Cover

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Table of Contents

Introduction

About This Book

Conventions Used in This Book

Foolish Assumptions

How This Book Is Organized

Part I: Small World, Huh? Essential Quantum Physics

Part II: Bound and Undetermined: Handling Particles in Bound States

Part III: Turning to Angular Momentum and Spin

Part IV: Multiple Dimensions: Going 3D with Quantum Physics

Part V: Group Dynamics: Introducing Multiple Particles

Part VI: The Part of Tens

Icons Used in This Book

Where to Go from Here

Part I: Small World, Huh? Essential Quantum Physics

Chapter 1: Discoveries and Essential Quantum Physics

Being Discrete: The Trouble with Black-Body Radiation

First attempt: Wien’s Formula

Second attempt: Raleigh-Jeans Law

An intuitive (quantum) leap: Max Planck’s spectrum

The First Pieces: Seeing Light as Particles

Solving the photoelectric effect

Scattering light off electrons: The Compton effect

Proof positron? Dirac and pair production

A Dual Identity: Looking at Particles as Waves

You Can’t Know Everything (But You Can Figure the Odds)

The Heisenberg uncertainty principle

Rolling the dice: Quantum physics and probability

Chapter 2: Entering the Matrix: Welcome to State Vectors

Creating Your Own Vectors in Hilbert Space

Making Life Easier with Dirac Notation

Abbreviating state vectors as kets

Writing the Hermitian conjugate as a bra

Multiplying bras and kets: A probability of 1

Covering all your bases: Bras and kets as basis-less state vectors

Understanding some relationships using kets

Grooving with Operators

Hello, operator: How operators work

I expected that: Finding expectation values

Looking at linear operators

Going Hermitian with Hermitian Operators and Adjoints

Forward and Backward: Finding the Commutator

Commuting

Finding anti-Hermitian operators

Starting from Scratch and Ending Up with Heisenberg

Eigenvectors and Eigenvalues: They’re Naturally Eigentastic!

