Quantum mechanics for mathematicians:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Mathematical Society
2008
|
| Series: | Graduate studies in mathematics
95 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XV, 387 S. |
| ISBN: | 9780821846308 |
Staff View
MARC
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| 100 | 1 | |a Takhtajan, Leon A. |d 1950- |e Verfasser |0 (DE-588)123335868 |4 aut | |
| 245 | 1 | 0 | |a Quantum mechanics for mathematicians |c Leon A. Takhtajan |
| 264 | 1 | |a Providence, R.I. |b American Mathematical Society |c 2008 | |
| 300 | |a XV, 387 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 95 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Mathematische Physik | |
| 650 | 4 | |a Quantentheorie | |
| 650 | 4 | |a Quantum theory | |
| 650 | 4 | |a Mathematical physics | |
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| 689 | 0 | 1 | |a Mathematische Physik |0 (DE-588)4037952-8 |D s |
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| 830 | 0 | |a Graduate studies in mathematics |v 95 |w (DE-604)BV009739289 |9 95 | |
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Record in the Search Index
| _version_ | 1804138041756876800 |
|---|---|
| adam_text | Contents
Preface
xiii
Part
1.
Foundations
Chapter
1.
Classical Mechanics
3
§1.
Lagrangian Mechanics
4
1.1.
Generalized coordinates
4
1.2.
The principle of the least action
4
1.3.
Examples of Lagrangian systems
8
1.4.
Symmetries and Noether s theorem
15
1.5.
One-dimensional motion
20
1.6.
The motion in a central field and the Kepler problem
22
1.7.
Legendre transform
26
§2.
Hamiltonian Mechanics
31
2.1.
Hamilton s equations
31
2.2.
The action functional in the phase space
33
2.3.
The action as a function of coordinates
35
2.4.
Classical
observables
and
Poisson
bracket
38
2.5.
Canonical transformations and generating functions
39
2.6.
Symplectic manifolds
42
2.7.
Poisson
manifolds
51
2.8.
Hamilton s and Liouville s representations
56
§3.
Notes and references
60
Chapter
2.
Basic Principles of Quantum Mechanics
63
§1.
Observables,
states, and dynamics
65
1.1.
Mathematical formulation
66
vii
viii
Contents
1.2.
Heisenberg s uncertainty relations
74
1.3.
Dynamics
75
§2.
Quantization
79
2.1. Heisenberg
commutation relations
80
2.2.
Coordinate and momentum representations
86
2.3.
Free quantum particle
93
2.4.
Examples of quantum systems
98
2.5.
Old quantum mechanics
102
2.6.
Harmonic oscillator
103
2.7.
HolomorpMc representation and Wick symbols
111
§3.
Weyl relations
118
3.1.
Stone-
von
Neumann theorem
119
3.2.
Invariant formulation
125
3.3.
Weyl quantization
128
3.4.
The •-product
136
3.5.
Deformation quantization
139
§4.
Notes and references
146
Chapter
3. Schrödinger
Equation
149
§1.
General properties
149
1.1.
Self-adjointness
150
1.2.
Characterization of the spectrum
152
1.3.
The virial theorem
154
§2.
One-dimensional
Schrödinger
equation
155
2.1.
Jost functions and transition coefficients
155
2.2.
Eigenfunction expansion
163
2.3.
S-matrix and scattering theory
170
2.4.
Other boundary conditions
177
§3.
Angular momentum and SO(3)
180
3.1.
Angular momentum operators
180
3.2.
Representation theory of S0(3)
182
§4.
Two-body problem
185
4.1.
Separation of the center of mass
185
4.2.
Three-dimensional scattering theory
186
4.3.
Particle in a central potential
188
§5.
Hydrogen atom and SO
(4)
193
5.1.
Discrete spectrum
193
5.2.
Continuous spectrum
197
5.3.
Hidden SO(4) symmetry
199
§6.
Semi-classical asymptotics
-
I
205
6.1.
Time-dependent asymptotics
206
Contents ix
6.2.
Time independent asymptotics
209
6.3. Bohr-Wilson-Sommerfeld
quantization rules
212
§7.
Notes and references
213
Chapter
4.
Spin and Identical Particles
217
§1.
Spin
217
1.1.
Spin operators
217
1.2.
Spin and representation theory of SU(2)
218
§2.
Charged spin particle in the magnetic field
221
2.1. Pauli Hamiltonian 221
2.2.
Particle in a uniform magnetic field
223
§3.
System of identical particles
224
3.1.
The symmetrization postulate
225
3.2.
