Essential quantum mechanics:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2008
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 208 S. graph. Darst. |
| ISBN: | 9780199228935 0199228930 9780199228928 |
Staff View
MARC
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| 100 | 1 | |a Bowman, Gary E. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Essential quantum mechanics |c Gary E. Bowman |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2008 | |
| 300 | |a XI, 208 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
Preface
ix
1
Introduction:
Three Worlds
1
1.1
Worlds
1
and
2 1
1.2
World
З З
1.3
Problems
4
2
The Quantum Postulates
5
2.1
Postulate
1:
The Quantum. State
6
2.2
Postulate
2:
Observables,
Operators, and Eigenstates
8
2.3
Postulate
3:
Quantum Superpositions
10
2.3.1
Discrete Eigenvalues
11
2.3.2
Continuous Eigenvalues
12
2.4
Closing Comments
15
2.5
Problems
16
3
What Is a Quantum State?
19
3.1
Probabilities, Averages, and Uncertainties
19
3.1.1
Probabilities
19
3.1.2
Averages
22
3.1.3
Uncertainties
24
3.2
The Statistical Interpretation
26
3.3
Bohr, Einstein, and Hidden Variables
28
3.3.1
Background
28
3.3.2 Fundamental
Issues
30
3.3.3
Einstein Revisited
32
3.4
Problems
33
4
The Structure of Quantum States
36
4.1
Mathematical Preliminaries
36
4.1.1
Vector Spaces
36
4.1.2
Function Spaces
39
4.2
Dirac s Bra-ket Notation
41
4.2.1
Bras and
Kets
41
4.2.2
Labeling States
42
4.3
The Scalar Product
43
4.3.1
Quantum Scalar Products
43
4.3.2
Discussion
45
vi
Contents
4.4
Representations
47
4.4.1
Basics
47
4.4.2
Superpositions and Representations
48
4.4.3
Representational Freedom
50
4.5
Problems
52
Operators
53
5.1
Introductory Comments
54
5.2
Hermitian Operators
56
5.2.1
Adjoint Operators
56
5.2.2
Hermitian Operators: Definition and Properties
57
5.2.3
Wavefunctions and Hermitian Operators
59
5.3
Projection and Identity Operators
61
5.3.1
Projection Operators
61
5.3.2
The Identity Operator
62
5.4
Unitary Operators
62
5.5
Problems
64
Matrix
Mechanics
68
6.1
Elementary Matrix Operations
68
6.1.1
Vectors and Scalar Products
68
6.1.2
Matrices and Matrix Multiplication
69
6.1.3
Vector Transformations
70
6.2
States as Vectors
71
6.3
Operators as Matrices
72
6.3.1
An Operator in Its
Eigenbasis
72
6.3.2
Matrix Elements and Alternative Bases
73
6.3.3
Change of Basis
75
6.3.4
Adjoint, Hermitian, and Unitary Operators
75
6.4
Eigenvalue Equations
77
6.5
Problems
78
Commutators and Uncertainty Relations
82
7.1
The Commutator
83
7.1.1
Definition and Characteristics
83
7.1.2
Commutators in Matrix Mechanics
85
7.2
The Uncertainty Relations
86
7.2.1
Uncertainty Products
86
7.2.2
General Form of the Uncertainty Relations
87
7.2.3
Interpretations
88
7.2.4
Reflections
91
7.3
Problems
93
Angular Momentum
95
8.1
Angular Momentum in Classical Mechanics
95
8.2
Basics of Quantum Angular Momentum
97
8.2.1
Operators and Commutation Relations
97
Contents
vii
8.2.2 Eigenstates
and Eigenvalues
99
8.2.3
Raising and Lowering Operators
100
8.3
Physical Interpretation
101
8.3.1
Measurements
101
8.3.2
Relating L2 and Lz
104
8.4
Orbital and Spin Angular Momentum
106
8.4.1
Orbital Angular Momentum
106
8.4.2
Spin Angular Momentum
107
8.5
Review
107
8.6
Problems
108
9
The Time-Independent
Schrödinger
Equation 111
9.1
An Eigenvalue Equation for Energy
112
9.2
Using the
Schrödinger
Equation
114
9.2.1
Conditions on Wavefunctions
114
9.2.2
An Example: the Infinite Potential Well
115
9.3
Interpretation
117
9.3.1
Energy Eigenstates in Position Space
117
9.3.2
Overall and Relative Phases
118
9.4
Potential Barriers and Tunneling
120
9.4.1
The Step Potential
120
9.4.2
The Step Potential and Scattering
122
9.4.3
Tunneling
124
9.5
What s Wrong with This Picture?
