric
  73. 6.6: Orbital Angular Momentum and the p-orbitals
  74. 6.7: The Helium Atom Cannot Be Solved Exactly
  75. 6.E: The Hydrogen Atom (Exercises)
  76. 7: Approximation Methods
  77. 7.1: Slater's Rules for Shielding
  78. 7.2: The Variational Method
  79. 7.3: Linear Variational Method and the Secular Determinant
  80. 7.4: Trial Functions Can Be Linear Combinations of Functions with Variational Parameters
  81. 7.5: Perturbation Theory Expresses the Solutions in Terms of Solved Problems
  82. 7.E: Approximation Methods (Exercises)
  83. 8: Multielectron Atoms
  84. 8.1: Atomic and Molecular Calculations are Expressed in Atomic Units
  85. 8.2: Perturbation Theory and the Variational Method for Helium
  86. 8.3: Hartree-Fock Equations are Solved by the Self-Consistent Field Method
  87. 8.4: An Electron Has An Intrinsic Spin Angular Momentum
  88. 8.5: Wavefunctions must be Antisymmetric to Interchange of any Two Electrons
  89. 8.6: Antisymmetric Wave Functions can be Represented by Slater Determinants
  90. 8.7: Hartree-Fock Calculations Give Good Agreement with Experimental Data
  91. 8.8: Term Symbols Gives Detailed Descriptions of an Electron Configuration
  92. 8.9: Multi-electron Considerations - A Closer Look at Helium
  93. 8.10: The Allowed Values of J - the Total Angular Momentum Quantum Number
  94. 8.11: Hund's Rules Determine the Term Symbols of the Ground Electronic States
  95. 8.12: Using Atomic Term Symbols to Describe Atomic Spectra
  96. 8.E: Multielectron Atoms (Exercises)
  97. 9: The Chemical Bond- Diatomic Molecules
  98. 9.1: The Born-Oppenheimer Approximation Simplifies the Molecular Schrödinger Equation
  99. 9.2: The H₂⁺ Prototypical Species
  100. 9.3: Solving the H₂⁺ System Exactly (Optional)
  101. 9.4: The Overlap Integral
  102. 9.5: Evaluating the Overlap Integral (Optional)
  103. 9.6: Chemical Bond Stability
  104. 9.7: Bonding and Antibonding Orbitals
  105. 9.8: A Simple Molecular-Orbital Treatment of H₂ Places Both Electrons in a Bonding Orbital
  106. 9.9: Molecular Orbitals Can Be Ordered According to Their Energies
  107. 9.10: Molecular-Orbital Theory Does not Predict a Stable Diatomic Helium Molecule
  108. 9.11: Electrons Populate Molecular Orbitals According to the Pauli Exclusion Principle
  109. 9.12: Molecular Orbital Theory Predicts that Molecular Oxygen is Paramagnetic
  110. 9.E: The Chemical Bond- Diatomic Molecules (Exercises)
  111. 10: MathChapters
  112. 10.1: A- Complex Numbers
  113. 10.2: B- Probability and Statistics
  114. 10.3: C- Vectors
  115. 10.4: D- Spherical Coordinates
  116. 10.5: E- Determinants
  117. 10.6: F- Matrices
  118. 10.7: G- Numerical Methods
  119. 10.8: H- Partial Differentiation
  120. 10.9: I- Series and Limits
  121. 10.10: J- The Binomial Distribution and Stirling's Appromixation
  122. Back Matter
  123. Index
  124. Glossary
  125. Detailed Licensing
  126. Lectures
  127. 1: The Rise of Quantum Mechanics (Lecture)
  128. 2: The Rise of Quantum Mechanics (Lecture)
  129. 3: The Rise of Quantum Mechanics (Lecture)
  130. 4: Bohr atom and Heisenberg Uncertainty (Lecture)
  131. 5: Classical Wave Equations and Solutions (Lecture)
  132. 6: Schrödinger Equation (Lecture)
  133. 7: Operators, Free Particles and the Quantum Superposition Principle (Lecture)
  134. 8: Topical Overview of PIB and Postulates QM (Lecture)
  135. 9: More on PIB and Orthonormality (Lecture)
  136. 10: Expectation values, 2D-PIB and Heisenberg Uncertainty Principle (Lecture)
  137. 11: Postulates of Quantum Mechanics (Lecture)
  138. 12: Vibrational Spectroscopy of Diatomic Molecules
  139. 13: Harmonic Oscillators and Rotation of Diatomic Molecules
  140. 14: Harmonic Oscillators and IR Spectroscopy
  141. 15: 3D Rotations and Microwave Spectroscopy
  142. 16: Linear Momentum and Electronic Spectroscopy
  143. 17: Hydrogen-like Solutions
  144. 18: Orbital Angular Momentum, Spectroscopy and Multi-Electron Atoms
  145. 19: More on Hydrogen Wavefunction, Electronic Spectroscopy and Effective Charge
  146. 20: Variational Method Approximation and Linear Varational Method
  147. 21: Linear Variational Theory
  148. 22: Perturbation Theory
  149. 23: Electron Spin, Indistinguishability and Slater Determinants
  150. 24: Total Spin, Molecules and Molecular Orbital Theory
  151. 25: Populating Molecular Orbitals: σ and π Orbitals
  152. 26: Molecular Orbitals and Diatomics
  153. 27: Molecular Orbitals with higher Energy Atomic Orbitals (Extra Lecture)
  154. Expanded Lecture 25: Molecules and Molecular Orbital Theory
  155. Lecture Extra: Hartree vs. Hartree-Fock, SCF, and Koopman's Theorem
  156. Lecture Extra III: Coupling Angular Momenta and Atomic Term Symbols
  157. Worksheets
  158. 0: Introduction to Complex Numbers
  159. 00: Vectors and Coordinate Systems (Worksheet)
  160. Worksheet 00 Solutions
  161. 1: Continuous Distributions (Worksheet)
  162. Worksheet 1 Solutions
  163. 2: Separation of Variables (Worksheet)
  164. Worksheet 2 Solutions
  165. 3A: Operators and Eigenvalues (Worksheet)
  166. Worksheet 3A Solutions
  167. 3B: Particle in a Box (Worksheet)
  168. Worksheet 3B Solutions
  169. 4A: Introduction to the Quantum Harmonic Oscillator (Worksheet)
  170. 4B: Even and Odd Functions (Worksheet)
  171. 5: Vibrational Spectroscopy (Worksheet)
  172. Worksheet 5A Solutions
  173. 6A: Rotational Spectroscopy (Worksheet)
  174. Worksheet 6A Solutions
  175. 6B: Angular Momentum and Commutators (Worksheet)
  176. Worksheet 6B Solutions
  177. 7A: The Probabilistic Interpretation of Atomic Orbitals (Worksheet)
  178. Worksheet 7A: The Probabilistic Interpretation of Atomic Orbitals
  179. 7B: Variational Method I (Worksheet)
  180. Worksheet 7B Solutions
  181. 8: Variational Method II (Worksheet)
  182. Worksheet 8 Solutions
  183. 9A: Determinants (Worksheet)
  184. Worksheet 9A Solutions
  185. 9B: Multi-Electron Wavefunctions (Worksheet)
  186. Worksheet 9B Solutions
  187. 10A: Adding Angular Momenta in Multi-Electron Atoms (Worksheet)
  188. Worksheet 10A Solutions
  189. 10B: The Dihydrogen Cation (Worksheet)
  190. Worksheet 10B Solutions
  191. Extra Credit
  192. Extra Credit 1
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