A Mastering Quantum Mechanics

Essentials, Theory, and Applications

by Zwiebach

| ISBN: 9780262366908 | Copyright 2022

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A complete overview of quantum mechanics, covering essential concepts and results, theoretical foundations, and applications.

This undergraduate textbook offers a comprehensive overview of quantum mechanics, beginning with essential concepts and results, proceeding through the theoretical foundations that provide the field's conceptual framework, and concluding with the tools and applications students will need for advanced studies and for research. Drawn from lectures created for MIT undergraduates and for the popular MITx online course, “Mastering Quantum Mechanics,” the text presents the material in a modern and approachable manner while still including the traditional topics necessary for a well-rounded understanding of the subject. As the book progresses, the treatment gradually increases in difficulty, matching students' increasingly sophisticated understanding of the material.

Part 1, on essentials, offers a sound introduction to the subject, touching on such topics as states and probability amplitudes, the Schrödinger equation, energy eigenstates of particles in potentials, the hydrogen atom, and spin one-half particles. Part 2, on theoretical foundations, covers mathematical tools, the pictures of quantum mechanics and the axioms of quantum mechanics, entanglement and tensor products, angular momentum, and identical particles. Part 3, on applications, introduces tools and techniques that help students master the theoretical concepts with a focus on approximation methods. About 240 exercises appear throughout the text, and nearly 300 end-of-chapter problems support the understanding of the subject. After mastering the material in this book, students will have the strong foundation in quantum mechanics that is required for graduate work in physics.

