Hilbert space and quantum mechanics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2015
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Item Description: | Literaturverz. S. 739 - 740 |
| Physical Description: | XIV, 746 S. graph. Darst. |
| ISBN: | 9789814635837 |
Staff View
MARC
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| 650 | 4 | |a Quantum theory | |
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Record in the Search Index
| _version_ | 1804153070029897728 |
|---|---|
| adam_text | Contents
Preface vii
1. Sets, Mappings, Groups 1
1.1 Symbols, sets, relations........................................ 1
1.1.1 Sets of numbers.......................................... 1
1.1.2 Proofs................................................... 2
1.1.3 Symbols and shorthand.................................... 3
1.1.4 Sets..................................................... 3
1.1.5 Relations................................................ 5
1.2 Mappings........................................................ 7
1.3 Groups......................................................... 18
2. Metric Spaces 21
2.1 Distance, convergent sequences................................. 21
2.2 Open sets...................................................... 23
2.3 Closed sets ................................................... 25
2.4 Continuous mappings............................................ 31
2.5 Characteristic functions of closed and of open sets ........... 32
2.6 Complete metric spaces ..................................... 35
2.7 Product of two metric spaces................................... 37
2.8 Compactness.................................................... 40
2.9 Connectedness ................................................. 47
3. Linear Operators in Linear Spaces 51
3.1 Linear spaces.................................................. 51
3.2 Linear operators............................................... 59
3.3 The algebra of linear operators................................ 65
4. Linear Operators in Normed Spaces 69
4.1 Normed spaces.................................................. 69
xi
K Hilbert Space and Quantum Mechanics
4.2 Bounded operators.............................-................ 74
4.3 The normed algebra of bounded operators........................
4.4 Closed operators.................................................. 8/
4.5 The spectrum of a linear operator................................. 91
4.6 Isomorphisms of normed spaces..................................... 94
5. The Extended Real Line 191
5.1 The extended real line as an ordered set..........................101
5.2 The extended real line as a metric space..........................102
5.3 Algebraic operations in R*........................................107
5.4 Series in [0, oo] ................................................HO
6. Measurable Sets and Measurable Functions 117
6.1 Semialgebras, algebras, a-algebras ...............................117
6.2 Measurable mappings...............................................133
6.3 Borel functions ..................................................147
7. Measures 151
7.1 Additive functions, premeasures, measures......................151
7.2 Outer measures....................................................158
7.3 Extension theorems.............................................162
7.4 Finite measures in metric spaces .................................168
8. Integration 177
8.1 Integration of positive functions..............................177
8.2 Integration of complex functions .................................191
8.3 Integration with respect to measures constructed from other
measures..........................................................201
8.4 Integration on product spaces..................................210
8.5 The Riesz-Markov theorem..........................................227
9. Lebesgue Measure 233
9.1 Lebesgue-Stieltjes and Lebesgue measures..........................233
9.2 Invariance properties of Lebesgue measure.........................239
9.3 The Lebesgue integral as an extension of the Riemann integral . . 243
10. Hilbert Spaces 247
10.1 Inner product spaces..............................................247
10.2 Orthogonality in inner product spaces...257
10.3 Completions, direct sums, unitary and antiunitary operators in
Hilbert spaces....................................................268
10.4 Orthogonality in Hilbert spaces...................................276
Contents xiii
10.5 The Riesz-Frechet theorem .........................................284
10.6 Complete orthonormal systems.......................................287
10.7 Separable Hilbert spaces...........................................294
10.8 The finite-dimensional case .......................................301
10.9 Projective Hilbert spaces and Wigner’s theorem....................304
11. L2 Hilbert Spaces 319
11.1 L2{X,A^)...........................................................319
11.2 L2(a,b) 325
11.3 L2( R).............................................................333
11.4 The Fourier transform on L2(R).....................................337
12. Adjoint Operators 355
12.1 Basic properties of adjoint operators..............................355
12.2 Adjoints and boundedness...........................................361
12.3 Adjoints and algebraic operations..................................362
12.4 Symmetric and self-adjoint operators...............................364
12.5 Unitary operators and adjoints.....................................377
12.6 The (7*-algebra of bounded operators in Hilbert space..............380
13. Orthogonal Projections and Projection Valued Measures 387
13.1 Orthogonal projections.............................................387
13.2 Orthogonal projections and subspaces ..............................393
13.3 Projection valued measures ........................................407
13.4 Extension theorems for projection valued mappings..................411
13.5 Product of commuting projection valued measures....................415
13.6 Spectral families and projection valued measures...................419
14. Integration with respect to a Projection Valued Measure 425
14.1 Integrals of bounded measurable functions..........................425
14.2 Integrals of general measurable functions..........................429
14.3 Sum, product, inverse, self-adjoint ness, unitarity of integrals .... 443
