Physical & Theoretical Chemistry
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- Supplemental Modules (Physical and Theoretical Chemistry)
- Fundamentals
- Atomic Theory
- Physical Properties of Matter
- Atomic and Molecular Properties
- Atomic and Ionic Radius
- Atomic Radii
- Dipole Moments
- Electronegativity
- Electron Affinity
- Formal Charges
- Intermolecular Forces
- Ionization Energy
- Lewis Structures
- Magnetic Properties
- Molecular Polarity
- Polarizability
- States of Matter
- Solutions and Mixtures
- All About Water
- Material Properties
- Atomic and Molecular Properties
- Acids and Bases
- Kinetics
- Introduction to Reaction Kinetics
- Reaction Rates
- Rate Laws
- Reaction Mechanisms
- Experimental Methods
- Modeling Reaction Kinetics
- Case Studies: Kinetics
- Review of Chemical Kinetics
- Diffusion
- Using logarithms: Log vs. Ln
- Equilibria
- Dynamic Equilibria
- Le Chatelier's Principle
- Physical Equilibria
- Fractional Distillation of Ideal Mixtures
- Fractional Distillation of Non-ideal Mixtures (Azeotropes)
- Immiscible Liquids and Steam Distillation
- Liquid-Solid Phase Diagrams: Salt Solutions
- Liquid-Solid Phase Diagrams: Tin and Lead
- Non-Ideal Mixtures of Liquids
- Phases and Their Transitions
- Phase Diagrams for Pure Substances
- Raoult's Law and Ideal Mixtures of Liquids
- Chemical Equilibria
- Balanced Equations and Equilibrium Constants
- Calculating an Equilibrium Concentration
- Calculating An Equilibrium Concentrations
- Calculating an Equilibrium Constant, Kp, with Partial Pressures
- Determining the Equilibrium Constant
- Difference Between K And Q
- Dissociation Constant
- Effect of Pressure on Gas-Phase Equilibria
- Equilibrium Calculations
- Kc
- Kp
- Law of Mass Action
- Mass Action Law
- Principles of Chemical Equilibria
- The Equilibrium Constant
- The Reaction Quotient
- Heterogeneous Equilibria
- Solubility
- An Introduction to Solubility Products
- Common Ion Effect
- Pressure Effects On the Solubility of Gases
- Relating Solubility to \(K_{sp}\)
- Solubility
- Solubility and Factors Affecting Solubility
- Solubility Product Constant, Ksp
- Solubility Rules
- Temperature Effects on Solubility
- Temperature Effects on the Solubility of Gases
- Types of Saturation
- Acid-Base Equilibria
- Thermodynamics
- Fundamentals of Thermodynamics
- A System and Its Surroundings
- Bond Energies
- Enthalpy Changes in Reactions
- Enthalpy Changes in Reactions II
- Entropy Changes in Reactions
- Free Energy and Equilibrium
- Free Energy Changes in Reactions
- Introduction to Thermodynamics
- Reversibility and Le Chatelier
- State vs. Path Functions
- TD7. Solutions to Selected Problems
- Temperature
- Trouton's rule
- Advanced Thermodynamics
- The Four Laws of Thermodynamics
- Energies and Potentials
- Calorimetry
- Chemical Energetics
- Ideal Systems
- Real (Non-Ideal) Systems
- Path Functions
- Thermochemistry
- Thermodynamic Cycles
- Basics Thermodynamics (General Chemistry)
- Fundamentals of Thermodynamics
- Statistical Mechanics
- Fundamentals of Statistical Mechanics
- Boltzmann Average
- Statistical Mechanical Ensembles
- Advanced Statistical Mechanics
- Calculation of observables from path integrals
- Calculation of spectra from perturbation theory
- Characterizing the Structure of Liquids
- Classical linear response theory
- Classical microstates, Newtonian, Lagrangian and Hamiltonian mechanics
- Classical virial theorem; Legendre transforms; the canonical ensemble
- Discretized and Continuous Path Integrals
- Distribution Function Theory
- Estimators, energy fluctuations, the isothermal-isobaric ensemble
- Expansion about the classical path and stationary phase
- Free-energy calculations
- Fundamentals of quantum statistical mechanics
- Liouville's Theorem, non-Hamiltonian systems, the microcanonical ensemble
- Perturbation theory and the van der Waals equation
- Postulates of Quantum Mechanics
- Quantum linear response theory
- Quantum time-dependent perturbation theory
- The Grand Canonical Ensemble
- The Langevin and Generalized Langevin equations
- Quantum Mechanics
- 1: Waves and Particles
- 2: Fundamentals of Quantum Mechanics
- 3: The Tools of Quantum Mechanics
- 4: The 7 Postulates of Quantum Mechanics
- 5: Particle in Boxes
- 6: One Dimensional Harmonic Oscillator
- 7: Angular Momentum
- 9: The Hydrogen Atom
- 10: Multi-electron Atoms
- 11: Angular Momenta Coupling
- 11: Molecules
- 12: Electron Scattering
- 12: Stationary Perturbation Theory
- 13: Fine and Hyperfine Structure
- 14: Systems of Identical Particles
- 15: Time-dependent Quantum Dynamics
- 16: Molecular Spectroscopy
- 17: Quantum Calculations
- Chapter 14. Nuclear Magnetic Resonance
- Zeeman Effect
- Gen Chem Quantum Theory
- Chemical Bonding
- Fundamentals of Chemical Bonding
- Band Structure
- Bond Energies
- Bond Order and Lengths
- Chemical Bonds
- Contrasting MO and VB theory
- Coordinate (Dative Covalent) Bonding
- Covalent Bonding
- Covalent Bonds
