Mastering Quantum Mechanics: Essentials, Theory, and Applications

1556 Pages·2022·68.552 MB·English

by  Barton Zwiebach

Mastering Quantum Mechanics
Essentials, Theory, and Applications
Barton Zwiebach
The MIT Press
Cambridge, Massachusetts
London, England
© 2022 Barton Zwiebach
All rights reserved. No part of this book may be reproduced in any form by any electronic or
mechanical means (including photocopying, recording, or information storage and retrieval)
without permission in writing from the publisher.
The MIT Press would like to thank the anonymous peer reviewers who provided comments on
drafts of this book. The generous work of academic experts is essential for establishing the
authority and quality of our publications. We acknowledge with gratitude the contributions of
these otherwise uncredited readers.
Library of Congress Cataloging-in-Publication Data.
Names: Zwiebach, Barton, 1954– author.
Title: Mastering quantum mechanics: essentials, theory, and applications / Barton Zwiebach,
Massachusetts Institute of Technology, Cambridge, MA.
Description: Cambridge, MA: The MIT Press, [2022] | Includes bibliographical references and
index.
Identifiers: LCCN 2021016609 | ISBN 9780262046138 (hardcover)
Subjects: LCSH: Quantum theory.
Classification: LCC QC174.12.Z85 2022 | DDC 530.12—dc23
LC record available at https://lccn.loc.gov/2021016609
d_r0
To my wife, Gaby, with love
Contents
Preface
I Essentials
1 Key Features of Quantum Mechanics
1.1 Linearity of the Equations of Motion
1.2 Complex Numbers Are Essential
1.3 Loss of Determinism
1.4 Quantum Superpositions
1.5 Entanglement
1.6 Making Atoms Possible
Problems
2 Light, Particles, and Waves
2.1 Mach-Zehnder Interferometer
2.2 Elitzur-Vaidman Bombs
2.3 Toward Perfect Bomb Detection
2.4 Photoelectric Effect
2.5 Compton Scattering
2.6 Matter Waves
2.7 De Broglie Wavelength and Galilean Transformations
2.8 Stationary Phase and Group Velocity
Problems
3 Schrödinger Equation
3.1 The Wave Function for a Free Particle
3.2 Equations for a Wave Function
3.3 Schrödinger Equation for a Particle in a Potential
3.4 Interpreting the Wave Function
3.5 Normalization and Time Evolution
3.6 The Wave Function as a Probability Amplitude
3.7 The Probability Current
3.8 Probability Current in Three Dimensions and Current Conservation
Problems
4 Wave Packets, Uncertainty, and Momentum Space
4.1 Wave Packets and Uncertainty
4.2 Wave Packet Shape Changes
4.3 Time Evolution of a Free Wave Packet
4.4 Uncovering Momentum Space
Problems
5 Expectation Values and Hermitian Operators
5.1 Expectation Values of Operators
5.2 Time Dependence of Expectation Values
5.3 Hermitian Operators and Axioms of Quantum Mechanics
5.4 Free Particle on a Circle—a First Look
5.5 Uncertainty
Problems
6 Stationary States I: Special Potentials
6.1 Stationary States
6.2 Solving for Energy Eigenstates
6.3 Free Particle on a Circle—a Second Look
6.4 The Infinite Square Well
6.5 The Finite Square Well
6.6 The Delta Function Potential
6.7 The Linear Potential
Problems
7 Stationary States II: General Features
7.1 General Properties
7.2 Bound States in Slowly Varying Potentials
7.3 Sketching Wave Function Behavior
7.4 The Node Theorem
7.5 Shooting Method
7.6 Removing Units from the Schrödinger Equation
7.7 Virial Theorem
7.8 Variational Principle
7.9 Hellmann-Feynman Lemma
Problems
8 Stationary States III: Scattering
8.1 The Step Potential
8.2 Wave Packets in the Step Potential
8.3 Resonant Transmission in a Square Well
Problems
9 Harmonic Oscillator
9.1 Harmonic Oscillator
9.2 Solving the Harmonic Oscillator Differential Equation
9.3 Algebraic Solution for the Spectrum
9.4 Excited States of the Oscillator
Problems
10 Angular Momentum and Central Potentials
10.1 Angular Momentum in Quantum Mechanics
10.2 Schrödinger Equation in Three Dimensions and Angular Momentum
10.3 The Angular Momentum Operator
10.4 Commuting Operators and Rotations
10.5 Eigenstates of Angular Momentum
10.6 The Radial Equation
Problems
11 Hydrogen Atom
11.1 The Two-Body Problem
11.2 Hydrogen Atom: Potential and Scales
11.3 Hydrogen Atom: Bound State Spectrum
11.4 Rydberg Atoms
