Quantum Mechanics: An Accessible Introduction

Table of Contents

1. The Origins of Quantum Mechanics

1.1 Introduction

1.2 Blackbody Radiation

The Problem with Blackbody Radiation

1.3 The Nature of Light

The Photoelectric Effect

The Compton Effect

Is it a Particle or a Wave?

1.4 TheWave Nature of Matter

1.5 The Bohr Atom

1.6 Where do we Stand?

2. Math Interlude A: Complex Numbers and Linear Operators

2.1 Complex Numbers

2.2 Operators

Definition of an Operator

Eigenfunctions and Eigenvalues

3. The Schrödinger Equation

3.1 Derivation of the Schrödinger Equation

3.2 The Meaning of theWave Function

3.3 The Time-Independent Schrödinger Equation

Derivation of the Time-Independent Schrödinger Equation

Qualitative Solutions and the Origin of Quantization

4. One-Dimensional Time-Independent

Schrödinger Equation

4.1 Unbound States: Scattering and Tunneling

Scattering From Step-Function Potentials

4.2 Bound Systems

The Infinite SquareWell

The Harmonic Oscillator Potential

5. Math Interlude B: Linear Algebra

5.1 Properties of Linear Operators

5.2 Vector Spaces

Inner Products

Adjoint and Hermitian Operators

Basis Sets

6. The Three-Dimensional Time-Independent

Schrödinger Equation

6.1 Solution in Rectangular Coordinates

6.2 Angular Momentum

6.3 The Schrödinger Equation in Spherical Coordinates

6.4 The Hydrogen Atom

7. Math Interlude C: Matrices, Dirac Notation, and the Dirac Delta Function

7.1 The Matrix Formulation of Linear Operators

7.2 Dirac Notation

7.3 The Dirac Delta Function

8. Spin Angular Momentum

8.1 Spin Operators

8.2 Evidence for Spin

8.3 Adding Angular Momentum

8.4 The Matrix Representation of Spin

8.5 The Stern–Gerlach Experiment

8.6 Spin Precession

8.7 Spin Systems with Two Particles

Noninteracting Spins

Interacting Spins

8.8 Measurement Theory

Hidden Variables

The ManyWorlds Interpretation of Quantum Mechanics

9. Time-Independent Perturbation Theory

9.1 Derivation of Time-Independent Perturbation Theory

9.2 Perturbations to the Atomic Energy Levels

Fine Structure

The Lamb Shift

9.3 The Atom in External Electric or Magnetic Fields

The Atom in an Electric Field: The Stark Effect

The Atom in a Magnetic Field: The Zeeman Effect

10. The Variational Principle

10.1 Variational Principle: Theory

10.2 Variational Principle: Application to the Helium Atom

11. Time-Dependent Perturbation Theory

11.1 Derivation of Time-Dependent Perturbation Theory

11.2 Application: Selection Rules for Electromagnetic Radiation

12. Scattering Theory

12.1 Definition of the Cross Section

12.2 The Born Approximation

12.3 PartialWaves

13. The Multiparticle Schrödinger Equation

13.1 Wave Function for Identical Particles

13.2 Multielectron Atoms

14. Some Modern Applications of Quantum Mechanics

14.1 Magnetic Resonance Imaging

14.2 Quantum Computing

15. What Comes Next? Relativistic Quantum Mechanics

15.1 The Klein–Gordon Equation

Derivation of the Klein–Gordon Equation

Probability Densities and Currents

15.2 The Dirac Equation

Answers and Hints for Selected End-of-Chapter Exercises

Index