- Covalent Bonds vs Ionic Bonds
- Covalent Bond Distance, Radius and van der Waals Radius
- Electrostatic Potential maps
- Ionic Bonds
- Metallic Bonding
- Non-Singular Covalent Bonds
- Valence-Shell Electron-Pair Repulsion Models
- Lewis Theory of Bonding
- Valence Bond Theory
- Molecular Orbital Theory
- Bonding and antibonding orbitals
- How to Build Molecular Orbitals
- MO. Molecular Orbitals
- MO1. Introduction
- MO10. S and P Mixing in HF
- MO11. Polyatomic Geometry and Symmetry
- MO12. Approximations in Complicated Structures
- MO13. Building MO from Smaller Pieces
- MO14. Delocalization
- MO15. Polyenes
- MO16. Aromatics
- MO17. Heteroaromatics
- MO18. Frontier Orbitals
- MO19. Solutions to Selected Problems
- MO2. Dihydrogen
- MO3. Lessons from Dihydrogen
- MO4. Sigma Bonding with P Orbitals
- MO5. Pi Bonding with P Orbitals
- MO6. Electron Population
- MO7. Experimental Evidence
- MO8. Symmetry & Mixing
- MO9. Bonding Between Different Atoms
- Molecular Orbitals of Li₂ to F₂
- MO bonding in F2 and O2
- MO for HF
- Pictorial Molecular Orbital Theory
- Fundamentals of Chemical Bonding
- Electronic Structure of Atoms and Molecules
- Atomic Orbitals
- Bohr Diagrams of Atoms and Ions
- Electronic Configurations
- Aufbau Principle
- Basic Electronic Structure of Atoms
- Case Study: Electron Configuration of Mn vs. Cu
- d-orbital Occupation and Electronic Configurations
- Electronic Configurations Intro
- Electronic Structure and Orbitals
- Hund's Rules
- Pauli Exclusion Principle
- Spin Pairing Energy
- Spin Quantum Number
- The Aufbau Process
- The Octet Rule
- The Order of Filling 3d and 4s Orbitals
- Evaluating Spin Multiplicity
- Hydrogen's Atomic Emission Spectrum
- Magnetic Behavior of Diatomic Species
- Predicting the Bond-Order of Diatomic Species
- Predicting the Bond-Order of Oxides based Acid Radicals
- Predicting the Hybridization of Simple Molecules
- Prediction of Aromatic, Anti Aromatic and Non Aromatic Character of Heterocyclic Compounds along with their Omission Behavior- Innovative Mnemonics
- Spectroscopy
- Fundamentals of Spectroscopy
- Magnetic Resonance Spectroscopies
- Electron Paramagnetic Resonance
- Nuclear Magnetic Resonance
- NMR: Background Physics and Mathematics
- NMR: Experimental
- NMR: Structural Assignment
- (n+1) Rule
- Background to C-13 NMR
- Determine Structure with Combined Spectra
- High Resolution Proton NMR Spectra
- Integration in Proton NMR
- Interpreting C-13 NMR Spectra
- Introduction to Proton NMR
- Low Resolution Proton NMR Spectra
- More About Electronics
- Multiplicity in Proton NMR
- NMR11. More About Multiplicity
- NMR14. More Practice with NMR Spectroscopy
- NMR2. Carbon-13 NMR
- NMR3. Symmetry in NMR
- NMR4. 13C NMR and Geometry
- NMR5. 13C NMR and Electronics
- NMR8. Chemical Shift in 1H NMR
- NMR Appendix. Useful Charts for NMR identification
- NMR: Theory
- NMR: Introduction
- Rotational Spectroscopy
- Vibrational Spectroscopy
- Vibrational Modes
- Infrared Spectroscopy
- How an FTIR Spectrometer Operates
- Identifying the Presence of Particular Groups
- Infrared: Application
- Infrared: Interpretation
- Infrared: Theory
- Interpreting Infrared Spectra
- Carbon Nitrogen Bonds
- IR10. More Practice with IR Spectra
- IR11. Appendix: IR Table of Organic Compounds
- IR2. Hydrocarbon Spectra
- IR3. Subtle Points of IR Spectroscopy
- IR4. Carbon Carbon Multiple Bonds
- IR5. Carbon Oxygen Single Bonds
- IR6. Carbon Oxygen Double Bonds
- IR8. More Complicated IR Spectra
- IR9. Misleading Peaks
- What Does an IR Spectrum Look Like?
- IR Spectroscopy Background
- The Fingerprint Region
- Raman Spectroscopy
- Electronic Spectroscopy
- Circular Dichroism
- Electronic Spectroscopy: Application
- Electronic Spectroscopy: Interpretation
- Electronic Spectroscopy Basics
- Fluorescence and Phosphorescence
- Jablonski diagram
- Metal to Ligand and Ligand to Metal Charge Transfer Bands
- Radiative Decay
- Selection Rules for Electronic Spectra of Transition Metal Complexes
- Spin-orbit Coupling
- Two-photon absorption
- Photoelectron Spectroscopy
- X-ray Spectroscopy
- Photoacoustic Spectroscopy
- Mössbauer Spectroscopy
- Nuclear Chemistry
- Radioactivity
- Nuclear Energetics and Stability
- Nuclear Kinetics
- Fission and Fusion
- Applications of Nuclear Chemistry
- Nuclear Chemistry (General Chemistry)
- Surface Science
- Group Theory
- Physical Organic Chemistry
- Quantum Tutorials (Rioux)
- Quantum Fundamentals
- 1: An Approach to Quantum Mechanics
- 2.1: Atomic and Molecular Stability
- 2.2: Atomic and Molecular Stability
- 2.3: Atomic and Molecular Stability
- 3.1: Quantum Computation - A Short Course
- 3.2: Quantum Computation: A Short Course
- 3.3: Quantum Computation: A Short Course
- 3.4: Quantum Computation: A Short Course
- 3.5: Quantum Computation: A Short Course
- 3.6: Quantum Computation: A Short Course
- 3.7: Quantum Computation: A Short Course
- 3.8: Quantum Computation: A Short Course
- 4.1: Quantum Mechanics and the Fourier Transform