Understanding how they work

Finding eigenvectors and eigenvalues

Preparing for the Inversion: Simplifying with Unitary Operators

Comparing Matrix and Continuous Representations

Going continuous with calculus

Doing the wave

Part II: Bound and Undetermined: Handling Particles in Bound States

Chapter 3: Getting Stuck in Energy Wells

Looking into a Square Well

Trapping Particles in Potential Wells

Binding particles in potential wells

Escaping from potential wells

Trapping Particles in Infinite Square Potential Wells

Finding a wave-function equation

Determining the energy levels

Normalizing the wave function

Adding time dependence to wave functions

Shifting to symmetric square well potentials

Limited Potential: Taking a Look at Particles and Potential Steps

Assuming the particle has plenty of energy

Assuming the particle doesn’t have enough energy

Hitting the Wall: Particles and Potential Barriers

Getting through potential barriers when E > V0

Getting through potential barriers, even when E < V0

Particles Unbound: Solving the Schrödinger Equation for Free Particles

Getting a physical particle with a wave packet

Going through a Gaussian example

Chapter 4: Back and Forth with Harmonic Oscillators

Grappling with the Harmonic Oscillator Hamiltonians

Going classical with harmonic oscillation

Understanding total energy in quantum oscillation

Creation and Annihilation: Introducing the Harmonic Oscillator Operators

Mind your p’s and q’s: Getting the energy state equations

Finding the Eigenstates

Using a and a† directly

Finding the harmonic oscillator energy eigenstates

Putting in some numbers

Looking at Harmonic Oscillator Operators as Matrices

A Jolt of Java: Using Code to Solve the Schrödinger Equation Numerically

Making your approximations

Building the actual code

Running the code

Part III: Turning to Angular Momentum and Spin

Chapter 5: Working with Angular Momentum on the Quantum Level

Ringing the Operators: Round and Round with Angular Momentum

inding Commutators of Lx, Ly, and Lz

Creating the Angular Momentum Eigenstates

Finding the Angular Momentum Eigenvalues

Deriving eigenstate equations with βmax and βmin

Getting rotational energy of a diatomic molecule

Finding the Eigenvalues of the Raising and Lowering Operators

Interpreting Angular Momentum with Matrices

Rounding It Out: Switching to the Spherical Coordinate System

The eigenfunctions of Lz in spherical coordinates

The eigenfunctions of L2 in spherical coordinates

Chapter 6: Getting Dizzy with Spin

The Stern-Gerlach Experiment and the Case of the Missing Spot

Getting Down and Dirty with Spin and Eigenstates

Halves and Integers: Saying Hello to Fermions and Bosons

Spin Operators: Running Around with Angular Momentum

Working with Spin 1/2 and Pauli Matrices

Spin 1/2 matrices

Pauli matrices

Part IV: Multiple Dimensions: Going 3D with Quantum Physics

Chapter 7: Rectangular Coordinates: Solving Problems in Three Dimensions

The Schrödinger Equation: Now in 3D!

Solving Three-Dimensional Free Particle Problems

The x, y, and z equations

Finding the total energy equation

Adding time dependence and getting a physical solution

Getting Squared Away with 3D Rectangular Potentials

Determining the energy levels

Normalizing the wave function

Using a cubic potential

Springing into 3D Harmonic Oscillators

Chapter 8: Solving Problems in Three Dimensions: Spherical Coordinates

A New Angle: Choosing Spherical Coordinates Instead of Rectangular

Taking a Good Look at Central Potentials in 3D

Breaking down the Schrödinger equation

The angular part of ψ(r, θ, ϕ)

The radial part of ψ(r, θ, ϕ)

Handling Free Particles in 3D with Spherical Coordinates

The spherical Bessel and Neumann functions

The limits for small and large ρ

Handling the Spherical Square Well Potential

Inside the square well: 0 < r < a

Outside the square well: r > a

Getting the Goods on Isotropic Harmonic Oscillators

Chapter 9: Understanding Hydrogen Atoms

Coming to Terms: The Schrödinger Equation for the Hydrogen Atom

Simplifying and Splitting the Schrödinger Equation for Hydrogen

Solving for ψ(R)

Solving for ψ(r)

