Bra–ket notation

Abstract: The mathematical methods used in quantum mechanics are developed, with emphasis on linear algebra and complex variables. Dirac notation for vectors in Hilbert space is introduced. The representation of coordinates and momenta in quantum mechanics is analyzed and applied to the Heisenberg uncertainty principle.

Abstract: Quantum mechanics, which by its very nature is highly mathematical (and therefore extremely abstract), is one of the most difficult areas of physics to master. In these pages we hope to help pierce the veil of obscurity by demonstrating, with explicit examples, how to do quantum mechanics. This book is divided into three main parts. After a brief historical review, we cover the basics of quantum theory from the perspective of wave mechanics. This includes a discussion of the wavefunction, the probability interpretation, operators, and the Schrodinger equation. We then consider simple one-dimensional scattering and bound state problems. In the second part of the book we cover the mathematical foundations needed to do quantum mechanics from a more modern perspective. We review the necessary elements of matrix mechanics and linear algebra, such as finding eigenvalues and eigenvectors, computing the trace of a matrix, and finding out if a matrix is Hermitian or unitary. We then cover Dirac notation and Hilbert spaces. The postulates of quantum mechanics are then formalized and illustrated with examples. In the chapters that cover these topics, we attempt to “demystify” quantum mechanics by providing a large number of solved examples. The final part of the book provides an illustration of the mathematical foundations of quantum theory with three important cases that are typically taught in a first semester course: angular momentum and spin, the harmonic oscillator, and an introduction to the physics of the hydrogen atom. Other topics covered at some level with examples include the density operator, the Bloch vector, and two-state systems. Unfortunately, due to the large amount of space that explicitly solved examples from quantum mechanics require, it is not possible to include everything about the theory in a volume of this size. As a result we hope to prepare a second volume to cover advanced topics from non-relativistic quantum theory such as scattering, identical particles, addition of angular momentum, higher Z atoms, and the WKB approximation.

Abstract: This text presents a rigorous mathematical account of the principles of quantum mechanics, in particular as applied to chemistry and chemical physics. Applications are used as illustrations of the basic theory. The first two chapters serve as an introduction to quantum theory, although it is assumed that the reader has been exposed to elementary quantum mechanics as part of an undergraduate physical chemistry or atomic physics course. Following a discussion of wave motion leading to Schrodinger's wave mechanics, the postulates of quantum mechanics are presented along with essential mathematical concepts and techniques. The postulates are rigorously applied to the harmonic oscillator, angular momentum, the hydrogen atom, the variation method, perturbation theory, and nuclear motion. Modern theoretical concepts such as hermitian operators, Hilbert space, Dirac notation, and ladder operators are introduced and used throughout. This text is appropriate for beginning graduate students in chemistry, chemical physics, molecular physics and materials science.