- 4.2: Quantum Mechanics and the Fourier Transform
- 4.3: Quantum Mechanics and the Fourier Transform
- 4.4: Quantum Mechanics and the Fourier Transform
- 4.5: Quantum Mechanics and the Fourier Transform
- 5: Exploring the Origin of Schrödinger’s Equations
- 6: Basic Quantum Mechanics in Coordinate, Momentum and Phase Space
- 7: The Repackaging of Quantum Weirdness
- 8: Quantum Principles Illuminated with Polarized Light
- 9: Relationship Between the Coordinate and Momentum Representations
- 10: Very Brief Relationship Between the Coordinate and Momentum Representations
- 11: Getting Accustomed to the Superposition Principle
- 12: The Dirac Delta Function
- 13: Elements of Dirac Notation
- 14: The Dirac Notation Applied to Variational Calculations
- 15: Raising and Lowering; Creating and Annihilating
- 16: Single Slit Diffraction and the Fourier Transform
- 17: From Coordinate Space to Momentum Space and Back
- 18: The Position and Momentum Commutation Relation in Coordinate and Momentum Space
- 19: Simulating the Aharonov-Bohm Effect
- 20: Basic Matrix Mechanics
- 21: Rudimentary Matrix Mechanics
- 22: Matrix Mechanics
- 23: Aspects of Diracʹs Relativistic Matrix Mechanics
- 24: The Double-Slit Experiment
- 25: Double‐Slit Experiment with Polarized Light
- 26: The Consequences of Path Information in a Mach-Zehnder Interferometer
- 27: Another look at the Consequences of Path Information in a Mach-Zehnder Interferometer
- 28: The Double‐Slit Experiment with Polarized Light
- 29: The Quantum Eraser
- 30: Which Way Did It Go? ‐ The Quantum Eraser
- 31: Which Path Information and the Quantum Eraser
- 32: Terse Analysis of Triple-slit Diffraction with a Quantum Eraser
- 32: Which Path Information and the Quantum Eraser (Brief)
- 71: Quantum Corrals - Electrons within a Ring
- 83: Quantum Mechanical Pressure
- Atomic Structure
- Chemical Bonding
- 149: The Covalent Bond and Quantum Mechanics
- 150: The Covalent Bond in the Hydrogen Molecule
- 151: The Covalent Bond Clarified Through the Use of the Virial Theorem
- 152: Brief Version of the Covalent Bond Clarified Through the Use of the Virial Theorem
- 153: The H2 Covalent Bond and the Virial Theorem
- 154: The Covalent Bond According to Slater and Ruedenberg
- 155: Slater's Analysis of the Covalent Bond using the Virial Theorem
- 156: Two Analyses of the Covalent Bond using the Virial Theorem
- 157: A Simple Charge Cloud Model for Molecular Hydrogen: Or, Is It a DFT Model?
- 158: Three Mechanisms for Bond Formation in the Hydrogen Molecule Ion
- Spectroscopy
- 185: Rudiments of Atomic Spectroscopy Using Mathcad
- 186: A Particle-in-a-Box Model for Color Centers
- 187: Cyanine Dyes as Two-State Electronic Systems
- 188: The Ammonia Inversion and the Maser
- 189: A Symmetric Double Well Potential Illustrating Tunneling
- 190: Analyses of the Pure Rotational Spectrum of HCl
- 191: A Rudimentary Analysis of the Vibrational-Rotational HCl Spectrum
- 192: Visualizing the Formally Forbidden Overtone Vibrational Transitions in HCl
- 193: The Quantum Jump
- 194: Another Look at the Quantum Jump
- 195: Quantum Beats
- 196: The 1s-2s Electronic Transition in the 1D Hydrogen Atom
- 197: The Quantum Jump in Momentum Space
- 198: The Harmonic Oscillator Quantum Jump
- 199: Coherent States of the Harmonic Oscillator
- 201: Analysis of the Electronic Spectrum of Ti(H2O)3+
- 200: Quantum Jumps for an Electron on a Ring
- 202: Analysis of the Vibrational and Electronic Spectrum of Benzene
- 203: NMR - Quantum Mechanics of a Three Proton System
- 204: AB Proton NMR Using Tensor Algebra
- 205: Calculating the AB Proton NMR Using Tensor Algebra
- 206: AB Proton NMR Analysis for 2,3-dibromothiophene
- 207: ABC Proton NMR Using Tensor Algebra
- 208: AB2 Proton NMR Using Tensor Algebra
- 209: AB3 Proton NMR Using Tensor Algebra
- 210: HD-Like NMR Spectrum Calculated Using Tensor Algebra
- 211: The Michelson Interferometer and Fourier Transform Spectroscopy
- 212: A Sum Over Histories Approach to Fourier Transform Infrared Spectroscopy
- 213: Modeling the Pi-electrons of Benzene as Particles on a Ring
- 214: Modeling the Pi-electrons of Benzene as Particles on a Ring - Version 2
- 215: Calculating the Pi-electron HOMO-LUMO Electronic Transition for Benzene
- 216: Modeling the Pi-electrons of Benzene as Particles in a Ring
- 217: Modeling the Pi-electrons of Corannulene as Particles in a Ring
- 218: The Vibrational and Electronic States of C60
- Diffraction Phenomena
- 218: Using Optical Transforms to Teach Quantum Mechanics
- 219: Single-slit Diffraction and the Uncertainty Principle
- 220: Single-slit Diffraction and the Uncertainty Principle (Mathcad Version)
- 221: Simulating DNA's Diffraction Pattern
- 222: Simulating DNA's Diffraction Pattern with a More Realistic Model
- 223: Simulating DNA's Diffraction Pattern - Short Version
- 224: A Model Graphene Diffraction Pattern
- 225: Is a Two-dimensional Fibonacci Array a Quasilattice?
