Mathematical concepts of quantum mechanics:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2006
|
| Edition: | Enlarged 2. print. |
| Series: | Universitext
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 286 S. graph. Darst. |
| ISBN: | 3540441603 9783540441601 |
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MARC
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| 100 | 1 | |a Gustafson, Stephen J. |d 1972- |e Verfasser |0 (DE-588)124948987 |4 aut | |
| 245 | 1 | 0 | |a Mathematical concepts of quantum mechanics |c Stephen J. Gustafson ; Israel Michael Sigal |
| 250 | |a Enlarged 2. print. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2006 | |
| 300 | |a XI, 286 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 700 | 1 | |a Sigal, Israel Michael |d 1945- |e Verfasser |0 (DE-588)110612043 |4 aut | |
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| _version_ | 1804137163003002880 |
|---|---|
| adam_text | Contents
1
Physical Background
...................................... 1
1.1
The Double-Slit Experiment
............................. 1
1.2
Wave Functions
........................................ 3
1.3
State Space
............................................ 4
1.4
The
Schrödinger
Equation
............................... 4
1.5
Mathematical Supplement: Operators on Hilbert Spaces
..... 6
2
Dynamics
.................................................. 11
2.1
Conservation of Probability
.............................. 11
2.2
Existence of Dynamics
.................................. 12
2.3
The Free Propagator
.................................... 16
2.4
Mathematical Supplement: Operator
Adjoints
.............. 17
2.5
Mathematical Supplement: the Fourier Transform
.......... 20
2.5.1
Definition of the Fourier Transform
.................. 20
2.5.2
Properties of the Fourier Transform
.................. 21
2.5.3
Functions of the Derivative
......................... 22
3
Observables
................................................ 25
3.1
Mean Values and the Momentum Operator
................ 25
3.2
Observables
........................................... 26
3.3
The
Heisenberg
Representation
.......................... 27
3.4
Quantization
.......................................... 28
3.5
Pseudodifferential Operators
............................. 29
4
The Uncertainty Principle
................................. 31
4.1
The
Heisenberg
Uncertainty Principle
..................... 31
4.2
A Refined Uncertainty Principle
.......................... 32
4.3
Application: Stability of Hydrogen
........................ 33
VIII Contents
5
Spectral
Theory
........................................... 35
5.1
The Spectrum of an Operator
............................ 35
5.2
Functions of Operators and the Spectral Mapping Theorem
. . 40
5.3
Applications to
Schrödinger
Operators
.................... 42
5.4
Spectrum and Evolution
................................ 47
5.5
Variational Characterization of Eigenvalues
................ 49
5.6
Number of Bound States
................................ 54
5.7
Mathematical Supplement: Integral Operators
............. 60
6
Scattering States
.......................................... 61
6.1
Short-range Interactions:
μ
> 1 .......................... 62
6.2
Long-range Interactions:
μ
< 1........................... 65
6.3
Wave Operators
........................................ 65
7
Special Cases
.............................................. 69
7.1
The Infinite Well
....................................... 69
7.2
The Torus
............................................. 70
7.3
A Potential Step
....................................... 70
7.4
The Square Well
....................................... 72
7.5
The Harmonic Oscillator
................................ 73
7.6
A Particle on a Sphere
.................................. 76
7.7
The Hydrogen Atom
.................................... 76
7.8
A Particle in an External EM Field
....................... 79
8
Many-particle Systems
.................................... 81
8.1
Quantization of a Many-particle System
................... 81
8.2
Separation of the Centre-of-mass Motion
.................. 85
8.3
Break-ups
............................................. 87
8.4
The HVZ Theorem
..................................... 88
8.5
Intra-
vs. Inter-cluster Motion
........................... 90
8.6
Existence of Bound States for Atoms and Molecules
........ 92
8.7
Scattering States
....................................... 93
8.8
Mathematical Supplement: Tensor Products
............... 95
8.9
Appendix:
Hartree
and Gross-Pitaevski Equations
.......... 97
9
Density Matrices
..........................................103
9.1
Introduction
...........................................103
9.2
States and Dynamics
...................................103
9.3
Open Systems
.........................................105
9.4
The Thermodynamic Limit
..............................106
9.5
Equilibrium States
.....................................107
9.6
The
Γ
— 0
Limit
.......................................107
9.7
Example: a System of Harmonic Oscillators
................109
9.8
A Particle Coupled to a Reservoir
........................110
9.9
Quantum
Systems
......................................
