Understanding Quantum Science: A Concise Primer for Students of Chemistry, Biochemistry and Physics

Description

Students are naturally drawn to quantum science by the intriguing behaviors of small particles. However, they can also be intimidated by the lengthy and complicated treatment found in the classroom. Understanding Quantum Science: A Concise Primer for Students of Chemistry, Biochemistry, and Physics is a highly accessible book that offers students an opportunity to grasp the most fascinating of quantum topics, without the intimidation. To be sure, math is necessary, but it is introduced as needed and kept concise. The emphasis is on the science: a certain differential equation can be solved, and when it is, we find the energies that hydrogen atom electrons are allowed to have. Each concept is developed in this manner, keeping focus on how and why it arises, and on the intriguing consequences. This book provides a brief tour of some of the wonders of quantum science. But it is more than that, it is designed to be the most concise tour possible that truly explains how these wonders arise so that you can develop a working understanding of quantum concepts. If your goal is loftier and you wish to become a quantum specialist, the conceptual groundwork presented here, along with rationalization of the mathematics required, will position you well for higher level classes. Cover Half Title Title Page Copyright Page Dedication Table of Contents Preface About the Author The Basics 1 Introducing Quantum Mechanics 1.1 What Is Quantum Mechanics? 2 The Schrödinger Equation 2.1 Comparison With the Classical Wave Equation 2.2 Complex Solution to the Classical Wave Equation 2.3 What Are K and ω? 2.4 Why Does an Electron Need a Wave Equation? 2.5 What’s Different About the Schrödinger Equation? 2.6 Imaginary Probability? 2.7 Replacing K and ω With P and E (For Photons) 2.8 Extending These Two Equations to Electrons Summary Notes 3 Deriving the Schrödinger Equation 4 Operators, Oscillations, Uncertainty, and Quanta 4.1 Operators 4.2 Oscillating Probability 4.3 Uncertainty 4.4 Quanta 5 Separation of Variables 5.1 What Is “Separation of Variables”? 5.2 In Order to Separate Space From Time, the Potential Must Be Time-Independent 5.3 Plug the Separable Form of Ψ (X,t) Into the TDSE 5.4 Separate Into the TISE and SISE 5.5 Separable Solution ↔ Single Energy ↔ Eigenfunction 5.6 Solving the SISE (Once) 5.7 Solving the TISE (OVER AND OVER) 5.8 Superposition of Eigenfunctions Note Chapter 6 ψ(x): General Conditions, Normalization, Bra-Kets 6.1 General Conditions for ψ(x) 6.2 Normalization of ψ(x) 6.3 Dirac Bra-Ket Notation One-Dimensional Potentials 7 Solving the TISE for the Simplest Potentials 7.1 Free Particle 7.2 Constant Potential: V = V0 Everywhere 7.3 Potential Steps, Barriers, and Wells 7.4 Finite Potential Step: Case of E > V0 7.5 Finite Potential Step: Case of E V0 7.7 Finite Barrier of Finite Width: Case E < V0 7.8 Finite Barrier of Finite Width: Case E < V0 (Finite Well) 8 The One-Dimensional Particle in a Box 8.1 Find the PIB Eigenfunctions 8.2 Eigenfunction ↔ Separable ↔ Stationary State ↔ Pure State 8.3 Normalization 8.4 Orthogonality 8.5 Orthonormality 8.6 Expectation Values Notes 9 The Formal Postulates of Quantum Mechanics 9.1 Example: PIB Superposition of States 9.2 Time-Dependence of a Superposition of States 9.3 Hilbert Space 10 Simple Harmonic Oscillator (SHO): V = ½ Kx2 10.1 Classical SHO (Mass On a Spring, Ball in a Well) 10.2 Quantum SHO 10.3 Creation/Annihilation Operators Approximation Methods 11 Time-Independent Perturbation Theory (TIPT) 11.1 Three Approximation Methods 11.2 TIPT (Time-Independent Perturbation Theory) 11.3 Matrix Shorthand 11.4 TIPT Example: PIB Perturbed By AN Electric Field . Problem Solution 12 Time-Dependent Perturbation Theory (TDPT) 12.1 TDPT 12.2 TDPT Example 1: PIB Temporarily Perturbed By AN Electric Field Problem Solution 12.3 TDPT Example 2: Harmonic Perturbation (Perturbation Oscillates With Time) Summary Notes 13 Variational Method 13.1 Variational Method Recipe 13.2 Example: SHO Ground State Three-Dimensional Space: Atoms and Molecules 14 Generalization to 3D 14.1 Three-Dimensional Derivatives 14.2 Angular TISE 15 Angular Momentum 15.1 Quantization of Orbital Angular Momentum (L) 15.2 Alternate Forms of F(ϕ): Exponential and Trigonometric 15.3 Spin Angular Momentum (S) Note 16 H-Atom: Solving the Radial TISE 16.1 Radial Equation: H-Atom Electron 16.2 Four Radial Equations (L = 0,1,2,3) 16.3 Radial Distance and Radial Nodes 16.4 One Electron Ions 16.5 Full 3D Solutions ψ(r,θ,ϕ) 16.6 Energies Note 17 Introduction to Multi-Electron Atoms, Molecules, and Spectroscopy Index