Young diagrams and representation theory of
Syrn^
229
3.3. Schur-Weyl
duality and symmetry of the wave functions
232
§4.
Notes and references
234
Part
2.
Functional Methods and Supersymmetry
Chapter
5.
Path Integral Formulation of Quantum Mechanics
239
§1.
Feynman path integral
239
1.1.
The fundamental solution of the
Schrödinger
equation
239
1.2.
Feynman path integral in the phase space
242
1.3.
Feynman path integral in the configuration space
245
1.4.
Several degrees of freedom
247
§2.
Symbols of the evolution operator and path integrals
249
2.1.
Thepç-symbol
249
2.2.
The
çp-symbol
250
2.3.
The Weyl symbol
251
2.4.
The Wick symbol
252
§3.
Feynman path integral for the harmonic oscillator
255
3.1.
Gaussian integration
255
3.2.
Propagator of the harmonic oscillator
257
3.3.
Mehler identity
259
§4.
Gaussian path integrals
260
4.1.
Gaussian path integral for a free particle
261
4.2.
Gaussian path integral for the harmonic oscillator
264
§5.
Regularized determinants of differential operators
268
5.1.
Dirichlet boundary conditions
268
5.2.
Periodic boundary conditions
274
5.3.
First order differential operators
278
Contents
§6. Semi-classical asymptotics -
II
280
6.1.
Using the Feynman path integral
281
6.2.
Rigorous derivation
282
§7.
Notes and references
285
Chapter
6.
Integration in Functional Spaces
289
§1.
Gaussian measures
289
1.1.
Finite-dimensional case
289
1.2.
Infinite-dimensional case
291
§2.
Wiener measure and Wiener integral
293
2.1.
Definition of the Wiener measure
293
2.2.
Conditional Wiener measure and Feynman-Kac formula
296
2.3.
Relation between Wiener and Feynman integrals
299
§3.
Gaussian Wiener integrals
301
3.1.
Dirichlet boundary conditions
301
3.2.
Periodic boundary conditions
302
§4.
Notes and references
305
Chapter
7.
Fermion Systems
307
§1.
Canonical
anticommutation
relations
307
1.1.
Motivation
307
1.2.
Clifford algebras
311
§2. Grassmann
algebras
314
2.1.
Realization of canonical
anticommutation
relations
315
2.2.
Differential forms
317
2.3.
Berezin integral
319
§3.
Graded linear algebra
324
3.1.
Graded vector spaces and superalgebras
324
3.2.
Examples of superalgebras
326
3.3.
Supertrace
and Berezinian
328
§4.
Path integrals for anticommuting variables
330
4.1.
Wick and matrix symbols
330
4.2.
Path integral for the evolution operator
335
4.3.
Gaussian path integrals over
Grassmann
variables
338
§5.
Notes and references
340
Chapter
8.
Supersymmetry
343
§1.
Supermanifolds
343
§2.
Equivariant cohomology and localization
345
2.1.
Finite-dimensional case
345
2.2.
Infinite-dimensional case
349
Contents xi
§3.
Classical mechanics on supermanifolds
354
3.1.
Functions with anticommuting values
354
3.2.
Classical systems
356
§4.
Supersymmetry
359
4.1.
Total angular momentum
359
4.2.
Supersymmetry transformation
360
4.3.
Supersymmetric particle on a Riemannian manifold
362
§5.
Quantum mechanics on supermanifolds
364
§6.
Atiyah-Singer index formula
369
§7.
Notes and references
370
Bibliography
373
Index
383
|
| adam_txt |
Contents
Preface
xiii
Part
1.
Foundations
Chapter
1.
Classical Mechanics
3
§1.
Lagrangian Mechanics
4
1.1.
Generalized coordinates
4
1.2.
The principle of the least action
4
1.3.
Examples of Lagrangian systems
8
1.4.
Symmetries and Noether's theorem
15
1.5.
One-dimensional motion
20
1.6.
The motion in a central field and the Kepler problem
22
1.7.
Legendre transform
26
§2.
Hamiltonian Mechanics
31
2.1.
Hamilton's equations
31
2.2.
The action functional in the phase space
33
2.3.
The action as a function of coordinates
35
2.4.
Classical
observables
and
Poisson
bracket
38
2.5.
Canonical transformations and generating functions
39
2.6.
Symplectic manifolds
42
2.7.
Poisson
manifolds
51
2.8.
Hamilton's and Liouville's representations
56
§3.
Notes and references
60
Chapter
2.
Basic Principles of Quantum Mechanics
63
§1.
Observables,
states, and dynamics
65
1.1.