125
9.6
Problems
126
10
Why Is the State Complex?
128
10.1
Complex Numbers
129
10.1.1
Basics
129
10.1.2
Polar Form
130
10.1.3
Argand Diagrams and the Role of the Phase
131
10.2
The Phase in Quantum Mechanics
133
10.2.1
Phases and the Description of States
133
10.2.2
Phase Changes and Probabilities
135
10.2.3
Unitary Operators Revisited
136
10.2.4
Unitary Operators, Phases, and Probabilities
137
10.2.5
Example: A Spin System
139
10.3
Wavefunctions
141
10.4
Reflections
142
10.5
Problems
143
11
Time Evolution
145
11.1
The Time-Dependent
Schrödinger
Equation
145
11.2
How Time Evolution Works
146
11.2.1
Time Evolving a Quantum State
146
11.2.2
Unitarity and Phases Revisited
148
viii Contents
11.3
Expectation Values
149
11.3.1
Time Derivatives
149
11.3.2
Constants of the Motion
150
11.4
Energy-Time Uncertainty Relations
151
11.4.1
Conceptual Basis
151
11.4.2
Spin
:
An Example
153
11.5
Problems
154
12
Wavefunctions
157
12.1
What is a Wavefunction?
158
12.1.1
Eigenstates and Coefficients
158
12.1.2
Representations and Operators
159
12.2
Changing Representations
161
12.2.1
Change of Basis Revisited
161
12.2.2
From
χ
to
ρ
and Back Again
161
12.2.3
Gaussiane
and Beyond
163
12.3
Phases and Time Evolution
165
12.3.1
Free Particle Evolution
165
12.3.2
Wavepackets
167
12.4
Bra-ket Notation
168
12.4.1
Quantum States
168
12.4.2
Eigenstates and Transformations
170
12.5
Epilogue
171
12.6
Problems
172
A Mathematical Concepts
175
A.I Complex Numbers and Functions
175
A.2 Differentiation
176
A.3 Integration
178
A.4 Differential Equations
180
В
Quantum Measurement
183
С
The Harmonic Oscillator
186
C.I Energy Eigenstates and Eigenvalues
186
C.2 The Number Operator and its Cousins
188
C.3 Photons as Oscillators
189
D
Unitary Transformations
192
D.I Unitary Operators
192
D.2 Finite Transformations and Generators
195
D.3 Continuous Symmetries
197
D.3.1 Symmetry Transformations
197
D.3.
2
Symmetries of Physical Law
197
D.3.3 System Symmetries
199
Bibliography
201
Index
205
|
| adam_txt |
Contents
Preface
ix
1
Introduction:
Three Worlds
1
1.1
Worlds
1
and
2 1
1.2
World
З З
1.3
Problems
4
2
The Quantum Postulates
5
2.1
Postulate
1:
The Quantum. State
6
2.2
Postulate
2:
Observables,
Operators, and Eigenstates
8
2.3
Postulate
3:
Quantum Superpositions
10
2.3.1
Discrete Eigenvalues
11
2.3.2
Continuous Eigenvalues
12
2.4
Closing Comments
15
2.5
Problems
16
3
What Is a Quantum State?
