A mathematical journey to quantum mechanics:
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Cham
Springer
[2021]
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| Series: | Unitext for physics
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xv, 289 Seiten Diagramme |
| ISBN: | 9783030860974 |
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Record in the Search Index
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| adam_text | Contents 1 Introduction: How to Read This Book................................................... 1 2 Newtonian, Lagrangian and Hamiltonian Mechanics ....................... 2.1 Lecture 1: A Summary of the Principles of Newtonian Mechanics ........................................................................................ 2.2 Lecture 2: The Mechanica! Lagrangian ......................................... 2.3 Lecture 3: The Euler-Lagrange Equations ................................... 2.4 Lecture 4: The Mechanical Hamiltonian ....................................... 2.5 Lecture 5: The Hamilton Equations .............................................. 2.6 Lecture 6: Poisson’s Brackets in Hamiltonian Mechanics........... 11 3 4 Can Light Be Described by Classical Mechanics? ............................. 3.1 Lecture 7: The Michelson-Morley Experiment and the Principles of Special Relativity ......................................... 3.2 Lecture 8: Motion Among Inertial Frames. The Lorentz Transformations................................................................................ 3.3 Lecture 9: Addition of Velocities. The Relativistic Formula....... 3.4 Lecture 10: The Einstein Rest Energy Formula £ = me2............ 3.5 Lecture 11: The Relativistic Energy Formula E2 = p2c2 + m2c4 .......................................................................... 3.6 Lecture 12: Electromagnetic Waves by the Maxwell Equations .......................................................................................... 3.7 Lecture 13: The Invariance of Maxwell Equations Under the Lorentz
Transformations ........................................................... Why Quantum Mechanics? .................................................................... 4.1 Lecture 14: The Problem of the Nature of Matter ....................... 4.2 Lecture 15: Monochromatic Plane Waves—The One Dimensional Case ............................................................................ 4.3 Lecture 16: The Young Double Split Experiment. Light Seen as a Wave ................................................................................ 4.4 Lecture 17: The Planck-Einstein Formula E = hv ..................... 11 14 18 22 26 28 33 33 38 41 43 45 46 51 55 55 60 66 70 xiii
xiv Contents 4.5 4.6 4.7 4.8 5 6 7 8 Lecture 18: Light as Particles. The Einstein Photoelectric Effect.................................................................................................. Lecture 19: Atomic Spectra and Bohr’s Model of Hydrogen Atom.................................................................................................. Lecture 20: De Broglie’s Hypothesis. Material Objects as Waves ............................................................................................ Lecture 21: Strengthening the Einstein Idea of Photons. The Compton Effect ......................................................................... The Schrödinger Equations and Their Consequences ........................ 5.1 Lecture 22: The Schrödinger Equations. The One Dimensional Case ............................................................................. 5.2 Lecture 23: Solving the Schrödinger Equation for the Free Particle .............................................................................................. 5.3 Lecture 24: Solving the Schrödinger Equation for a Particle in a Box ............................................................................................ 5.4 Lecture 25 : Solving the Schrödinger Equation of Harmonic Oscillator. The Quantized Energies ............................................... The Mathematics Behind the HarmonicOscillator ............................. 6.1 Lecture 26: The Hermite Polynomials ............................................ 6.2 Lecture 27: Real and Complex Vector Structures..........................
6.2.1 Finite Dimensional Real and Complex Vector Spaces, Inner Product, Norm, Distance, Completeness ....................................................................... 6.3 Lecture 28: Pre-Hilbert and Hilbert Spaces.................................... 6.4 Lecture 29: Examples of Hilbert Spaces ........................................ 6.5 Lecture 30: Orthogonal and Orthonormal Systems in Hilbert Spaces .............................................................................. 6.6 Lecture 31: Linear Operators, Eigenvalues, Eigenvectors for the Schrödinger Equation........................................................... From Monochromatic Plane Waves to Wave Packets.......................... 7.1 Lecture 32: Again on the de Broglie Hypothesis. Wave-Particle Duality and Wave Packets....................................... 7.2 Lecture 33: More About Electron in an Atom............................... The Heisenberg Uncertainty Principle and the Mathematics Behind .......................................................................................................... 8.1 Lecture 34: Wave Packets and the Schrödinger Equation ............ 8.2 Lecture 35: The Wave Function Ψ Solution of the Schrödinger Equation ........................................................... 8.3 Lecture 36: The Gauss Wave Packet and the Heisenberg Uncertainty Principle ....................................................................... 8.4 Lecture 37: The Mathematics Behind the Wave Packets. The Fourier Series and the Fourier Transforms.............................. 75 77 81 83 89 89 92 94