- 226: Calculating Diffraction Patterns
- 227: Modeling the C60 Diffraction Pattern
- 228: Diffraction Pattern for Pentagonal Point Scatterers
- 229: Diffraction Pattern for Pentagonal Finite Point Scatterers
- 230: Pentagram Diffraction Pattern
- 231: Model Diffraction Pattern for Napthalene
- 232: Calculating the Airy Diffraction Pattern
- 233: Diffraction Pattern for Two Concentric Rings
- 234: Density Operator Approach to the Double-Slit Experiment
- 235: Another Look at the Double-Slit Experiment
- 236: A Quantum Mechanical Interpretation of Diffraction
- 237: Electron Diffraction at Multiple Slits
- 238: Multiple Slit Diffraction and the Fourier Transform
- 239: The Double-Slit Experiment with C60 Molecules
- 240: Crystal Structure, Rotational Symmetry, and Quasicrystals
- 241: X-ray Crystallography from a Quantum Mechanical Perspective
- 242: Holography Involves Single Photon Interference
- 242: X-ray Diffraction
- 244: Holography Involves Single Photon Interference
- Group Theory with Mathcad
- Quantum Optics
- 256: Single-Photon Interference - First Version
- 257: Single-Photon Interference - Second Version
- 258: Single-photon Interference - Third Version
- 259: Single Photon Interference - Fourth Version
- 260: Single Photon Interference - Mathcad version
- 261: The Polarizing Beam Splitter and the Superposition Principle
- 262: Mach-Zehner Polarization Interferometer Analyzed Using Tensor Algebra
- 263: Illustrating the Superposition Principle with Single Photon Interference
- 264: Pure States, Mixtures and the Density Operator
- 265: Using the Trace Function to Calculate Expectation Values
- 266: Polarized Light and Quantum Superposition
- 267: Polarized Light and Quantum Mechanics
- 268: The Three-Polarizer Paradox
- 269: Matrix Mechanics Approach to Polarized Light
- 270: Matrix Mechanics Approach to Polarized Light - Version 2
- 271: Matrix Mechanics Exercises Using Polarized Light
- 272: Polarized Light and Quantum Mechanics
- 273: Neutron Interferometry with Polarized Spin States
- 274: Interaction Free Measurement - Seeing in the Dark
- 275: Quantum Seeing in the Dark - A Matrix-Tensor Analysis
- 276: Two Analyses of the Michelson Interferometer
- 277: A Quantum Circuit for a Michelson Interferometer
- 278: The Ramsey Atomic Interferometer
- 279: Optical Activity - A Quantum Perspective
- 280: A Quantum Optical Cheshire Cat
- 281: Two Photon Interference - The Creation of an Entangled Superposition
- 282: Two-particle Interference for Bosons and Fermions
- 283: Analysis of a Two-photon Interferometer
- 284: Two-photon Interferometry
- 285: Another Two Photon Interference Experiment
- 286: Quantum Correlations Illuminated with Tensor Algebra
- 287: Two Photon Entanglement - A Tensor Algebra Analysis
- 288: Two Photon Interference - Matrix Mechanics Approach
- 289: Two-electron Interference
- 290: Bosonic and Fermionic Photon Behavior at Beam Splitters
- 291: Bosonic and Fermionic Photon Behavior at Beam Splitters: A Tensor Algebra Analysis
- 292: Entangled Photons Can Behave Like Fermions
- 293: Analyzing Two-photon Interferometry Using Mathcad and Tensor Algebra
- 294: Analysis of a Two-photon Quantum Eraser
- 295: Another Example of a Two-photon Quantum Eraser
- 296: A Quantum Delayed-Choice Experiment
- 297: A Quantum Delayed-Choice Experiment
- Quantum Teleportation
- 298: A Single Page Summary of Quantum Teleportation
- 299: Quantum Teleportation - A Brief Introduction
- 300: Quantum Teleportation at a Glance
- 301: Another Look at Quantum Teleportation
- 302: Teleportation Using Quantum Gates
- 303: Another Example of Teleportation Using Quantum Gates
- 304: Yet Another Quantum Teleportation Circuit
- 305: Quantum Teleportation - Another Look
- 306: A Quantum Teleportation Experiment for Undergraduates
- 307: A Simple Teleportation Exercise
- 308: Teleportation as a Quantum Computation
- 309: Quantum Teleportation - Four Perspectives
- 311: Teleportation of Two Qubits
- 312.0: Greenaberger-Hrne-Zeilinger (GHZ) Entanglement and Local Realism
- 312.1: GHZ Math Appendix
- 313: GHZ Entanglement - A Tensor Algebra Analysis
- 314: Simulation of a GHZ Gedanken Experiment
- 315: Another Simulation of a GHZ Gedanken Experiment
- 316: A Surgical Refutation of the Local Realism Heresy
- 317: GHZ Four-Photon Entanglement Analyzed Using Tensor Algebra
- 318: Quantum v. Realism
- 319: Elements of Reality: Another GHZ Gedanken Experiment Analyzed
- 320: Brief Elements of Reality
- 321: A Brief Analysis of Mermin's GHZ Thought Experiment
- 322: Lucien Hardy's Paradox as Presented by N. David Mermin
- 323: Hardy's Paradox - An Algebraic Analysis
- 324: Quantum Entanglement Leads to Nonclassical Correlations
- 325: Nonclassial Correlations Revealed with Mermin's Pentagram
- 326: Spooky Action at a Distance: The EPR Experiment with Photons
- 328: David Bohm's EPR Gedanken Experiment
- 329: An Extension of Bohm's EPR Experiment
- 330: A Surgical Adjudication of the Conflict Between Quantum Theory and Local Realism
- 331: A Thought Experiment Reveals the Conflict Between Quantum Theory and Local Realism
- 332: Positronium Annihilation
- 333: Mermin's Version of Bohm's EPR Gedanken Experiment
- 334: A Concise Version of Mermin's EPR Gedanken Experiment
- 335: Another Look at Mermin's EPR Gedanken Experiment
- 336: Mermin's Version of Bohm's EPR Gedanken Experiment Using Tensor Algebra
- 337: A GHZ Gedanken Experiment Using Spatial Degrees of Freedom and Tensor Algebra