Abstract: 1 Experimental Basis of Quantum Theory.- 1-1. Introductory Remarks.- 1-2. Classical Concepts of Linear Momentum, Angular Momentum, and Energy.- 1-3. Energy Levels and Photons.- 1-4. Electron Impact Experiments.- 1-5. Atomic Spectra.- 1-6. Quantization of Angular Momentum.- 1-7. Momentum of a Photon.- 1-8. Wave-Particle Duality.- Problems.- 2 Vector Spaces and Linear Transformations.- 2-1. Vector Spaces.- 2-2. Linear Independence, Bases, and Dimensionality.- 2-3. Inner Product Spaces.- 2-4. Orthonormality and Complete Sets.- 2-5. Hilbert Space.- 2-6. Function Space and Generalized Fourier Series.- 2-7. Isomorphism between Hilbert Space and Function Space.- 2-8. Examples of Complete Sets of Functions.- 2-9. Extension to Continuum Functions.- 2-10. Function Minimization with Constraints.- 2-11. Linear Operators.- 2-12. Algebra of Linear Operators.- 2-13. Special Kinds of Linear Operators.- 2-14. Eigenvalues and Eigenvectors.- Problems.- 3 Matrix Theory.- 3-1. Elements of Matrix Algebra.- 3-2. Determinants.- 3-3. Characterization of Square Matrices.- 3-4. Matrix Inversion.- 3-5. Matrices Having Special Properties.- 3-6. Matrix Representations of Linear Operators and Matrix Transformations.- 3-7. Changes of Basis and Similarity Transformations.- 3-8. Matrix Eigenvalue Problems.- 3-9. Infinite Matrices and Linear Transformations on Hilbert Space.- 3-10. Dirac Notation.- Problems.- 4 Postulates of Quantum Mechanics and Initial Considerations.- 4-1. Quantum Mechanical States and Observables.- 4-2. Time Evolution of a Quantum State.- 4-3. Quantum Theory of Measurement and Expectation Values.- 4-4. Compatible Observables and Commuting Operators.- 4-5. Constants of Motion and Transition Probabilities.- 4-6. Different Pictures of Quantum Phenomena.- 4-7. Hamiltonian Operator Construction: Initial Considerations.- Problems.- 5 One-Dimensional Model Problems.- 5-1. General Comments.- 5-2. Wavefunction Criteria and Boundary Conditions.- 5-3. The Nondegeneracy Theorem.- 5-4. Particle on a Ring.- 5-5. Particle Trapped in a Box.- 5-6. Parity of Eigenfunctions.- 5-7. Square Well Potential.- 5-8. Double Wells and Tunneling.- 5-9. The Harmonic Oscillator.- 5-10. Zero Point Energy and the Uncertainty Principle.- Problems.- 6 Angular Momentum.- 6-1. Introduction.- 6-2. General Angular Momentum Considerations.- 6-3. Orbital Angular Momentum.- 6-4. Spin Angular Momentum.- Problems.- 7 The Hydrogen Atom, Rigid Rotor, and the H2+ Molecule.- 7-1. Separation of Motion of Center of Mass.- 7-2. Solution of Equation for Relative Electron Motion of the Hydrogen Atom and Hydrogen-Like Atoms.- 7-3. Wavefunction Shapes.- 7-4. Rigid Rotor.- 7-5. The H2+ Molecule.- Problems.- 8 The Molecular Hamiltonian.- 8-1. General Principles and Discussion.- 8-2. Introduction of External Fields.- 8-3. Introduction of Relativistic Effects.- 8-4. The Born-Oppenheimer Approximation.- Problems.- 9 Approximation Methods for Stationary States.- 9-1. The Variation Principle.- 9-2. Accuracy Considerations.- 9-3. Example: The Hydrogen Atom.- 9-4. Example: Variational Treatment of the Helium Atom.- 9-5. The Linear Variation Method.- 9-6. Example: The Hydrogen Atom Revisited.- 9-7. Lower Bounds.- 9-8. Rayleigh-Schrodinger Perturbation Theory.- 9-9. Brillouin-Wigner Perturbation Theory.- Problems.- 10 General Considerations for Many Electron Systems.- 10-1. Early Computational Concepts and Procedures.- 10-2. Symmetry Considerations and Group Theory.- 10-3. Antisymmetry and the Pauli Exclusion Principle.- 10-4. Multielectron Systems and Slater Determinants.- 10-5. Expansion Theorem and Slater Determinant Expansions.- 10-6. Matrix Elements between Slater Determinants.- 10-7. Virial Theorem, Hypervirial Theorem, and Hellmann-Feynman Theorem.- 10-8. Scaling.- 10-9. Coupling of Angular Momenta.- 10-10. Orbital Transformations.- Problems.- 11 Computational Techniques for Many-Electron Systems Using Single Configuration Wavefunctions.- 11-1. Hartree-Fock Theory for Closed Shell Systems.- 11-2. Hall-Roothaan LCAO-MO-SCF Theory for Closed Shell Systems.- 11-3. Hartree-Fock Theory for Open Shell Systems.- Problems.- 12 Beyond Hartree-Fock Theory.- 12-1. Electron Correlation: General Comments.- 12-2. Configuration Interaction.- 12-3. Specialized CI Approaches.- 12-4. Many-Body Perturbation Theory and Coupled Cluster Theory.- Problems.- Appendix 1.- References.