Quantum mechanics is one of the most fascinating elements of the physics curriculum, but its conceptual nuances and mathematical complexity can be daunting for beginning students. This user-friendly text is designed for a one-semester course which bridges the gap between sophomore-level treatments and advanced undergraduate/lower-graduate courses. Qualitative explanations and descriptions of historical background are combined with detailed mathematical analyses to help students establish a firm foundation for further study. Classical problems such potential wells, barrier penetration, alpha decay, the harmonic oscillator, and the hydrogen atom are examined in detail, and formalisms and techniques such as operators, expectation values, commutators, perturbation theory, numerical solutions, and the variational theorem are also covered. Particular emphasis is placed on providing numerous worked examples and exercises.
Author(s): Bruce Cameron Reed
Edition: 2
Publisher: Springer
Year: 2022
Language: English
Commentary: Publisher PDF
Pages: 406
City: Cham
Tags: Quantum Mechanics; Schrodinger's Equation; Potential Wells; Harmonic Oscillator; Hydrogen Atom; Perturbation Theory; Angular Momentum; WKB Method; Numerical Integration; Alpha Decay; Operators and Expectation Values; Variational Method
Preface
Contents
About the Author
1 Foundations
1.1 Faraday, Thomson, and Electrons
1.2 Spectra, Radiation, and Planck
1.3 The Rutherford-Bohr Atom
1.4 de Broglie Matter Waves
1.5 The Radiative Collapse Problem (Optional)
References
2 Schrödinger's Equation
2.1 The Classical Wave Equation
2.2 The Time-Independent Schrödinger Equation
2.3 The Time-Dependent Schrödinger Equation
2.4 Interpretation of ψ: Probabilities and Boundary Conditions
References
3 Solutions of Schrödinger's Equation in One Dimension
3.1 Concept of a Potential Well
3.2 The Infinite Potential Well
3.3 The Finite Potential Well
3.3.1 A Matrix Approach to the Finite Potential Well
3.4 Finite Potential Well-Even Solutions
3.5 Number of Bound States in a Finite Potential Well
3.6 Sketching Wavefunctions
3.7 Potential Barriers and Scattering
3.8 Penetration of Arbitrarily-Shaped Barriers
3.9 Alpha-Decay as a Barrier Penetration Effect
3.10 Scattering by One-Dimensional Potential Wells
References
4 Operators, Expectation Values, and Various Quantum Theories
4.1 Properties of Operators
4.2 Expectation Values
4.3 The Uncertainty Principle
4.4 Commutators and Uncertainty Relations
4.5 Ehrenfest's Theorem
4.6 The Orthogonality Theorem
4.7 The Superposition Theorem
4.8 Constructing a Time-Dependent Wave Packet
4.9 The Virial Theorem
4.10 Momentum-Space Wavefunctions
References
5 The Harmonic Oscillator
5.1 A Lesson in Dimensional Analysis
5.2 The Asymptotic Solution
5.3 The Series Solution
5.4 Hermite Polynomials and Harmonic Oscillator Wavefunctions
5.5 Comparing the Classical and Quantum Harmonic Oscillators
5.6 Raising and Lowering Operators
Reference
6 Schrödinger's Equation in Three Dimensions and the Quantum Theory of Angular Momentum
6.1 Separation of Variables: Cartesian Coordinates
6.2 Spherical Coordinates
6.3 Angular Momentum Operators
6.4 Separation of Variables in Spherical Coordinates: Central Potentials
6.5 Angular Wavefunctions and Spherical Harmonics
6.5.1 Solution of the Φ Equation
6.5.2 Solution of the Θ Equation
6.5.3 Spherical Harmonics
References
7 Central Potentials
7.1 Introduction
7.2 The Infinite Spherical Well
7.3 The Finite Spherical Well
7.4 The Coulomb Potential
7.5 Hydrogen Atom Probability Distributions
7.5.1 The (1, 0, 0) State of Hydrogen
7.5.2 The (2, 0, 0) and Other States of Hydrogen
7.6 The Effective Potential
7.7 Some Philosophical Remarks
References
8 Further Developments with Angular Momentum and Multiparticle Systems
8.1 Angular Momentum Raising and Lowering Operators
8.2 The Stern-Gerlach Experiment: Evidence for Qunatized Angular …
8.3 Diatomic Molecules and Angular Momentum
8.4 Identical Particles, Indistinguishability, and the Pauli Exclusion Principle
References
9 Approximation Methods
9.1 The WKB Method
9.2 The Superposition Theorem Revisited
9.3 Perturbation Theory
9.4 The Variational Method
9.5 Improving the Variational Method
References
10 Numerical Solution of Schrödinger's Equation
10.1 Atomic Units
10.2 A Straightforward Numerical Integration Method
References
11 A Few Results from Time-Dependent Quantum Mechanics: Transition Rates and Probabilities
11.1 Transition Frequencies
11.2 Transition Rules
11.3 The Sudden Approximation
References
Appendix A Miscellaneous Derivations
A.1 Heisenberg's Uncertainty Principle
A.2 Normalization of Hermite Polynomials
A.3 Explicit Series Form for Associated Legendre Functions
A.4 Proof That upper Y Subscript script l comma negative m Baseline equals left parenthesis negative 1 right parenthesis Superscript m Baseline upper Y Subscript script l comma m Superscript asteriskYell,-m = (-1)mYell,m*
A.5 Radial Nodes in Hydrogen Wavefunctions
Appendix B Answers to Selected Odd-Numbered Problems
Appendix C Integrals and Trigonometric Identities
Appendix D Physical Constants
Index