- 338: Another GHZ Example Using Spin-1/2 Particles
- 339: Entanglement Reveals a Conflict Between Local Realism and Quantum Theory
- 340: A Summary of Feynman's "Simulating Physics with Computers"
- 341: Another Summary of Fenyman's "Stimulating Physics with Computers"
- 342: Yet Another Assault on Local Realism
- 343: Yet Another Assault on Local Realism - A Matrix/Tensor Algebra Approach
- 344: Jim Baggott's Bell Theorem Analysis
- 345: Another Bell Theorem Analysis
- 346: Another Bell Theorem Analysis - Shorter Version
- 347: EPR Analysis for a Composite Singlet Spin System
- 348: EPR Analysis for a Composite Singlet Spin System - Short Version
- 349: Analysis of the Stern-Gerlach Experiment
- 350: Hardy's Paradox
- 351: Bell State Exercises
- 352: Expressing Bell and GHZ States in Vector Format Using Mathcad
- 353: Quantum Circuit for the Generation of GHZ States
- 354: A Brief Description of Aspect's Experiment
- 355: The Kochen-Specker Theorem Illustrated Using a Three-Qubit GHZ System
- 356: A Brief Introduction to Entanglement Swapping
- 357: An Entanglement Swapping Protocol
- 358: Quantum Correlations Simplified
- 359: Simulating Quantum Correlations with a Quantum Computer
- 360: Quantum Computer Simulation of Photon Correlations
- 361: Quantum Correlations Illustrated with Photons
- 362: Examining the Local States of an Entangled Bipartate Superposition
- 363: A Brief Introduction to the Quantum Computer
- 364: A Very Simple Example of Parallel Quantum Computation
- 365: Another Simple Example of Parallel Quantum Computation
- 366: A Simple Solution to Deutsch's Problem
- 367: Another Example of Deutsch's Algorithm
- 368: Evaluating a Function Using a Quantum Circuit and a Demonstration of Parallel Computation
- 369: A Simple Quantum Circuit for Parallel Computation
- 370: An Illustration of the Deutsch-Jozsa Algorithm
- 371: Another Illustration of the Deutsch-Jozsa Algorithm
- 372: Aspects of Simon's Algorithm
- 373: Qubit Quantum Mechanics
- 374: Implementation of Deutsch's Algorithm Using Mathcad
- 375: Using Quantum Gates to Create Superpositions and Entangled States
- 376: A Simple Quantum Computer
- 377: Solving Equations Using a Quantum Circuit
- 378: Introduction to Superdense Coding
- 379: A Brief Introduction to Quantum Dense Coding
- 380: The Discrete or Quantum Fourier Transform
- 381: Factoring Using Shor's Quantum Algorithm
- 382: Shor's Quantum Algorithm - A Summary
- 383: Simulating the Deutsch-Jozsa Algorithm with a Double-Slit Apparatus
- 384: Simulating a Quantum Computer with a Mach-Zehnder Interferometer
- 385: Quantum Restrictions on Cloning
- 386: Quantum Error Correction
- 387: Matrix Mechanics Analysis of the BB84 Key Distribution
- 388. The Quantum Math Behind Ekert's Key Distribution Scheme
- 389: A Shorter Version of the Quantum Math Behind Ekert's Key Distribution Scheme
- 390: Quantum Key Distribution Using a Mach-Zehnder Interferometer
- 391: Coding and Decoding Venus
- 392: Grover's Quantum Search Algorithm
- 393: Grover's Search Algorithm: Implementation for Two Items
- 394: Grover's Search Algorithm: Four-Card Monte
- Numerical Solutions for Schrödinger's Equation
- 395: Introduction to Numerical Solutions of Schödinger's Equation
- 396: Particle in an Infinite Potential Well
- 397: Particle in a Gravitational Field
- 398: Particle in a One-dimensional Egg Carton
- 399: Particle in a Finite Potential Well
- 400: Particle in a Semi-infinite Potential Well
- 401: Particle in a Slanted Well Potential
- 402: Numerical Solutions for a Particle in a V-Shaped Potential Well
- 403: Numerical Solutions for the Harmonic Oscillator
- 404: Numerical Solutions for a Double-Minimum Potential Well
- 405: Numerical Solutions for the Quartic Oscillator
- 406: Numerical Solutions for Morse Oscillator
- 407: Numerical Solutions for the Lennard-Jones Potential
- 408: Numerical Solutions for the Double Morse Potential
- 409: Particle in a Box with an Internal Barrier
- 410: Another Look at the in a Box with an Internal Barrier
- 411: Particle in a Box with Multiple Internal Barriers
- 412: Particle in an Infinite Spherical Potential Well
- 413: Numerical Solutions for the Two-Dimensional Harmonic Oscillator
- 414: Numerical Solutions for the Three-Dimensional Harmonic Oscillator
- 415: Numerical Solutions for the Hydrogen Atom Radial Equation
- 416: Numerical Solutions for a Modified Harmonic Potential
- Approximate Quantum Mechanical Methods
- 417: Trial Wavefunctions for Various Potentials
- 418: Energy Minimization - Four Methods Using Mathcad
- 419: The Variation Theorem in Dirac Notation
- 420: A Rudimentary Model for Alpha Particle Decay
- 421: Variational Method for a Particle in a Finite Potential Well
- 422: Variation Method for a Particle in a Symmetric 1D Potential Well
- 423: Variation Method for the Rydberg Potential
- 424: Variation Method for the Quartic Oscillator
- 425: Momentum-Space Variation Method for the Quartic Oscillator
- 426: Variation Method for a Particle in a Gravitational Field
- 427: Linear Variational Method for a Particle in a Slanted 1D Box
- 428: Variation Method for a Particle in a Semi-Infinite Potential Well
- 429: Variation Method for a Particle in a Box with an Internal Barrier
- 430: Variation Method for a Particle in a 1D Ice Cream Cone
- 431: Variation Method for a Particle in an Ice Cream Cone
- 432: Variation Method for a Particle in a Finite 3D Spherical Potential Well