97 105 105 112 113 116 120 128 129 137 137 142 145 145 148 150 155
Contents 9 The Principles of Quantum Mechanics ................................................. 173 9.1 9.2 173 9.3 9.4 10 Lecture 38: Operators in Quantum Mechanics ............................... Lecture 39: The Relation Փ* у2 Ψ - Ψ V2 Փ* — di {ф* v Ψ - Φ V φ*) and Its Consequences ............................................................... .. Lecture 40. Similarities with Hamiltonian Formalism of Classical Mechanics....................................................................... Lecture 41 : From the Wave Function to the Quantum State. The Postulates of Quantum Mechanics ........................................... Consequences of Quantum Mechanics Principles................................. 10.1 10.2 10.3 10.4 11 XV Lecture 42: The Ehrenfest Theorem................................................. Lecture 43: The Heisenberg General Uncertainty Principle ........ Lecture 44: The Dirac Notation and the Meaning of a Quantum Mechanics Experiment ............................................. Lecture 45: The Photon Polarization by Dirac’s Notation............ Quantum Mechanics at the Next Level ................................................... 11.1 11.2 11.3 11.4 11.5 Lecture 46: The Electron Spin........................................................... Lecture 47: Revisiting the Harmonic Oscillator. The Ladder Operators.................................................................................. Lecture 48: The Angular Momentum in Quantum Mechanics ............................................................................................ Lecture 49:
From Differential Operators in Spherical Coordinates to the Hydrogen Atom ................................................. Lecture 50: The Pauli Matrices and the Dirac Equation. Towards the Relativistic Quantum Mechanics ............................... 180 185 188 201 201 205 211 215 225 225 237 247 254 275 12 Conclusions.................................................................................................. 281 Bibliography ....................................................................................................... 285 Index..................................................................................................................... 287
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| adam_txt |
Contents 1 Introduction: How to Read This Book. 1 2 Newtonian, Lagrangian and Hamiltonian Mechanics . 2.1 Lecture 1: A Summary of the Principles of Newtonian Mechanics . 2.2 Lecture 2: The Mechanica! Lagrangian . 2.3 Lecture 3: The Euler-Lagrange Equations . 2.4 Lecture 4: The Mechanical Hamiltonian . 2.5 Lecture 5: The Hamilton Equations . 2.6 Lecture 6: Poisson’s Brackets in Hamiltonian Mechanics. 11 3 4 Can Light Be Described by Classical Mechanics? . 3.1 Lecture 7: The Michelson-Morley Experiment and the Principles of Special Relativity . 3.2 Lecture 8: Motion Among Inertial Frames. The Lorentz Transformations. 3.3 Lecture 9: Addition of Velocities. The Relativistic Formula. 3.4 Lecture 10: The Einstein Rest Energy Formula £ = me2. 3.5 Lecture 11: The Relativistic Energy Formula E2 = p2c2 + m2c4 . 3.6 Lecture 12: Electromagnetic Waves by the Maxwell Equations . 3.7 Lecture 13: The Invariance of Maxwell Equations Under the Lorentz
Transformations . Why Quantum Mechanics? . 4.1 Lecture 14: The Problem of the Nature of Matter . 4.2 Lecture 15: Monochromatic Plane Waves—The One Dimensional Case . 4.3 Lecture 16: The Young Double Split Experiment. Light Seen as a Wave . 4.4 Lecture 17: The Planck-Einstein Formula E = hv . 11 14 18 22 26 28 33 33 38 41 43 45 46 51 55 55 60 66 70 xiii
xiv Contents 4.5 4.6 4.7 4.8 5 6 7 8 Lecture 18: Light as Particles. The Einstein Photoelectric Effect. Lecture 19: Atomic Spectra and Bohr’s Model of Hydrogen Atom. Lecture 20: De Broglie’s Hypothesis. Material Objects as Waves . Lecture 21: Strengthening the Einstein Idea of Photons. The Compton Effect . The Schrödinger Equations and Their Consequences . 5.1 Lecture 22: The Schrödinger Equations. The One Dimensional Case . 5.2 Lecture 23: Solving the Schrödinger Equation for the Free Particle . 5.3 Lecture 24: Solving the Schrödinger Equation for a Particle in a Box . 5.4 Lecture 25 : Solving the Schrödinger Equation of Harmonic Oscillator. The Quantized Energies . The Mathematics Behind the HarmonicOscillator . 6.1 Lecture 26: The Hermite Polynomials . 6.2 Lecture 27: Real and Complex Vector Structures.
