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Field Guide to Quantum Mechanics
Author(s): Brian P. Anderson
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Book Description

This Field Guide serves as a reference guide to the primary results, explanations, and interpretations of quantum mechanics that foregoes the introductions, derivations, and conceptual discussions found in most courses and textbooks on the subject. Written primarily for physicists and engineers in quantum-mechanics-related fields, this book may also aid students, particularly as the concepts and methods of quantum mechanics find increasing applicability in more technologies.

Book Details

Date Published: 12 September 2019
Pages: 146
ISBN: 9781510622821
Volume: FG44

Table of Contents
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Glossary of Symbols

Glossary of Operators

Quantum Mechanics Formalism
Quantum States, Kets, and State Space
Elements of Dirac Notation
Vector Spaces and Scalar Products
Linear Operators and Commutators
Hermitian Conjugation
Eigenvalue Equations
Closure Relations
Functions of Operators

Postulates of Quantum Mechanics
Eigenvalue and Collapse Postulates
Probability Postulate
Examples of Observables: Ĥ, &Xcirc;, and &Pcirc;x

Bases and Representations
Calculating Quantities in Quantum Mechanics
Calculating Quantities Using a Discrete Basis
Discrete Representations
Transformation of Discrete Representation
Continuous Representations: &Xcirc;, &Pcirc;x
Continuous Representations: Wavefunctions
Calculating Quantities Using a Continuous Basis
Operators in the &Xcirc; Representation
&Xcirc; and &Pcirc;x Representations: Eigenkets and Operators
&Xcirc; and &Pcirc;x Representations: Fourier Transforms
Tensor Products: Merging State Spaces

Operator Definitions and Operator Manipulation
Expectation Values
Commutation Relations
Non-Commuting Operators, Uncertainty Relations
Complete Sets of Commuting Observables
CSCOs for Specific Problems

Properties of Wavefunctions
Wave-Like Properties of Matter
Graphical Interpretation of the Schrodinger Equation
Graphical Interpretation: Example
Superpositions, Relative and Global Phases
Probability Currents

Time Dependence, Transformations, "Pictures"
Unitary Transformations
Common Unitary Operators
Conservative Systems
Stationary States
Time-Dependent Reference Frames
Schrodinger, Heisenberg, and Interaction Pictures
Schrodinger Picture: Expectation Value Dynamics
Heisenberg Picture: Operators and Dynamics
Interaction Picture

Exactly Solvable Problems
Exactly Solvable Problems in One Dimension
Free Particle and Delta Function Potential Well
Infinite Square Well
Potential Barrier: Transmission and Reflection
Potential Barrier: Energy above Barrier
Potential Barrier: Tunneling
Potential Barrier: Examples
1D Quantum Harmonic Oscillator
Harmonic Oscillator: Energy Eigenfunctions
Harmonic Oscillator: Ladder Operators
Harmonic Oscillator: Properties and Dynamics
Harmonic Oscillator: Fourier Transforms
Coherent States (Quasi-Classical States)
Phase-Space Diagrams
Phase-Space Diagrams: Examples
3D Quantum Harmonic Oscillator

Angular Momentum
Angular Momentum: Definitions
Angular Momentum: Eigenvalues and Eigenstates
Orbital Angular Momentum: Operators
Orbital Angular Momentum: Position Representation
Spin Angular Momentum
Spin Angular Momentum: s = 1/2
Pauli Spin Operators
Angular Momentum j = 1
Magnetic Dipole Moments and Magnetic Fields
Gyromagnetic Ratios and g-Factors
Magnetic Moment Dynamics: Uniform Fields
Magnetic Moment Dynamics: Gradient Fields

Two-Level Systems and Spin 1/2
Two-Level Systems
Rabi Oscillations
The Bloch Vector
The Bloch Sphere
Spin 1/2 in a Uniform Magnetic Field
Spin 1/2 in a Uniform Magnetic Field: Dynamics
Bloch Vector Dynamics: Examples

Addition of Angular Momentum
Addition of Two Angular Momenta
Total Angular Momentum Basis
Addition of Angular Momentum: Example
Addition of Angular Momentum: Comments
Clebsch-Gordan Coefficients
Clebsch-Gordan Coefficients: Usage
Clebsch-Gordan Coefficients: Examples

Approximation Methods
Ritz Variational Method
Stationary Perturbation Theory
Degenerate Stationary Perturbation Theory
Time-Dependent Perturbation Theory
TDPT: First-Order Solution
Fourier Transform Pairs for Pulse Perturbations
TDPT: Harmonic Perturbations

Hydrogen and Atomic Structure
Central Potential Problems
"Spinless" Hydrogen: Energy Eigenvalues
Spinless Hydrogen: Energy Eigenfunctions
Hydrogen Radial Wavefunctions
Spherical Harmonics
Atomic Angular Momentum Quantum Numbers
Fine Structure of Hydrogen: Perturbation Terms
Fine Structure of Hydrogen: Solutions
Hyperfine Structure of Hydrogen
Zeeman Effect in Hydrogen: n = 1
Zeeman Effect in Hydrogen: n = 1 Solutions
Spectroscopic Notation and Term Symbols

Identical Particles
Identical Particles: Two Particles
Identical Particles: Three or More Particles
Identical Particles: Occupation Number Basis
Identical Particles: Occupation Number Basis States

Appendix: Mathematics Reference, Tables, and Constants
Miscellaneous Symbols and Notation
Linear Algebra Basics
Eigenvalue Equations in Linear Algebra
Spherical Coordinates
Operators in Spherical Coordinates
Properties of 1D Gaussian Wavefunctions
Clebsch-Gordan Coefficient Tables: J1 × 1/2
Clebsch-Gordan Coefficient Tables: J1 × 1
Clebsch-Gordan Coefficient Tables: J1 × 3/2
Clebsch-Gordan Coefficient Tables: 2 × 2
Integrals of Exponential Forms
Identities and Series Expansions
Dimensional Units (SI)
Common Physical Constants (to Four Significant Digits)

This Field Guide is a condensed reference to the concepts, definitions, formalism, equations, and problems of quantum mechanics. Many topics covered in quantum mechanics courses are included, while numerous details and derivations are necessarily omitted. This Field Guide is envisioned to appeal to undergraduate and graduate students engaged in quantum mechanics research or courses; to professors, as an aid in teaching and research; and to professional physicists and engineers pursuing cutting-edge applications of quantum mechanics. The mathematical formalism used here involves Dirac notation, with which the reader should be (or become) familiar to make the most of this Field Guide. Nevertheless, readers who are not yet familiar with this formalism should be able to utilize various aspects of this Field Guide, especially with extra attention directed to the basic concepts addressed in the first few sections.

I owe sincere thanks to mentors, professors, colleagues, collaborators, and friends too numerous to single out by name who have taught, motivated, and encouraged me throughout more than three decades of studying quantum physics. Since joining the University of Arizona faculty, the unwavering support and partnership of local and international colleagues and collaborators has been indispensable in learning and appreciating many of the numerous facets of this fascinating subject.

I am especially grateful to two physicists in particular who set in motion the trajectory of my eventual career while I was still in high school: the late Jeff Chalk, who first introduced me to Schrodinger's equation and quantum mechanics; and Al Rosenberger, my first laboratory mentor, who launched my interest in lasers, optics, and experimental physics.

This Field Guide is dedicated to Jeff and Al, and to the students who have worked in my labs, sat through my courses, and made my career as a mentor and educator profoundly fulfilling.

Brian P. Anderson
June 2019

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