11.5 Degeneracies and Semiclassical Electron Orbits
Problems
12 The Simplest Quantum System: Spin One-Half
12.1 A System with Two States
12.2 The Stern-Gerlach Experiment
12.3 Spin States
12.4 Quantum Key Distribution
Problems
II Theory
13 Vector Spaces and Operators
13.1 Vector Spaces
13.2 Subspaces, Direct Sums, and Dimensionality
13.3 Linear Operators
13.4 Null Space, Range, and Inverses of Operators
13.5 Matrix Representation of Operators
13.6 Eigenvalues and Eigenvectors
13.7 Functions of Linear Operators and Key Identities
Problems
14 Inner Products, Adjoints, and Bra-kets
14.1 Inner Products
14.2 Orthonormal Bases
14.3 Orthogonal Projectors
14.4 Linear Functionals and Adjoint Operators
14.5 Hermitian and Unitary Operators
14.6 Remarks on Complex Vector Spaces
14.7 Rotation Operators for Spin States
14.8 From Inner Products to Bra-kets
14.9 Operators, Projectors, and Adjoints
14.10 Nondenumerable Basis States
Problems
15 Uncertainty Principle and Compatible Operators
15.1 Uncertainty Defined
15.2 The Uncertainty Principle
15.3 Energy-Time Uncertainty
15.4 Lower Bounds for Ground State Energies
15.5 Diagonalization of Operators
15.6 The Spectral Theorem
15.7 Simultaneous Diagonalization of Hermitian Operators
15.8 Complete Set of Commuting Observables
Problems
16 Pictures of Quantum Mechanics
16.1 Schrödinger Picture and Unitary Time Evolution
16.2 Deriving the Schrödinger Equation
16.3 Calculating the Time Evolution Operator
16.4 Heisenberg Picture
16.5 Heisenberg Equations of Motion
16.6 Axioms of Quantum Mechanics
Problems
17 Dynamics of Quantum Systems
17.1 Basics of Coherent States
17.2 Heisenberg Picture for Coherent States
17.3 General Coherent States
17.4 Photon States
17.5 Spin Precession in a Magnetic Field
17.6 Nuclear Magnetic Resonance
17.7 Two-State System Viewed as a Spin System
17.8 The Factorization Method
Problems
18 Multiparticle States and Tensor Products
18.1 Introduction to the Tensor Product
18.2 Operators on the Tensor Product Space
18.3 Inner Products for Tensor Spaces
18.4 Matrix Representations and Traces
18.5 Entangled States
18.6 Bell Basis States
18.7 Quantum Teleportation
18.8 EPR and Bell Inequalities
18.9 No-Cloning Property
Problems
19 Angular Momentum and Central Potentials: Part II
19.1 Angular Momentum and Quantum Vector Identities
19.2 Properties of Angular Momentum
19.3 Multiplets of Angular Momentum
19.4 Central Potentials and Radial Equation
19.5 Free Particle and Spherical Waves
19.6 Rayleigh’s Formula
19.7 The Three-Dimensional Isotropic Oscillator
19.8 The Runge-Lenz Vector
Problems
20 Addition of Angular Momentum
20.1 Adding Apples to Oranges?
20.2 Adding Two Spin One-Half Angular Momenta
20.3 A Primer in Perturbation Theory
20.4 Hyperfine Splitting
20.5 Computation of
20.6 Spin-Orbit Coupling
20.7 General Aspects of Addition of Angular Momentum
20.8 Hydrogen Atom and Hidden Symmetry
Problems
21 Identical Particles
21.1 Identical Particles and Exchange Degeneracy
21.2 Permutation Operators
21.3 Complete Symmetrizer and Antisymmetrizer
21.4 The Symmetrization Postulate
21.5 Building Symmetrized States and Probabilities
21.6 Particles with Two Sets of Degrees of Freedom
21.7 States of Two-Electron Systems
21.8 Occupation Numbers
Problems
III Applications
22 Density Matrix and Decoherence
22.1 Ensembles and Mixed States
22.2 The Density Matrix
22.3 Dynamics of Density Matrices
22.4 Subsystems and Schmidt Decomposition
22.5 Open Systems and Decoherence
22.6 The Lindblad Equation
22.7 A Theory of Measurement?
Problems
23 Quantum Computation
23.1 Qubits and Gates
23.2 Deutsch’s Computation
23.3 Grover’s Algorithm
Problems
24 Charged Particles in Electromagnetic Fields
24.1 Electromagnetic Potentials
24.2 Schrödinger Equation with Electromagnetic Potentials
24.3 Heisenberg Picture
24.4 Magnetic Fields on a Torus
24.5 Particles in Uniform Magnetic Field: Landau Levels
24.6 The Pauli Equation
24.7 The Dirac Equation
Problems
25 Time-Independent Perturbation Theory
25.1 Time-Independent Perturbations
25.2 Nondegenerate Perturbation Theory
25.3 The Anharmonic Oscillator
25.4 Degenerate Perturbation Theory
25.5 Degeneracy Lifted at Second Order