Ill
Contents
IX
9.10 Problems..............................................
Ill
9.11 Hubert Space
Approach
.................................
Ill
9.12
ВЕС
at
Т=0
..........................................
Ш
9.13
Appendix: the Ideal
Bose
Gas
............................114
9.14
Appendix: Bose-Einstein Condensation
....................119
9.15
Mathematical Supplement: the Trace,
and Trace Class Operators
..............................122
9.16
Mathematical Supplement: Projections
....................127
10
Perturbation Theory: Feshbach Method
...................131
10.1
The Feshbach Method
..................................132
10.2
Example: The
Zeeman
Effect
............................135
10.3
Example: Time-dependent Perturbations
..................137
10.4
Appendix: Proof of Theorem
10.1 ........................142
11
The Feynman Path Integral
...............................145
11.1
The Feynman Path Integral
.............................145
11.2
Generalizations of the Path Integral
......................148
11.3
Mathematical Supplement: The Trotter
Product Formula
.......................................149
12
Quasi-classical Analysis
....................................151
12.1
Quasi-classical Asymptotics of the Propagator
.............152
12.2
Quasi-classical Asymptotics of Green s Function
............
15(i
12.2.1
Appendix
........................................159
12.3
Bohr-Sommerfeld Semi-classical Quantization
..............15!)
12.4
Quasi-classical Asymptotics for the Ground State Energy.
...
1(>1
12.5
Mathematical Supplement: Operator Determinants
.........
1(>3
13
Mathematical Supplement: The Calculus of Variations
.....107
13.1
Funcţionale
............................................{(>
13.2
The First Variation and Critical Points
...................
Ш)
13.3
Constrained Variational Problems
........................173
13.4
The Second Variation
...................................174
13.5
Conjugate Points and Jacobi Fields
.......................176
13.6
The Action of the Critical Path
..........................179
13.7
Appendix: Connection to Geodesies
.......................182
14
Resonances
................................................185
14.1
Tunneling and Resonances
...............................185
14.2
The Free Resonance Energy
............................. 187
14.3
Instantons
.............................................189
14.4
Positive Temperatures
..................................
H)l
14.5
Pre-exponential Factor for the Bounce
....................193
14.6
Contribution of the Zero-mode
...........................195
14.7
Bohr-Sommerfeld Quantization for Resonances
.............195
X
Contents
15
Introduction
to
Quantum
Field Theory
....................199
15.1
The Place of QFT
......................................199
15.1.1
Physical Theories
..................................200
15.1.2
The Principle of Minimal Action
....................200
15.2
Klein-Gordon Theory as a Hamiltonian System
............201
15.2.1
The Legendre Transform
...........................201
15.2.2
Hamiltonians
.....................................202
15.2.3
Poisson
Brackets
..................................202
15.2.4
Hamilton s Equations
..............................204
15.3
Maxwell s Equations as a Hamiltonian System
.............205
15.4
Quantization of the Klein-Gordon and Maxwell Equations
... 207
15.4.1
The Quantization Procedure
........................208
15.4.2
Creation and Annihilation Operators
................212
15.4.3
Wick Ordering
....................................214
15.4.4
Quantizing Maxwell s Equations
.....................215
15.5
Fock Space
............................................216
15.6
Generalized Free Theory
................................219
15.7
Interactions
...........................................220
15.8
Quadratic Approximation
...............................222
15.8.1
Further Discussion
.................................228
15.8.2
A Brief Remark on Many-body Hamiltonians in
Second Quantization and the
Hartree
Approximation
.. 229
16
Quantum Electrodynamics of Non-relativistic Particles:
The Theory of Radiation
..................................231
16.1
The Hamiltonian
.......................................231
16.2
Perturbation Set-up
....................................234
16.3
Results
...............................................236
16.4
Mathematical Supplements
..............................239
16.4.1
Spectral Projections
...............................239
16.4.2
Projecting-out Procedure
...........................240
17
Supplement: Renormalization Group
......................241
17.1
The Decimation Map
...................................242
17.2
Relative Bounds
.......................................243
17.3
Elimination of Particle and High Photon Energy Degrees of
Freedom
..............................................244
17.4
Generalized Normal Form of Operators on Fock Space
......248
17.5
The Hamiltonian
Η0(ε, ζ)...............................