Solving the radial Schrödinger equation for small r

Solving the radial Schrödinger equation for large r

You got the power: Putting together the solution for the radial equation

Fixing f(r) to keep it finite

Finding the allowed energies of the hydrogen atom

Getting the form of the radial solution of the Schrödinger equation

Some hydrogen wave functions

Calculating the Energy Degeneracy of the Hydrogen Atom

Quantum states: Adding a little spin

On the lines: Getting the orbitals

Hunting the Elusive Electron

Chapter 10: Handling Many Identical Particles

Many-Particle Systems, Generally Speaking

Considering wave functions and Hamiltonians

A Nobel opportunity: Considering multi-electron atoms

A Super-Powerful Tool: Interchange Symmetry

Order matters: Swapping particles with the exchange operator

Classifying symmetric and antisymmetric wave functions

Floating Cars: Tackling Systems of Many Distinguishable Particles

Juggling Many Identical Particles

Losing identity

Symmetry and antisymmetry

Exchange degeneracy: The steady Hamiltonian

Name that composite: Grooving with the symmetrization postulate

Building Symmetric and Antisymmetric Wave Functions

Working with Identical Noninteracting Particles

Wave functions of two-particle systems

Wave functions of three-or-more-particle systems

It’s Not Come One, Come All: The Pauli Exclusion Principle

Figuring out the Periodic Table

Part V: Group Dynamics: Introducing Multiple Particles

Chapter 11: Giving Systems a Push: Perturbation Theory

Introducing Time-Independent Perturbation Theory

Working with Perturbations to Nondegenerate Hamiltonians

A little expansion: Perturbing the equations

Matching the coefficients of λ and simplifying

Finding the first-order corrections

Finding the second-order corrections

Perturbation Theory to the Test: Harmonic Oscillators in Electric Fields

Finding exact solutions

Applying perturbation theory

Working with Perturbations to Degenerate Hamiltonians

Testing Degenerate Perturbation Theory: Hydrogen in Electric Fields

Chapter 12: Wham-Blam! Scattering Theory

Introducing Particle Scattering and Cross Sections

Translating between the Center-of-Mass and Lab Frames

Framing the scattering discussion

Relating the scattering angles between frames

Translating cross sections between the frames

Trying a lab-frame example with particles of equal mass

Tracking the Scattering Amplitude Of Spinless Particles

The incident wave function

The scattered wave function

Relating the scattering amplitude and differential cross section

Finding the scattering amplitude

The Born Approximation: Rescuing the Wave Equation

Exploring the far limits of the wave function

Using the first Born approximation

Putting the Born approximation to work

Part VI: The Part of Tens

Chapter 13: Ten Quantum Physics Tutorials

An Introduction to Quantum Mechanics

Quantum Mechanics Tutorial

Grains of Mystique: Quantum Physics for the Layman

Quantum Physics Online Version 2.0

Todd K. Timberlake’s Tutorial

Physics 24/7’s Tutorial

Stan Zochowski’s PDF Tutorials

Quantum Atom Tutorial

College of St. Benedict’s Tutorial

A Web-Based Quantum Mechanics Course

Chapter 14: Ten Quantum Physics Triumphs

Wave-Particle Duality

The Photoelectric Effect

Postulating Spin

Differences between Newton’s Laws and Quantum Physics

Heisenberg Uncertainty Principle

Quantum Tunneling

Discrete Spectra of Atoms

Harmonic Oscillator

Square Wells

Schrödinger’s Cat

Glossary

Cheat Sheet

Quantum Physics For Dummies®, Revised Edition

by Steven Holzner

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Quantum Physics For Dummies®, Revised Edition

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About the Author

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

Author’s Acknowledgments

I’d particularly like to thank the people at Wiley: Tracy Boggier, Tim Gallan, and Danielle Voirol.

Dedication

To Nancy, of course!

Publisher’s Acknowledgments

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Introduction

Physics as a general discipline has no limits, from the very huge (galaxy-wide) to the very small (atoms and smaller). This book is about the very small side of things — that’s the specialty of quantum physics. When you quantize something, you can’t go any smaller; you’re dealing with discrete units.

Classical physics is terrific at explaining things like heating cups of coffee or accelerating down ramps or cars colliding, as well as a million other things, but it has problems when things get very small. Quantum physics usually deals with the micro world, such as what happens when you look at individual electrons zipping around. For example, electrons can exhibit both particle and wave-like properties, much to the consternation of experimenters — and it took quantum physics to figure out the full picture.

Quantum physics also introduced the uncertainty principle, which says you can’t know a particle’s exact position and momentum at the same time. And the field explains the way that the energy levels of the electrons bound in atoms work. Figuring out those ideas all took quantum physics, as physicists probed ever deeper for a way to model reality. Those topics are all coming up in this book.

About This Book

Because uncertainty and probability are so important in quantum physics, you can’t fully appreciate the subject without getting into calculus. This book presents the need-to-know concepts, but you don’t see much in the way of thought experiments that deal with cats or parallel universes. I focus on the math and how it describes the quantum world.

I’ve taught physics to many thousands of students at the university level, and from that experience, I know most of them share one common trait: Confusion as to what they did to deserve such torture.

Quantum Physics For Dummies largely maps to a college course, but this book is different from standard texts. Instead of writing it from the physicist’s or professor’s point of view, I’ve tried to write it from the reader’s point of view. In other words, I’ve designed this book to be crammed full of the good stuff — and only the good stuff. Not only that, but you can discover ways of looking at things that professors and teachers use to make figuring out problems simple.

Although I encourage you to read this book from start to finish, you can also leaf through this book as you like, reading the topics that you find interesting. Like other For Dummies books, this one lets you skip around as you like as much as possible. You don’t have to read the chapters in order if you don’t want to. This is your book, and quantum physics is your oyster.