Mathematical formulation
66
vii
viii
Contents
1.2.
Heisenberg's uncertainty relations
74
1.3.
Dynamics
75
§2.
Quantization
79
2.1. Heisenberg
commutation relations
80
2.2.
Coordinate and momentum representations
86
2.3.
Free quantum particle
93
2.4.
Examples of quantum systems
98
2.5.
Old quantum mechanics
102
2.6.
Harmonic oscillator
103
2.7.
HolomorpMc representation and Wick symbols
111
§3.
Weyl relations
118
3.1.
Stone-
von
Neumann theorem
119
3.2.
Invariant formulation
125
3.3.
Weyl quantization
128
3.4.
The •-product
136
3.5.
Deformation quantization
139
§4.
Notes and references
146
Chapter
3. Schrödinger
Equation
149
§1.
General properties
149
1.1.
Self-adjointness
150
1.2.
Characterization of the spectrum
152
1.3.
The virial theorem
154
§2.
One-dimensional
Schrödinger
equation
155
2.1.
Jost functions and transition coefficients
155
2.2.
Eigenfunction expansion
163
2.3.
S-matrix and scattering theory
170
2.4.
Other boundary conditions
177
§3.
Angular momentum and SO(3)
180
3.1.
Angular momentum operators
180
3.2.
Representation theory of S0(3)
182
§4.
Two-body problem
185
4.1.
Separation of the center of mass
185
4.2.
Three-dimensional scattering theory
186
4.3.
Particle in a central potential
188
§5.
Hydrogen atom and SO
(4)
193
5.1.
Discrete spectrum
193
5.2.
Continuous spectrum
197
5.3.
Hidden SO(4) symmetry
199
§6.
Semi-classical asymptotics
-
I
205
6.1.
Time-dependent asymptotics
206
Contents ix
6.2.
Time independent asymptotics
209
6.3. Bohr-Wilson-Sommerfeld
quantization rules
212
§7.
Notes and references
213
Chapter
4.
Spin and Identical Particles
217
§1.
Spin
217
1.1.
Spin operators
217
1.2.
Spin and representation theory of SU(2)
218
§2.
Charged spin particle in the magnetic field
221
2.1. Pauli Hamiltonian 221
2.2.
Particle in a uniform magnetic field
223
§3.
System of identical particles
224
3.1.
The symmetrization postulate
225
3.2.
Young diagrams and representation theory of
Syrn^
229
3.3. Schur-Weyl
duality and symmetry of the wave functions
232
§4.
Notes and references
234
Part
2.
Functional Methods and Supersymmetry
Chapter
5.
Path Integral Formulation of Quantum Mechanics
239
§1.
Feynman path integral
239
1.1.
The fundamental solution of the
Schrödinger
equation
239
1.2.
Feynman path integral in the phase space
242
1.3.
Feynman path integral in the configuration space
245
1.4.
Several degrees of freedom
247
§2.
Symbols of the evolution operator and path integrals
249
2.1.
Thepç-symbol
249
2.2.
The
çp-symbol
250
2.3.
The Weyl symbol
251
2.4.
The Wick symbol
252
§3.
Feynman path integral for the harmonic oscillator
255
3.1.
Gaussian integration
255
3.2.
Propagator of the harmonic oscillator
257
3.3.
Mehler identity
259
§4.
Gaussian path integrals
260
4.1.
Gaussian path integral for a free particle
261
4.2.
Gaussian path integral for the harmonic oscillator
264
§5.
Regularized determinants of differential operators
268
5.1.
Dirichlet boundary conditions
268
5.2.
Periodic boundary conditions
274
5.3.
First order differential operators
278
Contents
§6. Semi-classical asymptotics -
II
280
6.1.
Using the Feynman path integral
281
6.2.
Rigorous derivation
282
§7.
Notes and references
285
Chapter
6.
Integration in Functional Spaces
289
§1.
Gaussian measures
289
1.1.
Finite-dimensional case
289
1.2.
Infinite-dimensional case
291
§2.
Wiener measure and Wiener integral
293
2.1.
Definition of the Wiener measure
293
2.2.
Conditional Wiener measure and Feynman-Kac formula
296
2.3.
Relation between Wiener and Feynman integrals
299
§3.
Gaussian Wiener integrals
301
3.1.
Dirichlet boundary conditions
301
3.2.
Periodic boundary conditions
302
§4.
Notes and references
305
Chapter
7.
Fermion Systems
307
§1.
Canonical
anticommutation
relations
307
1.1.
Motivation
307
1.2.
Clifford algebras
311
§2. Grassmann
algebras
314
2.1.