250
17.6
A Banach Space of Operators
............................254
17.7
Rescaling
..............................................255
17.8
The Renormalization Map
...............................257
17.9
Linearized Flow
........................................258
17.10
Central-stable Manifold for RG and Spectra of Hamiltonians
. 261
17.11
Appendix
.............................................265
Contents
XI
18
Comments on Missing Topics, Literature,
and Further Reading
......................................267
References
.....................................................273
Index
..........................................................281
|
| adam_txt |
Contents
1
Physical Background
. 1
1.1
The Double-Slit Experiment
. 1
1.2
Wave Functions
. 3
1.3
State Space
. 4
1.4
The
Schrödinger
Equation
. 4
1.5
Mathematical Supplement: Operators on Hilbert Spaces
. 6
2
Dynamics
. 11
2.1
Conservation of Probability
. 11
2.2
Existence of Dynamics
. 12
2.3
The Free Propagator
. 16
2.4
Mathematical Supplement: Operator
Adjoints
. 17
2.5
Mathematical Supplement: the Fourier Transform
. 20
2.5.1
Definition of the Fourier Transform
. 20
2.5.2
Properties of the Fourier Transform
. 21
2.5.3
Functions of the Derivative
. 22
3
Observables
. 25
3.1
Mean Values and the Momentum Operator
. 25
3.2
Observables
. 26
3.3
The
Heisenberg
Representation
. 27
3.4
Quantization
. 28
3.5
Pseudodifferential Operators
. 29
4
The Uncertainty Principle
. 31
4.1
The
Heisenberg
Uncertainty Principle
. 31
4.2
A Refined Uncertainty Principle
. 32
4.3
Application: Stability of Hydrogen
. 33
VIII Contents
5
Spectral
Theory
. 35
5.1
The Spectrum of an Operator
. 35
5.2
Functions of Operators and the Spectral Mapping Theorem
. . 40
5.3
Applications to
Schrödinger
Operators
. 42
5.4
Spectrum and Evolution
. 47
5.5
Variational Characterization of Eigenvalues
. 49
5.6
Number of Bound States
. 54
5.7
Mathematical Supplement: Integral Operators
. 60
6
Scattering States
. 61
6.1
Short-range Interactions:
μ
> 1 . 62
6.2
Long-range Interactions:
μ
< 1. 65
6.3
Wave Operators
. 65
7
Special Cases
. 69
7.1
The Infinite Well
. 69
7.2
The Torus
. 70
7.3
A Potential Step
. 70
7.4
The Square Well
. 72
7.5
The Harmonic Oscillator
. 73
7.6
A Particle on a Sphere
. 76
7.7
The Hydrogen Atom
. 76
7.8
A Particle in an External EM Field
. 79
8
Many-particle Systems
. 81
8.1
Quantization of a Many-particle System
. 81
8.2
Separation of the Centre-of-mass Motion
. 85
8.3
Break-ups
. 87
8.4
The HVZ Theorem
. 88
8.5
Intra-
vs. Inter-cluster Motion
. 90
8.6
Existence of Bound States for Atoms and Molecules
. 92
8.7
Scattering States
. 93
8.8
Mathematical Supplement: Tensor Products
. 95
8.9
Appendix:
Hartree
and Gross-Pitaevski Equations
. 97
9
Density Matrices
.103
9.1
Introduction
.103
9.2
States and Dynamics
.103
9.3
Open Systems
.105
9.4
The Thermodynamic Limit
.106
9.5
Equilibrium States
.107
9.6
The
Γ
— 0
Limit
.107
9.7
Example: a System of Harmonic Oscillators
.109
9.8
A Particle Coupled to a Reservoir
.110
9.9
Quantum
Systems
.