19
3.1
Probabilities, Averages, and Uncertainties
19
3.1.1
Probabilities
19
3.1.2
Averages
22
3.1.3
Uncertainties
24
3.2
The Statistical Interpretation
26
3.3
Bohr, Einstein, and Hidden Variables
28
3.3.1
Background
28
3.3.2 Fundamental
Issues
30
3.3.3
Einstein Revisited
32
3.4
Problems
33
4
The Structure of Quantum States
36
4.1
Mathematical Preliminaries
36
4.1.1
Vector Spaces
36
4.1.2
Function Spaces
39
4.2
Dirac's Bra-ket Notation
41
4.2.1
Bras and
Kets
41
4.2.2
Labeling States
42
4.3
The Scalar Product
43
4.3.1
Quantum Scalar Products
43
4.3.2
Discussion
45
vi
Contents
4.4
Representations
47
4.4.1
Basics
47
4.4.2
Superpositions and Representations
48
4.4.3
Representational Freedom
50
4.5
Problems
52
Operators
53
5.1
Introductory Comments
54
5.2
Hermitian Operators
56
5.2.1
Adjoint Operators
56
5.2.2
Hermitian Operators: Definition and Properties
57
5.2.3
Wavefunctions and Hermitian Operators
59
5.3
Projection and Identity Operators
61
5.3.1
Projection Operators
61
5.3.2
The Identity Operator
62
5.4
Unitary Operators
62
5.5
Problems
64
Matrix
Mechanics
68
6.1
Elementary Matrix Operations
68
6.1.1
Vectors and Scalar Products
68
6.1.2
Matrices and Matrix Multiplication
69
6.1.3
Vector Transformations
70
6.2
States as Vectors
71
6.3
Operators as Matrices
72
6.3.1
An Operator in Its
Eigenbasis
72
6.3.2
Matrix Elements and Alternative Bases
73
6.3.3
Change of Basis
75
6.3.4
Adjoint, Hermitian, and Unitary Operators
75
6.4
Eigenvalue Equations
77
6.5
Problems
78
Commutators and Uncertainty Relations
82
7.1
The Commutator
83
7.1.1
Definition and Characteristics
83
7.1.2
Commutators in Matrix Mechanics
85
7.2
The Uncertainty Relations
86
7.2.1
Uncertainty Products
86
7.2.2
General Form of the Uncertainty Relations
87
7.2.3
Interpretations
88
7.2.4
Reflections
91
7.3
Problems
93
Angular Momentum
95
8.1
Angular Momentum in Classical Mechanics
95
8.2
Basics of Quantum Angular Momentum
97
8.2.1
Operators and Commutation Relations
97
Contents
vii
8.2.2 Eigenstates
and Eigenvalues
99
8.2.3
Raising and Lowering Operators
100
8.3
Physical Interpretation
101
8.3.1
Measurements
101
8.3.2
Relating L2 and Lz
104
8.4
Orbital and Spin Angular Momentum
106
8.4.1
Orbital Angular Momentum
106
8.4.2
Spin Angular Momentum
107
8.5
Review
107
8.6
Problems
108
9
The Time-Independent
Schrödinger
Equation 111
9.1
An Eigenvalue Equation for Energy
112
9.2
Using the
Schrödinger
Equation
114
9.2.1
Conditions on Wavefunctions
114
9.2.2
An Example: the Infinite Potential Well
115
9.3
Interpretation
117
9.3.1
Energy Eigenstates in Position Space
117
9.3.2
Overall and Relative Phases
118
9.4
Potential Barriers and Tunneling
120
9.4.1
The Step Potential
120
9.4.2
The Step Potential and Scattering
122
9.4.3
Tunneling
124
9.5
What's Wrong with This Picture?
125
9.6
Problems
126
10
Why Is the State Complex?
128
10.1
Complex Numbers
129
10.1.1
Basics
129
10.1.2
Polar Form
130
10.1.3
Argand Diagrams and the Role of the Phase
131
10.2
The Phase in Quantum Mechanics
133
10.2.1
Phases and the Description of States
133
10.2.2
Phase Changes and Probabilities
135
10.2.3
Unitary Operators Revisited
136
10.2.4
Unitary Operators, Phases, and Probabilities
137
10.2.5
Example: A Spin \ System
139
10.3
Wavefunctions
141
10.4
Reflections
142
10.5
Problems
143
11
Time Evolution
145
11.1
The Time-Dependent
Schrödinger
Equation
145
11.2
How Time Evolution Works
146
11.2.1
Time Evolving a Quantum State
146
11.2.2
Unitarity and Phases Revisited
148
viii Contents
11.3
Expectation Values
149
11.3.1
Time Derivatives
149
11.3.2
Constants of the Motion
150
11.4
Energy-Time Uncertainty Relations
151
11.4.1
Conceptual Basis
151
11.4.2
Spin \
:
An Example
153
11.5
Problems
154
12
Wavefunctions
157
12.1
What is a Wavefunction?