Conventions Used in This Book

Some books have a dozen dizzying conventions that you need to know before you can even start. Not this one. Here’s all you need to know:

I put new terms in italics, like this, the first time they’re discussed; I follow them with a definition.

Vectors — those items that have both a magnitude and a direction — are given in bold, like this: B.

Web addresses appear in monofont.

Foolish Assumptions

I don’t assume that you have any knowledge of quantum physics when you start to read this book. However, I do make the following assumptions:

You’re taking a college course in quantum physics, or you’re interested in how math describes motion and energy on the atomic and subatomic scale.

You have some math prowess. In particular, you know some calculus. You don’t need to be a math pro, but you should know how to perform integration and deal with differential equations. Ideally, you also have some experience with Hilbert space.

You have some physics background as well. You’ve had a year’s worth of college-level physics (or understand all that’s in Physics For Dummies) before you tackle this one.

How This Book Is Organized

Quantum physics — the study of very small objects — is actually a very big topic. To handle it, quantum physicists break the world down into different parts. Here are the various parts that are coming up in this book.

Part I: Small World, Huh? Essential Quantum Physics

Part I is where you start your quantum physics journey, and you get a good overview of the topic here. I survey quantum physics and tell you what it’s good for and what kinds of problems it can solve. You also get a good foundation in the math that you need for the rest of the book, such as state vectors and quantum matrix manipulations. Knowing this stuff prepares you to handle the other parts.

Part II: Bound and Undetermined: Handling Particles in Bound States

Particles can be trapped inside potentials; for instance, electrons can be bound in an atom. Quantum physics excels at predicting the energy levels of particles bound in various potentials, and that’s what Part II covers. You see how to handle particles bound in square wells and in harmonic oscillators.

Part III: Turning to Angular Momentum and Spin

Quantum physics lets you work with the micro world in terms of the angu- lar momentum of particles, as well as the spin of electrons. Many famous experiments — such as the Stern-Gerlach experiment, in which beams of particles split in magnetic fields — are understandable only in terms of quantum physics, and you get all the details here.

Part IV: Multiple Dimensions: Going 3D with Quantum Physics

In the first three parts, all the quantum physics problems are one-dimensional to make life a little easier while you’re understanding how to solve those problems. In Part IV, you branch out to working with three-dimensional problems in both rectangular and spherical coordinate systems. Taking things from 1D to 3D gives you a better picture of what happens in the real world.

Part V: Group Dynamics: Introducing Multiple Particles

In this part, you work with multiple-particle systems, such as atoms and gases. You see how to handle many electrons in atoms, particles interacting with other particles, and particles that scatter off other particles.

Dealing with multiple particles is all another step in modeling reality — after all, systems with only a single particle don’t take you very far in the real world, which is built of mega, mega systems of particles. In Part V, you see how quantum physics can handle the situation.

Part VI: The Part of Tens

You see the Part of the Tens in all For Dummies books. This part is made up of fast-paced lists of ten items each. You get to see some of the ten best online tutorials on quantum physics and a discussion of quantum physics’ ten greatest triumphs.

Icons Used in This Book

You find a handful of icons in this book, and here’s what they mean:

This icon flags particularly good advice, especially when you’re solving problems.

This icon marks something to remember, such as a law of physics or a particularly juicy equation.

This icon means that what follows is technical, insider stuff. You don’t have to read it if you don’t want to, but if you want to become a quantum physics pro (and who doesn’t?), take a look.

This icon helps you avoid mathematical or conceptual slip-ups.

Where to Go from Here

All right, you’re all set and ready to go. You can jump in anywhere you like. For instance, if you’re sure electron spin is going to be a big topic of conversation at a party this weekend, check out Chapter 6. And if your upcoming vacation to Geneva, Switzerland, includes a side trip to your new favorite particle accelerator — the Large Hadron Collider — you can flip to Chapter 12 and read up on scattering theory. But if you want to get the full story from the beginning, jump into Chapter 1 first — that’s where the action starts.