- 433: Variation Method for the Harmonic Oscillator
- 434: Trigonometric Trial Wave Function for the Harmonic Potential Well
- 435: Trigonometric Trial Wave Function for the 3D Harmonic Potential Well
- 436: Gaussian Trial Wavefunction for the Hydrogen Atom
- 437: Variation Calculation on the 1D Hydrogen Atom Using a Trigonometric Trial Wave Function
- 438: Variation Calculation on the 1D Hydrogen Atom Using a Gaussian Trial Wavefunction
- 439: Variational Calculation on the Two-dimensional Hydrogen Atom
- 440: Variational Calculation on Helium Using a Hydrogenic Wavefunction
- 441: Gaussian Trial Wave Function for the Helium Atom
- 442: Trigonometric Trial Wavefunction for the Helium Atom
- 443: Trigonometric Trial Wavefunction for the Hydrogen Atom
- 444: Hydrogen Atom Calculation Assuming the Electron is a Particle in a Sphere of Radius R
- 445: Electronic Structure - Variational Calculations on the Lithium Atom
- 446: The Variation Method in Momentum Space
- 447: Momentum-Space Variation Method for Particle in a Gravitational Field
- 448: Momentum-Space Variation Method for the Abs(x) Potential
- 449: Variational Method for the Feshbach Potential
- 450: Numerical Solution for the Feshbach Potential
- 451: First Order Degenerate Perturbation Theory - the Stark Effect of the Hydrogen Atom
- 452: Variational Calculation for the Polarizability of the Hydrogen Atom
- 453: Hybrid Variational Calculation for the 1D Hydrogen Atom with Delta Function Potential
- 454: Variation Method Using the Wigner Function: Finite Potential Well
- 455. Variation Method Using the Wigner Function: V(x) = |x|
- 456: Variation Method Using the Wigner Function: The Harmonic Oscillator
- 457: Variation Method Using the Wigner Function - The Quartic Oscillator
- 458: Variation Method Using the Wigner Function - The Feshbach Potential
- Miscellaneous
- 459: The Art of Science
- 460: Mass-Energy Equivalence
- 461: Commentary on “Probing the Orbital Energy of an Electron in an Atom”
- 462: The Use of Models in Introductory Chemistry
- 463: Reaction to Gillespie's Six Great Ideas in Chemistry - Another Great Idea
- 464: An Alternative Derivation of Gas Pressure Using the Kinetic Theory
- 465: Examining Fourier Synthesis with Dirac Notation
- 466: Finding Roots of Transcendental Equations
- 467: Calculation of the Composition of a Weak Polyprotic Acid Using Mathcad
- 468: Solving Linear Equations Using Mathcad
- 469: Let's Teach High School Students Computer Algebra Methods
- 470: Thermodynamics and Kinetics
- 471. Simple Kinetic Derivations of Thermodynamic Relations
- 472: The Global Approach to Thermodynamics
- 473: Global Thermodynamic Analyses of Heat Engines
- 474: Using Charles' Law to Determine Absolute Zero
- 475: The Origin of KE = 3/2 RT
- 476: Cosmic Background Radiation
- 477: Age of the Elements
- Quantum Fundamentals
- Map: Physical Chemistry (McQuarrie and Simon)
- 1: The Dawn of the Quantum Theory
- 1.1: Blackbody Radiation Cannot Be Explained Classically
- 1.2: Quantum Hypothesis Used for Blackbody Radiation Law
- 1.3: Photoelectric Effect Explained with Quantum Hypothesis
- 1.4: The Hydrogen Atomic Spectrum
- 1.5: The Rydberg Formula and the Hydrogen Atomic Spectrum
- 1.6: Matter Has Wavelike Properties
- 1.7: de Broglie Waves can be Experimentally Observed
- 1.8: The Bohr Theory of the Hydrogen Atom
- 1.9: The Heisenberg Uncertainty Principle
- 1.E: The Dawn of the Quantum Theory (Exercises)
- 2: The Classical Wave Equation
- 3: The Schrödinger Equation and a Particle in a Box
- 3.1: The Schrödinger Equation
- 3.2: Linear Operators in Quantum Mechanics
- 3.3: The Schrödinger Equation is an Eigenvalue Problem
- 3.4: Wavefunctions Have a Probabilistic Interpretation
- 3.5: The Energy of a Particle in a Box is Quantized
- 3.6: Wavefunctions Must Be Normalized
- 3.7: The Average Momentum of a Particle in a Box is Zero
- 3.8: The Uncertainty Principle - Estimating Uncertainties from Wavefunctions
- 3.9: A Particle in a Three-Dimensional Box
- 3.E: The Schrödinger Equation and a Particle in a Box (Exercises)
- 4: Postulates and Principles of Quantum Mechanics
- 4.1: The Wavefunction Specifies the State of a System
- 4.2: Quantum Operators Represent Classical Variables
- 4.3: Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators
- 4.4: The Time-Dependent Schrödinger Equation
- 4.5: Eigenfunctions of Operators are Orthogonal
- 4.6: Commuting Operators Allow Infinite Precision
- 4.E: Postulates and Principles of Quantum Mechanics (Exercises)
- 5: The Harmonic Oscillator and the Rigid Rotor
- 5.1: A Harmonic Oscillator Obeys Hooke's Law
- 5.2: The Equation for a Harmonic-Oscillator Model of a Diatomic Molecule Contains the Reduced Mass of the Molecule
- 5.3: The Harmonic Oscillator Approximates Vibrations
- 5.4: The Harmonic Oscillator Energy Levels
- 5.5: The Harmonic Oscillator and Infrared Spectra
- 5.6: The Harmonic-Oscillator Wavefunctions Involve Hermite Polynomials
- 5.7: Hermite Polynomials are either Even or Odd Functions
- 5.8: The Energy Levels of a Rigid Rotor
- 5.9: The Rigid Rotator is a Model for a Rotating Diatomic Molecule
- 5.E: The Harmonic Oscillator and the Rigid Rotor (Exercises)
- 6: The Hydrogen Atom
- 6.1: The Schrodinger Equation for the Hydrogen Atom Can Be Solved Exactly
- 6.2: The Wavefunctions of a Rigid Rotator are Called Spherical Harmonics
- 6.3: The Three Components of Angular Momentum Cannot be Measured Simultaneously with Arbitrary Precision