158
12.1.1
Eigenstates and Coefficients
158
12.1.2
Representations and Operators
159
12.2
Changing Representations
161
12.2.1
Change of Basis Revisited
161
12.2.2
From
χ
to
ρ
and Back Again
161
12.2.3
Gaussiane
and Beyond
163
12.3
Phases and Time Evolution
165
12.3.1
Free Particle Evolution
165
12.3.2
Wavepackets
167
12.4
Bra-ket Notation
168
12.4.1
Quantum States
168
12.4.2
Eigenstates and Transformations
170
12.5
Epilogue
171
12.6
Problems
172
A Mathematical Concepts
175
A.I Complex Numbers and Functions
175
A.2 Differentiation
176
A.3 Integration
178
A.4 Differential Equations
180
В
Quantum Measurement
183
С
The Harmonic Oscillator
186
C.I Energy Eigenstates and Eigenvalues
186
C.2 The Number Operator and its Cousins
188
C.3 Photons as Oscillators
189
D
Unitary Transformations
192
D.I Unitary Operators
192
D.2 Finite Transformations and Generators
195
D.3 Continuous Symmetries
197
D.3.1 Symmetry Transformations
197
D.3.
2
Symmetries of Physical Law
197
D.3.3 System Symmetries
199
Bibliography
201
Index
205 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Bowman, Gary E. |
| author_facet | Bowman, Gary E. |
| author_role | aut |
| author_sort | Bowman, Gary E. |
| author_variant | g e b ge geb |
| building | Verbundindex |
| bvnumber | BV022822502 |
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| callnumber-subject | QC - Physics |
| classification_rvk | UK 1000 |
| classification_tum | PHY 020f |
| ctrlnum | (OCoLC)156994574 (DE-599)GBV538552328 |
| dewey-full | 530.12 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
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| dewey-search | 530.12 |
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| discipline | Physik |
| discipline_str_mv | Physik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV022822502 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:40:37Z |
| indexdate | 2024-07-09T21:06:56Z |
| institution | BVB |
| isbn | 9780199228935 0199228930 9780199228928 |
| language | English |
| lccn | 007030092 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016027838 |
| oclc_num | 156994574 |
| open_access_boolean | |
| owner | DE-20 DE-703 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-634 DE-11 DE-91G DE-BY-TUM |
| owner_facet | DE-20 DE-703 DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-634 DE-11 DE-91G DE-BY-TUM |
| physical | XI, 208 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| spelling | Bowman, Gary E. Verfasser aut Essential quantum mechanics Gary E. Bowman 1. publ. Oxford [u.a.] Oxford Univ. Press 2008 XI, 208 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Quantum theory Quantentheorie Quantenmechanik (DE-588)4047989-4 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Quantenmechanik (DE-588)4047989-4 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016027838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bowman, Gary E. Essential quantum mechanics Quantum theory Quantentheorie Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4047989-4 (DE-588)4123623-3 |
| title | Essential quantum mechanics |
| title_auth | Essential quantum mechanics |
| title_exact_search | Essential quantum mechanics |
| title_exact_search_txtP | Essential quantum mechanics |
| title_full | Essential quantum mechanics Gary E. Bowman |
| title_fullStr | Essential quantum mechanics Gary E. Bowman |
| title_full_unstemmed | Essential quantum mechanics Gary E. Bowman |
| title_short | Essential quantum mechanics |
| title_sort | essential quantum mechanics |
| topic | Quantum theory Quantentheorie Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Quantum theory Quantentheorie Quantenmechanik Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016027838&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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