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Contents (pg. vii)
Preface (pg. xv)
I. Essentials (pg. 1)
1. Key Features of Quantum Mechanics (pg. 3)
1.1 Linearity of the Equations of Motion (pg. 4)
1.2 Complex Numbers Are Essential (pg. 7)
1.3 Loss of Determinism (pg. 10)
1.4 Quantum Superpositions (pg. 14)
1.5 Entanglement (pg. 22)
1.6 Making Atoms Possible (pg. 25)
Problems (pg. 27)
2. Light, Particles, and Waves (pg. 31)
2.1 Mach-Zehnder Interferometer (pg. 31)
2.2 Elitzur-Vaidman Bombs (pg. 37)
2.3 Toward Perfect Bomb Detection (pg. 39)
2.4 Photoelectric Effect (pg. 44)
2.5 Compton Scattering (pg. 47)
2.6 Matter Waves (pg. 51)
2.7 De Broglie Wavelength and Galilean Transformations (pg. 53)
2.8 Stationary Phase and Group Velocity (pg. 58)
Problems (pg. 60)
3. Schrödinger Equation (pg. 67)
3.1 The Wave Function for a Free Particle (pg. 67)
3.2 Equations for a Wave Function (pg. 69)
3.3 Schrödinger Equation for a Particle in a Potential (pg. 73)
3.4 Interpreting the Wave Function (pg. 78)
3.5 Normalization and Time Evolution (pg. 79)
3.6 The Wave Function as a Probability Amplitude (pg. 81)
3.7 The Probability Current (pg. 83)
3.8 Probability Current in Three Dimensions and Current Conservation (pg. 85)
Problems (pg. 87)
4. Wave Packets, Uncertainty, and Momentum Space (pg. 93)
4.1 Wave Packets and Uncertainty (pg. 93)
4.2 Wave Packet Shape Changes (pg. 98)
4.3 Time Evolution of a Free Wave Packet (pg. 100)
4.4 Uncovering Momentum Space (pg. 101)
Problems (pg. 106)
5. Expectation Values and Hermitian Operators (pg. 111)
5.1 Expectation Values of Operators (pg. 111)
5.2 Time Dependence of Expectation Values (pg. 114)
5.3 Hermitian Operators and Axioms of Quantum Mechanics (pg. 117)
5.4 Free Particle on a Circle—a First Look (pg. 123)
5.5 Uncertainty (pg. 125)
Problems (pg. 128)
6. Stationary States I: Special Potentials (pg. 133)
6.1 Stationary States (pg. 133)
6.2 Solving for Energy Eigenstates (pg. 137)
6.3 Free Particle on a Circle—a Second Look (pg. 140)
6.4 The Infinite Square Well (pg. 144)
6.5 The Finite Square Well (pg. 148)
6.6 The Delta Function Potential (pg. 156)
6.7 The Linear Potential (pg. 161)
Problems (pg. 165)
7. Stationary States II: General Features (pg. 171)
7.1 General Properties (pg. 171)
7.2 Bound States in Slowly Varying Potentials (pg. 175)
7.3 Sketching Wave Function Behavior (pg. 180)
7.4 The Node Theorem (pg. 185)
7.5 Shooting Method (pg. 187)
7.6 Removing Units from the Schrödinger Equation (pg. 189)
7.7 Virial Theorem (pg. 191)
7.8 Variational Principle (pg. 194)
7.9 Hellmann-Feynman Lemma (pg. 198)
Problems (pg. 201)
8. Stationary States III: Scattering (pg. 211)
8.1 The Step Potential (pg. 211)
8.2 Wave Packets in the Step Potential (pg. 218)
8.3 Resonant Transmission in a Square Well (pg. 223)
Problems (pg. 227)
9. Harmonic Oscillator (pg. 233)
9.1 Harmonic Oscillator (pg. 233)
9.2 Solving the Harmonic Oscillator Differential Equation (pg. 236)
9.3 Algebraic Solution for the Spectrum (pg. 242)
9.4 Excited States of the Oscillator (pg. 246)
Problems (pg. 251)
10. Angular Momentum and Central Potentials (pg. 255)
10.1 Angular Momentum in Quantum Mechanics (pg. 255)
10.2 Schrödinger Equation in Three Dimensions and Angular Momentum (pg. 259)
10.3 The Angular Momentum Operator (pg. 261)
10.4 Commuting Operators and Rotations (pg. 264)
10.5 Eigenstates of Angular Momentum (pg. 267)
10.6 The Radial Equation (pg. 272)
Problems (pg. 277)
11. Hydrogen Atom (pg. 281)
11.1 The Two-Body Problem (pg. 281)
11.2 Hydrogen Atom: Potential and Scales (pg. 285)
11.3 Hydrogen Atom: Bound State Spectrum (pg. 287)
11.4 Rydberg Atoms (pg. 295)
11.5 Degeneracies and Semiclassical Electron Orbits (pg. 298)
Problems (pg. 302)
12. The Simplest Quantum System: Spin One-Half (pg. 307)