- 6.4: Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers
- 6.5: s-orbitals are Spherically Symmetric
- 6.6: Orbital Angular Momentum and the p-Orbitals
- 6.7: The Helium Atom Cannot Be Solved Exactly
- 6.E: The Hydrogen Atom (Exercises)
- 7: Approximation Methods
- 7.1: The Variational Method Approximation
- 7.2: Linear Variational Method and the Secular Determinant
- 7.3: Trial Functions Can Be Linear Combinations of Functions That Also Contain Variational Parameters
- 7.4: Perturbation Theory Expresses the Solutions in Terms of Solved Problems
- 7.E: Approximation Methods (Exercises)
- 8: Multielectron Atoms
- 8.1: Atomic and Molecular Calculations are Expressed in Atomic Units
- 8.2: Perturbation Theory and the Variational Method for Helium
- 8.3: Hartree-Fock Equations are Solved by the Self-Consistent Field Method
- 8.4: An Electron Has an Intrinsic Spin Angular Momentum
- 8.5: Wavefunctions must be Antisymmetric to Interchange of any Two Electrons
- 8.6: Antisymmetric Wavefunctions can be Represented by Slater Determinants
- 8.7: Hartree-Fock Calculations Give Good Agreement with Experimental Data
- 8.8: A Term Symbol Gives a Detailed Description of an Electron Configuration
- 8.9: The Allowed Values of J - the Total Angular Momentum Quantum Number
- 8.10: Hund's Rules Determine the Term Symbols of the Ground Electronic States
- 8.11: Using Atomic Term Symbols to Describe Atomic Spectra
- 8.E: Multielectron Atoms (Exercises)
- 9: Chemical Bonding in Diatomic Molecules
- 9.1: The Born-Oppenheimer Approximation Simplifies the Schrödinger Equation for Molecules
- 9.2: The H₂⁺ Prototypical Species
- 9.3: The Overlap Integral
- 9.4: Chemical Bond Stability
- 9.5: Bonding and Antibonding Orbitals
- 9.6: A Simple Molecular-Orbital Treatment of H₂ Places Both Electrons in a Bonding Orbital
- 9.7: Molecular Orbitals Can Be Ordered According to Their Energies
- 9.8: Molecular-Orbital Theory Does not Predict a Stable Diatomic Helium Molecule
- 9.9: Electrons Populate Molecular Orbitals According to the Pauli Exclusion Principle
- 9.10: Molecular Orbital Theory Predicts that Molecular Oxygen is Paramagnetic
- 9.11: Photoelectron Spectra Support the Existence of Molecular Orbitals
- 9.12: Molecular-Orbital Theory Also Applies to Heteronuclear Diatomic Molecules
- 9.13: An SCF-LCAO-MO Wave Function Is a Molecular Orbital Formed from a Linear Combination of Atomic Orbitals and Whose Coefficients Are Determined Self-Consistently
- 9.14: Molecular Term Symbols Describe Electronic States of Molecules
- 9.15: Molecular Term Symbols Designate Symmetry
- 9.16: Most Molecules Have Excited Electronic States
- 9.E: Chemical Bond in Diatomic Molecules (Exercises)
- 10: Bonding in Polyatomic Molecules
- 10.1: Hybrid Orbitals Account for Molecular Shape
- 10.2: Hybrid Orbitals in Water
- 10.3: Why is BeH₂ Linear and H₂O Bent?
- 10.4: Photoelectron Spectroscopy
- 10.5: The \(\pi\)-Electron Approximation of Conjugation
- 10.6: Butadiene is Stabilized by a Delocalization Energy
- 10.7: Benzene and Aromaticity
- 10.E: Bonding in Polyatomic Molecules (Exercises)
- 11: Computational Quantum Chemistry
- 12: Group Theory - The Exploitation of Symmetry
- 12.1: The Exploitation of Symmetry
- 12.2: Symmetry Elements
- 12.3: Symmetry Operations Define Groups
- 12.4: Symmetry Operations as Matrices
- 12.5: The \(C_{3V}\) Point Group
- 12.6: Character Tables
- 12.7: Characters of Irreducible Representations
- 12.8: Using Symmetry to Solve Secular Determinants
- 12.9: Generating Operators
- 12.E: Group Theory - The Exploitation of Symmetry (Exercises)
- 13: Molecular Spectroscopy
- 13.1: The Electromagnetic Spectrum
- 13.2: Rotations Accompany Vibrational Transitions
- 13.3: Unequal Spacings in Vibration-Rotation Spectra
- 13.4: Unequal Spacings in Pure Rotational Spectra
- 13.5: Vibrational Overtones
- 13.6: Electronic Spectra Contain Electronic, Vibrational, and Rotational Information
- 13.7: The Franck-Condon Principle
- 13.8: Rotational Spectra of Polyatomic Molecules
- 13.9: Normal Modes in Polyatomic Molecules
- 13.10: Irreducible Representation of Point Groups
- 13.11: Time-Dependent Perturbation Theory
- 13.12: The Selection Rule for the Rigid Rotor
- 13.13: The Harmonic Oscillator Selection Rule
- 13.14: Group Theory Determines Infrared Activity
- 13.E: Molecular Spectroscopy (Exercises)
- 14: Nuclear Magnetic Resonance Spectroscopy
- 14.1: Nuclei Have Intrinsic Spin Angular Momenta
- 14.2: Magnetic Moments Interact with Magnetic Fields
- 14.3: Proton NMR Spectrometers Operate at Frequencies Between 60 MHz and 750 MHz
- 14.4: The Magnetic Field Acting upon Nuclei in Molecules Is Shielded
- 14.5: Chemical Shifts Depend upon the Chemical Environment of the Nucleus
- 14.6: Spin-Spin Coupling Can Lead to Multiplets in NMR Spectra
- 14.7: Spin-Spin Coupling Between Chemically Equivalent Protons is Not Observed
- 14.8: The n+1 Rule Applies Only to First-Order Spectra
- 14.9: Second-Order Spectra Can Be Calculated Exactly Using the Variational Method
- 14.E: Nuclear Magnetic Resonance Spectroscopy (Exercises)
- 15: Lasers, Laser Spectroscopy, and Photochemistry
- 15.1: Electronically Excited Molecules can Relax by a Number of Processes
- 15.2: The Dynamics of Transitions can be Modeled by Rate Equations
- 15.3: A Two-Level System Cannot Achieve a Population Inversion
- 15.4: Population Inversion can be Achieved in a Three-Level System
- 15.5: What is Inside a Laser?
