Theoretical Chemistry

2.

Quantum Mechanics

2.1.

Historical Development

2.1.1.

Failures of Classical Physics

2.1.1.1.

Blackbody Radiation Problem

2.1.1.2.

Ultraviolet Catastrophe

2.1.1.3.

Planck's Quantum Hypothesis

2.1.2.

Photoelectric Effect

2.1.2.1.

Experimental Observations

2.1.2.2.

Einstein's Photon Theory

2.1.2.3.

Work Function and Threshold Frequency

2.1.3.

Atomic Structure

2.1.3.1.

Rutherford Scattering

2.1.3.2.

Bohr Model of Hydrogen

2.1.3.3.

Limitations of Bohr Model

2.1.4.

Wave-Particle Duality

2.1.4.1.

de Broglie Hypothesis

2.1.4.2.

Davisson-Germer Experiment

2.1.4.3.

Double-Slit Experiment

2.2.

Fundamental Postulates

2.2.1.

The Wavefunction

2.2.1.1.

Physical Interpretation

2.2.1.2.

Born's Statistical Interpretation

2.2.1.3.

Normalization Condition

2.2.1.4.

Probability Current Density

2.2.2.

Operators and Observables

2.2.2.1.

Linear Operators

2.2.2.2.

Hermitian Operators

2.2.2.3.

Eigenvalue Equations

2.2.2.4.

Commutator Relations

2.2.2.5.

Compatible Observables

2.2.3.

The Schrödinger Equation

2.2.3.1.

Time-Dependent Form

2.2.3.2.

Time-Independent Form

2.2.3.3.

Continuity Equation

2.2.3.4.

Probability Conservation

2.2.4.

Measurement Theory

2.2.4.1.

Eigenvalues as Measurement Outcomes

2.2.4.2.

Expectation Values

2.2.4.3.

Standard Deviation and Uncertainty

2.2.4.4.

Collapse of the Wavefunction

2.2.5.

Uncertainty Principle

2.2.5.1.

Heisenberg Uncertainty Relations

2.2.5.2.

Position-Momentum Uncertainty

2.2.5.3.

Energy-Time Uncertainty

2.2.5.4.

Generalized Uncertainty Relations

2.3.

One-Dimensional Systems

2.3.1.

2.3.1.1.

Plane Wave Solutions

2.3.1.2.

2.3.1.3.

Group and Phase Velocity

2.3.2.

Particle in a Box

2.3.2.1.

Infinite Square Well

2.3.2.2.

Energy Eigenvalues

2.3.2.3.

Wavefunctions and Normalization

2.3.2.4.

Finite Square Well

2.3.2.5.

Tunneling Phenomenon

2.3.3.

Quantum Harmonic Oscillator

2.3.3.1.

Classical Harmonic Oscillator Review

2.3.3.2.

Schrödinger Equation Solution

2.3.3.3.

Energy Levels and Degeneracy

2.3.3.4.

Ladder Operators

2.3.3.5.

Coherent States

2.3.4.

Potential Barriers and Wells

2.3.4.1.

2.3.4.2.

Barrier Penetration

2.3.4.3.

Tunneling Probability

2.3.4.4.

Resonant Tunneling

2.4.

Three-Dimensional Systems

2.4.1.

Particle in a Three-Dimensional Box

2.4.1.1.

Separation of Variables

2.4.1.2.

Energy Levels and Degeneracy

2.4.1.3.

Density of States

2.4.2.

Central Force Problems

2.4.2.1.

Separation in Spherical Coordinates

2.4.2.2.

Radial Equation

2.4.2.3.

Angular Momentum Eigenvalue Problem

2.4.3.

The Hydrogen Atom

2.4.3.1.

Coulomb Potential

2.4.3.2.

Radial Wavefunctions

2.4.3.3.

Associated Laguerre Polynomials

2.4.3.4.

Quantum Numbers

2.4.3.5.

Energy Levels and Degeneracy

2.4.3.6.

Electron Probability Distributions

2.4.4.

2.4.4.1.

Classical Rigid Rotor

2.4.4.2.

Quantum Mechanical Treatment

2.4.4.3.

Rotational Energy Levels

2.4.4.4.

Spherical Harmonics

2.5.

Angular Momentum

2.5.1.

Orbital Angular Momentum

2.5.1.1.

Angular Momentum Operators

2.5.1.2.

Commutation Relations

2.5.1.3.

Eigenvalue Problem

2.5.1.4.

Ladder Operators

2.5.2.

Spin Angular Momentum

2.5.2.1.

Spin-1/2 Particles

2.5.2.2.

2.5.2.3.

2.5.2.4.

Stern-Gerlach Experiment

2.5.3.

Addition of Angular Momenta

2.5.3.1.

2.5.3.2.

Clebsch-Gordan Coefficients

2.5.3.3.

Wigner 3j Symbols

2.5.3.4.

Coupling Schemes

2.6.

Approximation Methods

2.6.1.

Time-Independent Perturbation Theory

2.6.1.1.

Non-Degenerate Perturbation Theory

2.6.1.2.

First-Order Energy Correction

2.6.1.3.

Second-Order Energy Correction

2.6.1.4.

First-Order Wavefunction Correction

2.6.1.5.

Degenerate Perturbation Theory

2.6.1.6.

Lifting of Degeneracy

2.6.2.

Time-Dependent Perturbation Theory

2.6.2.1.

Interaction Picture

2.6.2.2.

First-Order Time-Dependent Perturbation

2.6.2.3.

Transition Probabilities

2.6.2.4.

Fermi's Golden Rule

2.6.2.5.

Adiabatic Approximation

2.6.3.

Variational Method

2.6.3.1.

Variational Principle

2.6.3.2.

Trial Wavefunctions

2.6.3.3.

Linear Variational Method

2.6.3.4.

Application to Helium Atom

2.6.4.

WKB Approximation

2.6.4.1.

Semiclassical Approximation

2.6.4.2.

Connection Formulas

2.6.4.3.

Tunneling Applications

2.7.

Advanced Formalism

2.7.1.

2.7.1.1.

Ket and Bra Vectors

2.7.1.2.

2.7.1.3.

2.7.1.4.

Completeness Relations

2.7.1.5.

Matrix Elements

2.7.2.

Representation Theory

2.7.2.1.

Position Representation

2.7.2.2.

Momentum Representation

2.7.2.3.

Energy Representation

2.7.2.4.

Transformation Between Representations

2.7.3.

Scattering Theory

2.7.3.1.

Scattering Cross Section

2.7.3.2.

2.7.3.3.

Partial Wave Analysis

2.7.3.4.

Born Approximation

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1. Foundations of Theoretical Chemistry

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3. Electronic Structure Theory