Ill
Contents
IX
9.10 Problems.
Ill
9.11 Hubert Space
Approach
.
Ill
9.12
ВЕС
at
Т=0
.
Ш
9.13
Appendix: the Ideal
Bose
Gas
.114
9.14
Appendix: Bose-Einstein Condensation
.119
9.15
Mathematical Supplement: the Trace,
and Trace Class Operators
.122
9.16
Mathematical Supplement: Projections
.127
10
Perturbation Theory: Feshbach Method
.131
10.1
The Feshbach Method
.132
10.2
Example: The
Zeeman
Effect
.135
10.3
Example: Time-dependent Perturbations
.137
10.4
Appendix: Proof of Theorem
10.1 .142
11
The Feynman Path Integral
.145
11.1
The Feynman Path Integral
.145
11.2
Generalizations of the Path Integral
.148
11.3
Mathematical Supplement: The Trotter
Product Formula
.149
12
Quasi-classical Analysis
.151
12.1
Quasi-classical Asymptotics of the Propagator
.152
12.2
Quasi-classical Asymptotics of Green's Function
.
15(i
12.2.1
Appendix
.159
12.3
Bohr-Sommerfeld Semi-classical Quantization
.15!)
12.4
Quasi-classical Asymptotics for the Ground State Energy.
.
1(>1
12.5
Mathematical Supplement: Operator Determinants
.
1(>3
13
Mathematical Supplement: The Calculus of Variations
.107
13.1
Funcţionale
.{(>"
13.2
The First Variation and Critical Points
.
Ш)
13.3
Constrained Variational Problems
.173
13.4
The Second Variation
.174
13.5
Conjugate Points and Jacobi Fields
.176
13.6
The Action of the Critical Path
.179
13.7
Appendix: Connection to Geodesies
.182
14
Resonances
.185
14.1
Tunneling and Resonances
.185
14.2
The Free Resonance Energy
. 187
14.3
Instantons
.189
14.4
Positive Temperatures
.
H)l
14.5
Pre-exponential Factor for the Bounce
.193
14.6
Contribution of the Zero-mode
.195
14.7
Bohr-Sommerfeld Quantization for Resonances
.195
X
Contents
15
Introduction
to
Quantum
Field Theory
.199
15.1
The Place of QFT
.199
15.1.1
Physical Theories
.200
15.1.2
The Principle of Minimal Action
.200
15.2
Klein-Gordon Theory as a Hamiltonian System
.201
15.2.1
The Legendre Transform
.201
15.2.2
Hamiltonians
.202
15.2.3
Poisson
Brackets
.202
15.2.4
Hamilton's Equations
.204
15.3
Maxwell's Equations as a Hamiltonian System
.205
15.4
Quantization of the Klein-Gordon and Maxwell Equations
. 207
15.4.1
The Quantization Procedure
.208
15.4.2
Creation and Annihilation Operators
.212
15.4.3
Wick Ordering
.214
15.4.4
Quantizing Maxwell's Equations
.215
15.5
Fock Space
.216
15.6
Generalized Free Theory
.219
15.7
Interactions
.220
15.8
Quadratic Approximation
.222
15.8.1
Further Discussion
.228
15.8.2
A Brief Remark on Many-body Hamiltonians in
Second Quantization and the
Hartree
Approximation
. 229
16
Quantum Electrodynamics of Non-relativistic Particles:
The Theory of Radiation
.231
16.1
The Hamiltonian
.231
16.2
Perturbation Set-up
.234
16.3
Results
.236
16.4
Mathematical Supplements
.239
16.4.1
Spectral Projections
.239
16.4.2
Projecting-out Procedure
.240
17
Supplement: Renormalization Group
.241
17.1
The Decimation Map
.242
17.2
Relative Bounds
.243
17.3
Elimination of Particle and High Photon Energy Degrees of
Freedom
.244
17.4
Generalized Normal Form of Operators on Fock Space
.248
17.5
The Hamiltonian
Η0(ε, ζ).
