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.
Author(s): Steven M. Pascal
Publisher: CRC Press
Year: 2023
Language: English
Pages: 118
City: Boca Raton
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 (Tunneling)
7.6 Finite Barrier of Finite Width: Case 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
