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Butterfly in the quantum world : the story of the most fascinating quantum fractal /
Butterfly in the Quantum World is the first book ever to tell the story of the "Hofstadter butterfly", a beautiful and fascinating graph lying at the heart of the quantum theory of matter. The butterfly came out of a simple-sounding question: What happens if you immerse a crystal in a magn...
| Clasificación: | Libro Electrónico |
|---|---|
| Autores principales: | , |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
San Rafael [California] (40 Oak Drive, San Rafael, CA, 94903, USA) :
Morgan & Claypool Publishers,
[2016]
|
| Colección: | IOP (Series). Release 3.
IOP concise physics. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Summary
- Preface
- Prologue
- Prelude
- part I. The butterfly fractal
- 0. Kiss precise
- 0.1. Apollonian gaskets and integer wonderlands
- Appendix. An Apollonian sand painting--the world's largest artwork
- 1. The fractal family
- 1.1. The Mandelbrot set
- 1.2. The Feigenbaum set
- 1.3. Classic fractals
- 1.4. The Hofstadter set
- Appendix. Harper's equation as an iterative mapping
- 2. Geometry, number theory, and the butterfly : friendly numbers and kissing circles
- 2.1. Ford circles, the Farey tree, and the butterfly
- 2.2. A butterfly at every scale--butterfly recursions
- 2.3. Scaling and universality
- 2.4. The butterfly and a hidden trefoil symmetry
- 2.5. Closing words : physics and number theory
- Appendix A. Hofstadter recursions and butterfly generations
- Appendix B. Some theorems of number theory
- Appendix C. Continued-fraction expansions
- Appendix D. Nearest-integer continued fraction expansion
- Appendix E. Farey paths and some comments on universality
- 3 The Apollonian-butterfly connection (ABC)
- 3.1 Integral Apollonian gaskets (IAG) and the butterfly
- 3.2 The kaleidoscopic effect and trefoil symmetry
- 3.3 Beyond Ford Apollonian gaskets and fountain butterflies
- Appendix. Quadratic Diophantine equations and IAGs
- 4. Quasiperiodic patterns and the butterfly
- 4.1. A tale of three irrationals
- 4.2. Self-similar butterfly hierarchies
- 4.3. The diamond, golden, and silver hierarchies, and Hofstadter recursions
- 4.4. Symmetries and quasiperiodicities
- Appendix. Quasicrystals
- part II. Butterfly in the quantum world
- 5. The quantum world
- 5.1. Wave or particle--what is it?
- 5.2. Quantization
- 5.3. What is waving?--The Schrödinger picture
- 5.4. Quintessentially quantum
- 5.5. Quantum effects in the macroscopic world
- 6. A quantum-mechanical marriage and its unruly child
- 6.1. Two physical situations joined in a quantum-mechanical marriage
- 6.2. The marvelous pure number [phi]
- 6.3. Harper's equation, describing Bloch electrons in a magnetic field
- 6.4. Harper's equation as a recursion relation
- 6.5. On the key role of inexplicable artistic intuitions in physics
- 6.6. Discovering the strange eigenvalue spectrum of Harper's equation
- 6.7. Continued fractions and the looming nightmare of discontinuity
- 6.8. Polynomials that dance on several levels at once
- 6.9. A short digression on INT and on perception of visual patterns
- 6.10. The spectrum belonging to irrational values of [phi] and the "ten-martini problem"
- 6.11. In which continuity (of a sort) is finally established
- 6.12. Infinitely recursively scalloped wave functions : cherries on the doctoral sundae
- 6.13. Closing words
- Appendix. Supplementary material on Harper's equation
- part III. Topology and the butterfly
- 7. A different kind of quantization : the quantum Hall effect
- 7.1. What is the Hall effect? Classical and quantum answers
- 7.2. A charged particle in a magnetic field : cyclotron orbits and their quantization
- 7.3. Landau levels in the Hofstadter butterfly
- 7.4. Topological insulators
- Appendix A. Excerpts from the 1985 Nobel Prize press release
- Appendix B. Quantum mechanics of electrons in a magnetic field
- Appendix C. Quantization of the Hall conductivity
- 8. Topology and topological invariants : preamble to the topological aspects of the quantum Hall effect
- 8.1. A puzzle : the precision and the quantization of Hall conductivity
- 8.2. Topological invariants
- 8.3. Anholonomy : parallel transport and the Foucault pendulum
- 8.4. Geometrization of the Foucault pendulum
- 8.5. Berry magnetism--effective vector potential and monopoles
- 8.6. The ESAB effect as an example of anholonomy
- Appendix. Classical parallel transport and magnetic monopoles
- 9. The Berry phase and the quantum Hall effect
- 9.1. The Berry phase
- 9.2. Examples of Berry phase
- 9.3. Chern numbers in two-dimensional electron gases
- 9.4. Conclusion : the quantization of Hall conductivity
- 9.5. Closing words : topology and physical phenomena
- Appendix A. Berry magnetism and the Berry phase
- Appendix B. The Berry phase and 2 x 2 matrices
- Appendix C. What causes Berry curvature? Dirac strings, vortices, and magnetic monopoles
- Appendix D. The two-band lattice model for the quantum Hall effect
- 10. The kiss precise and precise quantization
- 10.1. Diophantus gives us two numbers for each swath in the butterfly
- 10.2. Chern labels not just for swaths but also for bands
- 10.3. A topological map of the butterfly
- 10.4. Apollonian-butterfly connection : where are the Chern numbers?
- 10.5. A topological landscape that has trefoil symmetry
- 10.6. Chern-dressed wave functions
- 10.7. Summary and outlook
- part IV. Catching the butterfly
- 11. The art of tinkering
- 11.1. The most beautiful physics experiments
- 12. The butterfly in the laboratory
- 12.1. Two-dimensional electron gases, superlattices, and the butterfly revealed
- 12.2. Magical carbon : a new net for the Hofstadter butterfly
- 12.3. A potentially sizzling hot topic in ultracold atom laboratories
- Appendix. Excerpts from the 2010 Physics Nobel Prize press release
- 13. The butterfly gallery : variations on a theme of Philip G Harper
- 14. Divertimento
- 15. Gratitude 15-1
- 16. Poetic math & science
- 17. Coda.