12.1 A System with Two States (pg. 307)
12.2 The Stern-Gerlach Experiment (pg. 314)
12.3 Spin States (pg. 322)
12.4 Quantum Key Distribution (pg. 326)
Problems (pg. 330)
II. Theory (pg. 335)
13. Vector Spaces and Operators (pg. 337)
13.1 Vector Spaces (pg. 337)
13.2 Subspaces, Direct Sums, and Dimensionality (pg. 344)
13.3 Linear Operators (pg. 349)
13.4 Null Space, Range, and Inverses of Operators (pg. 354)
Exercise 13.5. (pg. 356)
13.5 Matrix Representation of Operators (pg. 360)
13.6 Eigenvalues and Eigenvectors (pg. 366)
13.7 Functions of Linear Operators and Key Identities (pg. 370)
Problems (pg. 377)
14. Inner Products, Adjoints, and Bra-kets (pg. 381)
14.1 Inner Products (pg. 381)
14.2 Orthonormal Bases (pg. 387)
14.3 Orthogonal Projectors (pg. 390)
14.4 Linear Functionals and Adjoint Operators (pg. 395)
14.5 Hermitian and Unitary Operators (pg. 398)
14.6 Remarks on Complex Vector Spaces (pg. 403)
14.7 Rotation Operators for Spin States (pg. 404)
14.8 From Inner Products to Bra-kets (pg. 407)
14.9 Operators, Projectors, and Adjoints (pg. 411)
14.10 Nondenumerable Basis States (pg. 417)
Problems (pg. 421)
15. Uncertainty Principle and Compatible Operators (pg. 427)
15.1 Uncertainty Defined (pg. 427)
15.2 The Uncertainty Principle (pg. 431)
15.3 Energy-Time Uncertainty (pg. 435)
15.4 Lower Bounds for Ground State Energies (pg. 438)
15.5 Diagonalization of Operators (pg. 440)
15.6 The Spectral Theorem (pg. 442)
15.7 Simultaneous Diagonalization of Hermitian Operators (pg. 447)
15.8 Complete Set of Commuting Observables (pg. 451)
Problems (pg. 454)
16. Pictures of Quantum Mechanics (pg. 459)
16.1 Schrödinger Picture and Unitary Time Evolution (pg. 459)
16.2 Deriving the Schrödinger Equation (pg. 461)
16.3 Calculating the Time Evolution Operator (pg. 464)
16.4 Heisenberg Picture (pg. 467)
16.5 Heisenberg Equations of Motion (pg. 469)
16.6 Axioms of Quantum Mechanics (pg. 474)
Problems (pg. 478)
17. Dynamics of Quantum Systems (pg. 481)
17.1 Basics of Coherent States (pg. 481)
17.2 Heisenberg Picture for Coherent States (pg. 484)
17.3 General Coherent States (pg. 489)
17.4 Photon States (pg. 492)
17.5 Spin Precession in a Magnetic Field (pg. 494)
17.6 Nuclear Magnetic Resonance (pg. 498)
17.7 Two-State System Viewed as a Spin System (pg. 503)
17.8 The Factorization Method (pg. 504)
Problems (pg. 511)
18. Multiparticle States and Tensor Products (pg. 519)
18.1 Introduction to the Tensor Product (pg. 519)
18.2 Operators on the Tensor Product Space (pg. 522)
18.3 Inner Products for Tensor Spaces (pg. 526)
18.4 Matrix Representations and Traces (pg. 527)
18.5 Entangled States (pg. 530)
18.6 Bell Basis States (pg. 532)
18.7 Quantum Teleportation (pg. 536)
18.8 EPR and Bell Inequalities (pg. 538)
18.9 No-Cloning Property (pg. 545)
Problems (pg. 547)
19. Angular Momentum and Central Potentials: Part II (pg. 555)
19.1 Angular Momentum and Quantum Vector Identities (pg. 555)
19.2 Properties of Angular Momentum (pg. 559)
19.3 Multiplets of Angular Momentum (pg. 562)
19.4 Central Potentials and Radial Equation (pg. 573)
19.5 Free Particle and Spherical Waves (pg. 578)
19.6 Rayleigh’s Formula (pg. 581)
19.7 The Three-Dimensional Isotropic Oscillator (pg. 584)
19.8 The Runge-Lenz Vector (pg. 589)
Problems (pg. 593)
20. Addition of Angular Momentum (pg. 599)
20.1 Adding Apples to Oranges? (pg. 599)
20.2 Adding Two Spin One-Half Angular Momenta (pg. 601)
20.3 A Primer in Perturbation Theory (pg. 606)
20.4 Hyperfine Splitting (pg. 607)
20.5 Computation of 1⊗ 1/2 (pg. 612)
20.6 Spin-Orbit Coupling (pg. 615)
20.7 General Aspects of Addition of Angular Momentum (pg. 619)
20.8 Hydrogen Atom and Hidden Symmetry (pg. 625)
Problems (pg. 631)
21. Identical Particles (pg. 639)
21.1 Identical Particles and Exchange Degeneracy (pg. 640)