Realization of canonical
anticommutation
relations
315
2.2.
Differential forms
317
2.3.
Berezin integral
319
§3.
Graded linear algebra
324
3.1.
Graded vector spaces and superalgebras
324
3.2.
Examples of superalgebras
326
3.3.
Supertrace
and Berezinian
328
§4.
Path integrals for anticommuting variables
330
4.1.
Wick and matrix symbols
330
4.2.
Path integral for the evolution operator
335
4.3.
Gaussian path integrals over
Grassmann
variables
338
§5.
Notes and references
340
Chapter
8.
Supersymmetry
343
§1.
Supermanifolds
343
§2.
Equivariant cohomology and localization
345
2.1.
Finite-dimensional case
345
2.2.
Infinite-dimensional case
349
Contents xi
§3.
Classical mechanics on supermanifolds
354
3.1.
Functions with anticommuting values
354
3.2.
Classical systems
356
§4.
Supersymmetry
359
4.1.
Total angular momentum
359
4.2.
Supersymmetry transformation
360
4.3.
Supersymmetric particle on a Riemannian manifold
362
§5.
Quantum mechanics on supermanifolds
364
§6.
Atiyah-Singer index formula
369
§7.
Notes and references
370
Bibliography
373
Index
383 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Takhtajan, Leon A. 1950- |
| author_GND | (DE-588)123335868 |
| author_facet | Takhtajan, Leon A. 1950- |
| author_role | aut |
| author_sort | Takhtajan, Leon A. 1950- |
| author_variant | l a t la lat |
| building | Verbundindex |
| bvnumber | BV035086723 |
| callnumber-first | Q - Science |
| callnumber-label | QC174 |
| callnumber-raw | QC174.12 |
| callnumber-search | QC174.12 |
| callnumber-sort | QC 3174.12 |
| callnumber-subject | QC - Physics |
| classification_rvk | SK 950 UK 1200 |
| classification_tum | PHY 011f PHY 020f |
| ctrlnum | (OCoLC)613124775 (DE-599)BVBBV035086723 |
| dewey-full | 530.12 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.12 |
| dewey-search | 530.12 |
| dewey-sort | 3530.12 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| discipline_str_mv | Physik Mathematik |
| format | Book |
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| id | DE-604.BV035086723 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T22:09:10Z |
| indexdate | 2024-07-09T21:21:53Z |
| institution | BVB |
| isbn | 9780821846308 |
| language | English |
| lccn | 2008013072 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016754896 |
| oclc_num | 613124775 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-384 DE-20 DE-29T DE-83 DE-11 DE-188 DE-91G DE-BY-TUM DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-384 DE-20 DE-29T DE-83 DE-11 DE-188 DE-91G DE-BY-TUM DE-19 DE-BY-UBM |
| physical | XV, 387 S. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Takhtajan, Leon A. 1950- Verfasser (DE-588)123335868 aut Quantum mechanics for mathematicians Leon A. Takhtajan Providence, R.I. American Mathematical Society 2008 XV, 387 S. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 95 Includes bibliographical references and index Mathematische Physik Quantentheorie Quantum theory Mathematical physics Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Mathematische Physik (DE-588)4037952-8 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 s Mathematische Physik (DE-588)4037952-8 s DE-604 Graduate studies in mathematics 95 (DE-604)BV009739289 95 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754896&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Takhtajan, Leon A. 1950- Quantum mechanics for mathematicians Graduate studies in mathematics Mathematische Physik Quantentheorie Quantum theory Mathematical physics Quantenmechanik (DE-588)4047989-4 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| subject_GND | (DE-588)4047989-4 (DE-588)4037952-8 |
| title | Quantum mechanics for mathematicians |
| title_auth | Quantum mechanics for mathematicians |
| title_exact_search | Quantum mechanics for mathematicians |
| title_exact_search_txtP | Quantum mechanics for mathematicians |
| title_full | Quantum mechanics for mathematicians Leon A. Takhtajan |
| title_fullStr | Quantum mechanics for mathematicians Leon A. Takhtajan |
| title_full_unstemmed | Quantum mechanics for mathematicians Leon A. Takhtajan |
| title_short | Quantum mechanics for mathematicians |
| title_sort | quantum mechanics for mathematicians |
| topic | Mathematische Physik Quantentheorie Quantum theory Mathematical physics Quantenmechanik (DE-588)4047989-4 gnd Mathematische Physik (DE-588)4037952-8 gnd |
| topic_facet | Mathematische Physik Quantentheorie Quantum theory Mathematical physics Quantenmechanik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016754896&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
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