14.4 Spectral properties of integrals ..................................455
14.5 Multiplication operators......................................... 458
14.6 Change of variable. Unitary equivalence............................460
15. Spectral Theorems 463
15.1 The spectral theorem for unitary operators ........................463
15.2 The spectral theorem for self-adjoint operators....................475
15.3 Functions of a self-adjoint operator...............................483
15.4 Unitary equivalence................................................494
XIV
Hilbert Space and Quantum Mechanics
16. One-Parameter Unitary Groups and Stone’s Theorem
16.1 Continuous one-parameter unitary groups......................^5
16.2 Norm-continuous one-parameter unitary groups.................508
16.3 Unitary equivalence..........................................512
16.4 One-parameter groups of automorphisms........................5 i.2
17. Commuting Operators and Reducing Subspaces
U ^ t/-
17.1 Commuting operators .............................................529
17.2 Invariant and reducing subspaces...................................550
17.3 Irreducibility.....................................................567
18. Trace Class and Statistical Operators 571
18.1 Positive operators and polar decomposition..............................571
18.2 The trace class.........................................................578
18.3 Statistical operators...................................................596
19. Quantum Mechanics in Hilbert Space 611
19.1 Elements of a general statistical theory..............................612
19.2 States, propositions, observables in classical statistical theories . . 628
19.3 States, propositions, observables in quantum mechanics ...............636
19.4 State reduction in quantum mechanics..................................653
19.5 Compatible observables and uncertainty relations in quantum
mechanics.............................................................667
19.6 Time evolution in non-relativistic quantum mechanics..................688
20. Position and Momentum in Non-Relativistic Quantum Mechanics 697
20.1 The Weyl commutation relation....................................697
20.2 The Stone-von Neumann uniqueness theorem.........................703
20.3 Position and momentum as Galilei-covariant observables...........715
739
Bibliography
Index
741
The topics of this book are the mathematical foundations of
non-reiativistic quantum mechanics and the mathematical
theory they require. The main characteristic of the book is
that the mathematics is developed assuming familiarity with
elementary analysis only. Moreover, all the proofs are carried
out in detail. These features make the book easily accessible
to readers with only the mathematical training offered by
undergraduate education in mathematics or in physics, and
also ideal for individual study. The principles of quantum
mechanics are discussed with complete mathematical
accuracy and an effort is made to always trace them back to
the experimental reality that lies at their root. The treatment
of quantum mechanics is axiomatic, with definitions followed
by propositions proved in a mathematical fashion. No previous
knowledge of quantum mechanics is required. This book is
designed so that parts of it can be easily used for various
courses in mathematics and mathematical physics, as
suggested in the Preface.
The book is of interest to researchers and graduate students
in functional analysis, who can see how closely an important
part of their chosen field is linked with quantum mechanics,
and also to physicists, who can see how the abstract language
of functional analysis brings unity to the apparently distinct
approaches employed in quantum theory.
On the cover are portraits of David Hilbert (¡eft) and John von Neumann (right).
the founding fathers of the mathematical formulation of quantum mechanics.
|
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.733 |
| dewey-search | 515/.733 |
| dewey-sort | 3515 3733 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042409856 |
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| indexdate | 2024-07-10T01:20:45Z |
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| language | English |
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| spelling | Gallone, Franco 1944- Verfasser (DE-588)1068489952 aut Hilbert space and quantum mechanics Franco Gallone New Jersey [u.a.] World Scientific 2015 XIV, 746 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Literaturverz. S. 739 - 740 Mathematik Mathematische Physik Quantentheorie Hilbert space Quantum theory Mathematics Linear operators Mathematical physics Nonrelativistic quantum mechanics Hilbert-Raum (DE-588)4159850-7 gnd rswk-swf Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Hilbert-Raum (DE-588)4159850-7 s Quantenmechanik (DE-588)4047989-4 s DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027845397&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027845397&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Gallone, Franco 1944- Hilbert space and quantum mechanics Mathematik Mathematische Physik Quantentheorie Hilbert space Quantum theory Mathematics Linear operators Mathematical physics Nonrelativistic quantum mechanics Hilbert-Raum (DE-588)4159850-7 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4159850-7 (DE-588)4047989-4 |
| title | Hilbert space and quantum mechanics |
| title_auth | Hilbert space and quantum mechanics |
| title_exact_search | Hilbert space and quantum mechanics |
| title_full | Hilbert space and quantum mechanics Franco Gallone |
| title_fullStr | Hilbert space and quantum mechanics Franco Gallone |
| title_full_unstemmed | Hilbert space and quantum mechanics Franco Gallone |
| title_short | Hilbert space and quantum mechanics |
| title_sort | hilbert space and quantum mechanics |
| topic | Mathematik Mathematische Physik Quantentheorie Hilbert space Quantum theory Mathematics Linear operators Mathematical physics Nonrelativistic quantum mechanics Hilbert-Raum (DE-588)4159850-7 gnd Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Mathematik Mathematische Physik Quantentheorie Hilbert space Quantum theory Mathematics Linear operators Mathematical physics Nonrelativistic quantum mechanics Hilbert-Raum Quantenmechanik |
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