21.2 Permutation Operators (pg. 643)
21.3 Complete Symmetrizer and Antisymmetrizer (pg. 648)
21.4 The Symmetrization Postulate (pg. 652)
21.5 Building Symmetrized States and Probabilities (pg. 659)
21.6 Particles with Two Sets of Degrees of Freedom (pg. 665)
21.7 States of Two-Electron Systems (pg. 667)
21.8 Occupation Numbers (pg. 671)
Problems (pg. 677)
III. Applications (pg. 685)
22. Density Matrix and Decoherence (pg. 687)
22.1 Ensembles and Mixed States (pg. 687)
22.2 The Density Matrix (pg. 692)
22.3 Dynamics of Density Matrices (pg. 700)
22.4 Subsystems and Schmidt Decomposition (pg. 701)
22.5 Open Systems and Decoherence (pg. 709)
22.6 The Lindblad Equation (pg. 717)
22.7 A Theory of Measurement? (pg. 721)
Problems (pg. 730)
23. Quantum Computation (pg. 737)
23.1 Qubits and Gates (pg. 739)
23.2 Deutsch’s Computation (pg. 747)
23.3 Grover’s Algorithm (pg. 749)
Problems (pg. 756)
24. Charged Particles in Electromagnetic Fields (pg. 761)
24.1 Electromagnetic Potentials (pg. 761)
24.2 Schrödinger Equation with Electromagnetic Potentials (pg. 763)
24.3 Heisenberg Picture (pg. 767)
24.4 Magnetic Fields on a Torus (pg. 770)
24.5 Particles in Uniform Magnetic Field: Landau Levels (pg. 774)
24.6 The Pauli Equation (pg. 779)
24.7 The Dirac Equation (pg. 781)
Problems (pg. 783)
25. Time-Independent Perturbation Theory (pg. 793)
25.1 Time-Independent Perturbations (pg. 793)
25.2 Nondegenerate Perturbation Theory (pg. 796)
25.3 The Anharmonic Oscillator (pg. 804)
25.4 Degenerate Perturbation Theory (pg. 807)
25.5 Degeneracy Lifted at Second Order (pg. 814)
25.6 Review of Hydrogen Atom (pg. 817)
25.7 Fine Structure of Hydrogen (pg. 821)
25.8 Zeeman Effect (pg. 833)
Problems (pg. 838)
26. WKB and Semiclassical Approximation (pg. 849)
26.1 The Classical Limit (pg. 849)
26.2 WKB Approximation Scheme (pg. 852)
26.3 Using Connection Formulae (pg. 860)
26.4 Airy Functions and Their Expansions (pg. 863)
26.5 Connection Formulae Derived (pg. 868)
26.6 Tunneling through a Barrier (pg. 872)
26.7 Double-Well Potentials (pg. 876)
Problems (pg. 884)
27. Time-Dependent Perturbation Theory (pg. 891)
27.1 Time-Dependent Hamiltonians (pg. 891)
27.2 The Interaction Picture (pg. 893)
27.3 Perturbative Solution in the Interaction Picture (pg. 899)
27.4 Constant Perturbations (pg. 904)
27.5 Harmonic Perturbations (pg. 907)
27.6 Fermi’s Golden Rule (pg. 911)
27.7 Helium Atom and Autoionization (pg. 919)
27.8 Modeling the Decay of a Discrete State to the Continuum (pg. 921)
27.9 Ionization of Hydrogen (pg. 928)
27.10 Atoms and Light (pg. 935)
27.11 Atom-Light Dipole Interaction (pg. 939)
27.12 Selection Rules (pg. 943)
Problems (pg. 945)
28. Adiabatic Approximation (pg. 953)
28.1 Adiabatic Changes and Adiabatic Invariants (pg. 953)
28.2 From Classical to Quantum Adiabatic Invariants (pg. 955)
28.3 Instantaneous Energy Eigenstates (pg. 960)
28.4 Quantum Adiabatic Theorem (pg. 964)
28.5 Landau-Zener Transitions (pg. 969)
28.6 Berry’s Phase (pg. 975)
28.7 Born-Oppenheimer Approximation (pg. 981)
28.8 The Hydrogen Molecule Ion (pg. 987)
Problems (pg. 993)
29. Scattering in One Dimension (pg. 999)
29.1 Scattering on the Half Line (pg. 999)
29.2 Time Delay (pg. 1003)
29.3 Levinson’s Theorem (pg. 1009)
29.4 Resonances (pg. 1011)
29.5 Modeling Resonances (pg. 1015)
Problems (pg. 1020)
30. Scattering in Three Dimensions (pg. 1025)
30.1 Energy Eigenstates for Scattering (pg. 1026)
30.2 Cross Sections from Scattering Amplitudes (pg. 1029)
30.3 Scattering Amplitude in Terms of Phase Shifts (pg. 1031)
30.4 Computation of Phase Shifts (pg. 1036)
30.5 Integral Equation for Scattering (pg. 1042)
30.6 The Born Approximation (pg. 1046)
Problems (pg. 1050)
References (pg. 1057)
Index (pg. 1065)
Useful Formulae (pg. 1073)
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