6.2.1 Finite Dimensional Real and Complex Vector Spaces, Inner Product, Norm, Distance, Completeness . 6.3 Lecture 28: Pre-Hilbert and Hilbert Spaces. 6.4 Lecture 29: Examples of Hilbert Spaces . 6.5 Lecture 30: Orthogonal and Orthonormal Systems in Hilbert Spaces . 6.6 Lecture 31: Linear Operators, Eigenvalues, Eigenvectors for the Schrödinger Equation. From Monochromatic Plane Waves to Wave Packets. 7.1 Lecture 32: Again on the de Broglie Hypothesis. Wave-Particle Duality and Wave Packets. 7.2 Lecture 33: More About Electron in an Atom. The Heisenberg Uncertainty Principle and the Mathematics Behind . 8.1 Lecture 34: Wave Packets and the Schrödinger Equation . 8.2 Lecture 35: The Wave Function Ψ Solution of the Schrödinger Equation . 8.3 Lecture 36: The Gauss Wave Packet and the Heisenberg Uncertainty Principle . 8.4 Lecture 37: The Mathematics Behind the Wave Packets. The Fourier Series and the Fourier Transforms. 75 77 81 83 89 89 92 94
97 105 105 112 113 116 120 128 129 137 137 142 145 145 148 150 155
Contents 9 The Principles of Quantum Mechanics . 173 9.1 9.2 173 9.3 9.4 10 Lecture 38: Operators in Quantum Mechanics . Lecture 39: The Relation Փ* у2 Ψ - Ψ V2 Փ* — di\{ф* v Ψ - Φ V φ*) and Its Consequences . . Lecture 40. Similarities with Hamiltonian Formalism of Classical Mechanics. Lecture 41 : From the Wave Function to the Quantum State. The Postulates of Quantum Mechanics . Consequences of Quantum Mechanics Principles. 10.1 10.2 10.3 10.4 11 XV Lecture 42: The Ehrenfest Theorem. Lecture 43: The Heisenberg General Uncertainty Principle . Lecture 44: The Dirac Notation and the Meaning of a Quantum Mechanics Experiment . Lecture 45: The Photon Polarization by Dirac’s Notation. Quantum Mechanics at the Next Level . 11.1 11.2 11.3 11.4 11.5 Lecture 46: The Electron Spin. Lecture 47: Revisiting the Harmonic Oscillator. The Ladder Operators. Lecture 48: The Angular Momentum in Quantum Mechanics . Lecture 49:
From Differential Operators in Spherical Coordinates to the Hydrogen Atom . Lecture 50: The Pauli Matrices and the Dirac Equation. Towards the Relativistic Quantum Mechanics . 180 185 188 201 201 205 211 215 225 225 237 247 254 275 12 Conclusions. 281 Bibliography . 285 Index. 287 |
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| index_date | 2024-07-03T18:29:43Z |
| indexdate | 2024-07-10T09:15:07Z |
| institution | BVB |
| isbn | 9783030860974 |
| language | English |
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| spelling | Capozziello, Salvatore ca. 20./21. Jh. Verfasser (DE-588)142835277 aut A mathematical journey to quantum mechanics Salvatore Capozziello, Wladimir-Georges Boskoff Cham Springer [2021] © 2021 xv, 289 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Unitext for physics Quantum Physics Atomic/Molecular Structure and Spectra Theoretical, Mathematical and Computational Physics Functional Analysis Quantum physics Atomic structure Molecular structure Mathematical physics Functional analysis Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Mathematik (DE-588)4037944-9 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 s Mathematische Physik (DE-588)4037952-8 s Mathematik (DE-588)4037944-9 s DE-604 Boskoff, Wladimir-George 1958- Verfasser (DE-588)1116975017 aut Erscheint auch als Online-Ausgabe 978-3-030-86098-1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032954670&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Capozziello, Salvatore ca. 20./21. Jh Boskoff, Wladimir-George 1958- A mathematical journey to quantum mechanics Quantum Physics Atomic/Molecular Structure and Spectra Theoretical, Mathematical and Computational Physics Functional Analysis Quantum physics Atomic structure Molecular structure Mathematical physics Functional analysis Mathematische Physik (DE-588)4037952-8 gnd Mathematik (DE-588)4037944-9 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4037952-8 (DE-588)4037944-9 (DE-588)4047989-4 |
| title | A mathematical journey to quantum mechanics |
| title_auth | A mathematical journey to quantum mechanics |
| title_exact_search | A mathematical journey to quantum mechanics |
| title_exact_search_txtP | A mathematical journey to quantum mechanics |
| title_full | A mathematical journey to quantum mechanics Salvatore Capozziello, Wladimir-Georges Boskoff |
| title_fullStr | A mathematical journey to quantum mechanics Salvatore Capozziello, Wladimir-Georges Boskoff |
| title_full_unstemmed | A mathematical journey to quantum mechanics Salvatore Capozziello, Wladimir-Georges Boskoff |
| title_short | A mathematical journey to quantum mechanics |
| title_sort | a mathematical journey to quantum mechanics |
| topic | Quantum Physics Atomic/Molecular Structure and Spectra Theoretical, Mathematical and Computational Physics Functional Analysis Quantum physics Atomic structure Molecular structure Mathematical physics Functional analysis Mathematische Physik (DE-588)4037952-8 gnd Mathematik (DE-588)4037944-9 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Quantum Physics Atomic/Molecular Structure and Spectra Theoretical, Mathematical and Computational Physics Functional Analysis Quantum physics Atomic structure Molecular structure Mathematical physics Functional analysis Mathematische Physik Mathematik Quantenmechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032954670&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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