250
17.6
A Banach Space of Operators
.254
17.7
Rescaling
.255
17.8
The Renormalization Map
.257
17.9
Linearized Flow
.258
17.10
Central-stable Manifold for RG and Spectra of Hamiltonians
. 261
17.11
Appendix
.265
Contents
XI
18
Comments on Missing Topics, Literature,
and Further Reading
.267
References
.273
Index
.281 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Gustafson, Stephen J. 1972- Sigal, Israel Michael 1945- |
| author_GND | (DE-588)124948987 (DE-588)110612043 |
| author_facet | Gustafson, Stephen J. 1972- Sigal, Israel Michael 1945- |
| author_role | aut aut |
| author_sort | Gustafson, Stephen J. 1972- |
| author_variant | s j g sj sjg i m s im ims |
| building | Verbundindex |
| bvnumber | BV022931325 |
| classification_rvk | SK 950 UK 1200 |
| classification_tum | PHY 011f PHY 020f |
| ctrlnum | (OCoLC)476214257 (DE-599)BVBBV022931325 |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| edition | Enlarged 2. print. |
| format | Book |
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| genre | (DE-588)4123623-3 Lehrbuch gnd-content |
| genre_facet | Lehrbuch |
| id | DE-604.BV022931325 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:54:33Z |
| indexdate | 2024-07-09T21:07:55Z |
| institution | BVB |
| isbn | 3540441603 9783540441601 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016136140 |
| oclc_num | 476214257 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-29T DE-91G DE-BY-TUM DE-11 DE-355 DE-BY-UBR |
| owner_facet | DE-19 DE-BY-UBM DE-29T DE-91G DE-BY-TUM DE-11 DE-355 DE-BY-UBR |
| physical | XI, 286 S. graph. Darst. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Springer |
| record_format | marc |
| series2 | Universitext |
| spelling | Gustafson, Stephen J. 1972- Verfasser (DE-588)124948987 aut Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal Enlarged 2. print. Berlin [u.a.] Springer 2006 XI, 286 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Universitext Quantenmechanik Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Quantenmechanik (DE-588)4047989-4 s Mathematische Physik (DE-588)4037952-8 s DE-604 Sigal, Israel Michael 1945- Verfasser (DE-588)110612043 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016136140&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gustafson, Stephen J. 1972- Sigal, Israel Michael 1945- Mathematical concepts of quantum mechanics Quantenmechanik Quantenmechanik (DE-588)4047989-4 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4047989-4 (DE-588)4037952-8 (DE-588)4123623-3 |
| title | Mathematical concepts of quantum mechanics |
| title_auth | Mathematical concepts of quantum mechanics |
| title_exact_search | Mathematical concepts of quantum mechanics |
| title_exact_search_txtP | Mathematical concepts of quantum mechanics |
| title_full | Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal |
| title_fullStr | Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal |
| title_full_unstemmed | Mathematical concepts of quantum mechanics Stephen J. Gustafson ; Israel Michael Sigal |
| title_short | Mathematical concepts of quantum mechanics |
| title_sort | mathematical concepts of quantum mechanics |
| topic | Quantenmechanik Quantenmechanik (DE-588)4047989-4 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Quantenmechanik Mathematische Physik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016136140&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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