- 15.6: The Helium-Neon Laser
- 15.7: High-Resolution Laser Spectroscopy
- 15.8: Pulsed Lasers Can by Used to Measure the Dynamics of Photochemical Processes
- 15.E: Lasers, Laser Spectroscopy, and Photochemistry (Exercises)
- 16: The Properties of Gases
- 16.1: All Dilute Gases Behave Ideally
- 16.2: van der Waals and Redlich-Kwong Equations of State
- 16.3: A Cubic Equation of State
- 16.4: The Law of Corresponding States
- 16.5: The Second Virial Coefficient
- 16.6: The Repulsive Term in the Lennard-Jones Potential
- 16.7: Van der Waals Constants in Terms of Molecular Parameters
- 16.E: The Properties of Gases (Exercises)
- 17: Boltzmann Factor and Partition Functions
- 17.1: The Boltzmann Factor
- 17.2: The Thermal Boltzman Distribution
- 17.3: The Average Ensemble Energy
- 17.4: Heat Capacity at Constant Volume
- 17.5: Pressure in Terms of Partition Functions
- 17.6: Partition Functions of Distinguishable Molecules
- 17.7: Partition Functions of Indistinguishable Molecules
- 17.8: Partition Functions can be Decomposed
- 17.E: Boltzmann Factor and Partition Functions (Exercises)
- 18: Partition Functions and Ideal Gases
- 18.1: Translational Partition Functions of Monotonic Gases
- 18.2: Most Atoms are in the Ground Electronic State
- 18.3: The Energy of a Diatomic Molecule Can Be Approximated as a Sum of Separate Terms
- 18.4: Most Molecules are in the Ground Vibrational State
- 18.5: Most Molecules are Rotationally Excited at Ordinary Temperatures
- 18.6: Rotational Partition Functions of Diatomic Gases
- 18.7: Vibrational Partition Functions of Polyatomic Molecules
- 18.8: Rotational Partition Functions of Polyatomic Molecules
- 18.9: Molar Heat Capacities
- 18.E: Partition Functions and Ideal Gases (Exercises)
- Ortho and Para Hydrogen
- The Equipartition Principle
- 19: The First Law of Thermodynamics
- 19.0: Overview of Classical Thermodynamics
- 19.1: Pressure-Volume Work
- 19.2: Work and Heat are not State Functions, but Energy is a State Function
- 19.3: Energy is a State Function
- 19.4: An Adiabatic Process is a Process in which No Energy as Heat is Transferred
- 19.5: The Temperature of a Gas Decreases in a Reversible Adiabatic Expansion
- 19.6: Work and Heat Have a Simple Molecular Interpretation
- 19.7: Pressure-Volume Work
- 19.8: Heat Capacity is a Path Function
- 19.9: Relative Enthalpies Can Be Determined from Heat Capacity Data and Heats of Transition
- 19.10: Enthalpy Changes for Chemical Equations are Additive
- 19.11: Heats of Reactions Can Be Calculated from Tabulated Heats of Formation
- 19.12: The Temperature Dependence of ΔH
- 19.E: The First Law of Thermodynamics (Exercises)
- 20: Entropy and The Second Law of Thermodynamics
- 20.1: Energy Does not Determine Spontaneity
- 20.2: Nonequilibrium Isolated Systems Evolve in a Direction That Increases Their Probability
- 20.3: Unlike heat, Entropy Is a State Function
- 20.4: The Second Law of Thermodynamics
- 20.5: The Famous Equation of Statistical Thermodynamics
- 20.6: We Must Always Devise a Reversible Process to Calculate Entropy Changes
- 20.7: Thermodynamics Provides Insight into the Conversion of Heat into Work
- 20.8: Entropy Can Be Expressed in Terms of a Partition Function
- 20.9: The Molecular Formula S = kB in W is Analogous to the Thermodynamic Formula dS = deltaqrev
- 20.E: Entropy and The Second Law of Thermodynamics (Exercises)
- 21: Entropy & the Third Law of Thermodynamics
- 21.1: Entropy Increases With Increasing Temperature
- 21.2: Absolute Entropy
- 21.3: Temperatures at a Phase Transition
- 21.4: The Third Law of Thermodynamics
- 21.5: Practical Absolute Entropies Can Be Determined Calorimetrically
- 21.6: Practical Absolute Entropies of Gases Can Be Calculated from Partition Functions
- 21.7: Standard Entropies Depend Upon Molecular Mass and Structure
- 21.8: Spectroscopic Entropies sometimes disgree with Calorimetric Entropies
- 21.9: Standard Entropies Can Be Used to Calculate Entropy Changes of Chemical Reactions
- 21.E: Entropy & the Third Law of Thermodynamics (Exercises)
- 22: Helmholtz and Gibbs Energies
- 22.1: Helmholtz Energy
- 22.2: Gibbs Energy
- 22.3: The Maxwell Relations
- 22.4: The Enthalpy of an Ideal Gas
- 22.5: Thermodynamic Functions have Natural Variables
- 22.6: The Standard State for a Gas is Ideal Gas
- 22.7: The Gibbs-Helmholtz Equation
- 22.8: Fugacity Measures Nonideality of a Gas
- 22.E: Helmholtz and Gibbs Energies (Exercises)
- 23: Phase Equilibria
- 23.1: A Phase Diagram Summarizes the Solid-Liquid-Gas Behavior of a Substance
- 23.2: Gibbs Energies and Phase Diagrams
- 23.3: The Chemical Potentials of a Pure Substance in Two Phases in Equilibrium
- 23.4: The Clausius-Clapeyron Equation
- 23.5: Chemical Potential Can be Evaluated From a Partition Function
- 23.E: Phase Equilibria (Exercises)
- 24: Solutions I: Liquid-Liquid Solutions
- 1: The Dawn of the Quantum Theory