Non-relativistic quantum mechanics:
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| Format: | Book |
| Language: | English |
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Cambridge
Cambridge University Press
2017
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | xvii, 438 Seiten |
| ISBN: | 9781107164369 |
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Record in the Search Index
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| adam_text | Contents
A
Preface XV
1. History of Quantum Mechanics 1
1.1 Failure of Classical Physics 1
1.1.1 Black-Body Radiation 1
1.1.2 Photoelectric Effect 7
1.1.3 Einstein’s Theory of Photoelectric Effect 8
1.1.4 Atomic Structure 9
1.2 Birth of Matrix Mechanics 10
1.2.1 Heisenberg’s Theory 10
1.2.2 Bom and Jordon’s Theory 16
1.2.3 Dirac’s Formulation 21
1.3 Birth of Wave Mechanics 22
1.3.1 de Broglie’s Hypothesis 22
1.3.2 From Hamilton-Jacobi to Schrodinger Equation 24
1.3.3 Interpretation of Wave Function 29
2. Vectors and Operators 31
2.1 Vector Space 31
2.1.1 Subspaces 32
2.1.2 Dimension of a Vector Space 32
2.2 Scalar Product Space 34
2.2.1 Hilbert Space 36
2.2.2 Representation of Vectors 37
2.3 Operators 38
2.3.1 Representation of Operators 38
2.3.2 Operator Algebra 40
2.4 Operator Calculus 44
2.5 Lie Algebraic Similarity Transformation 48
2.5.1 Harmonic Oscillator Algebra 48
2.5.2 SU(2) Algebra 49
2.6 Disentangling an Exponential 51
2.6.1 Harmonic Oscillator Algebra 52
2.6.2 SU(2) Algebra 53
viii
Contents
2.7 Time-Ordered Exponential Integral 54
2.7.1 Harmonic Oscillator Algebra 55
2.7.2 SU(2) Algebra 55
2.8 Entangling Exponentials: Baker-Campbell-Hausdroff Formula 56
3. Finite Dimensional Spaces 61
3.1 Matrix Algebra 61
3.1.1 Triangular and Normal Matrices 62
3.2 Determinant 64
3.2.1 Inverse of a Matrix 65
3.2.2 System of Linear Equations 65
3.3 Eigenvalue Problem of a Matrix 66
3.3.1 Cayley-Hamilton Theorem 70
3.4 Canonical Forms of a Matrix 71
3.4.1 Schur Decomposition 72
3.4.2 Canonical Form of Normal Matrices 74
3.4.3 Canonical Form of Non-Normal Matrices 74
3.4.4 Spectral Decomposition 76
3.5 Polar and Singular Value Decompositions 77
3.5.1 Polar Decomposition 77
3.5.2 Singular Value Decomposition 79
3.6 Two-Dimensional Space 81
3.6.1 Representation of Vectors 81
3.6.2 Representation of Operators 82
4. Function Space 87
4.1 Space of Square Integrable Functions 87
4.2 Spectrum 88
4.2.1 Spectrum of Self-Adjoint Operators 89
4.3 Spectrum of Multiplication Operator X(x) 91
4.3.1 Eigenfunction as a Sequence of Square Integrable Functions 92
4.3.2 Delta Function Normalizable Eigenfunctions 93
4.4 Spectrum of id/dx 93
4.4.1 Eigenfunction as a Sequence of Square Integrable Functions 95
4.4.2 Delta Function Normalizable Eigenfunctions 95
4.5 Spectrum of-d2/dx2 96
5. Postulates of Quantum Mechanics 99
5.1 Postulate 1: On Representing an Isolated System 99
5.2 Postulate 2: On Representing Dynamical Variables 99
5.3 Postulate 3: On Commutation Relation 99
5.4 Postulate 4: On Measurement 103
5.4.1 Uncertainty Relation 104
5.4.2 Minimum Uncertainty States 105
5.4.3 No-Cloning Theorem 107
Contents
ix
5.5 Postulate 5: On Time-Evolution 108
5.5.1 Heisenberg Picture 109
5.5.2 Interaction Picture 110
5.5.3 Multi-time Joint Probability 110
5.5.4 Quantum Zeno Effect 111
5.6 Composite Systems 112
5.6.1 Representation of States and Operators 112
5.6.2 Schmidt Decomposition 114
5.6.3 Operators on Composite System 116
5.7 Feynman Path Integral 117
5.8 Positive Operator-Valued Measure (POVM) 118
5.8.1 Distinguishing Two Non-Orthogonal States of a Spin-1/2 120
5.8.2 Distinguishing States of M-Level System 122
6. Density Operator 125
6.1 State of a Subsystem 125
6.1.1 Spectrum of Density Operator 128
6.1.2 Reduced Density Operator 130
6.1.3 Constructing Schmidt Bases 131
6.1.4 Density Operator of a Spin-1/2 Particle 131
6.1.5 Density Operator of Two Spin-1/2s 133
6.2 Mixed States 137
6.2.1 Pure and Mixed States in Stern-Gerlach Experiment 138
6.2.2 Density Operator after Measurement 140
7. Measurement Postulate and Paradoxes of Quantum Mechanics 143
7.1 The Measurement Problem 143
7.2 Schrôdinger’s Cat Paradox 146
7.3 EPR Paradox 147
8* Position and Momentum Representations 149
8.1 Position Representation 149
8.1.1 Representation of Operators 150
8.2 Conditions on Functions Representing State Vectors 151
8.3 Momentum Space Representation 152
8.4 Generalization to Three Dimensions 154
8.4.1 Position Representation 154
8.4.2 Momentum Space Representation 155
8.4.3 Finding Expectation Values 156
8.5 Minimum Uncertainty States of x and p 158
8.6 Schrôdinger Equation in Position Representation 159
8.6.1 Probability Current 161
8.6.2 Time-Dependent Potentials 162
X
Contents
9. Schrôdinger Equation in One Dimension 165
9.1 Asymptotic Behaviour 165
9.2 One-dimensional Bound States 166
9.2.1 Non-Degeneracy 166
9.2.2 Parity 166
9.3 Continuity Conditions 167
9.4 Zeros of the Solutions 168
9.4.1 U(x) e 168
9.4.2 U(x) s 169
9.5 Homogeneous Boundary Conditions 172
10. One-Dimensional Piecewise Constant Potentials 177
10.1 Constant Potential 177
10.1.1 Motion in Free Space 178
10.2 Reflection and Transmission Coefficients 179
10.3 Step Potential 180
10.4 Square Well Potential 182
10.5 Potential Barrier 190
10.6 Particle in a Box 192
10.6.1 Variance in Energy 193
10.6.2 Momentum Distribution Function 195
10.6.3 Moving Boundary 197
10.7 Delta-Function Potential 198
10.7.1 Moving Delta-Function Potential 199
11. One-Dimensional Exactly Solvable Continuous Potentials 201
11.1 Potentials for which Schrôdinger Equation is Reducible to Hypergeometric Form 201
11.1.1 Rosen-Morse Potential 205
11.1.2 Pöschl-Teller Potential 208
11.2 Potentials for which Schrôdinger Equation is Reducible to Confluent
Hypergeometric Form 210
11.2.1 Morse Potential 213
11.2.2 One-Dimensional Harmonic Oscillator 214
11.3 Potentials Solvable Exactly by Energy-Dependent Transformation 217
11.3.1 Linear Potential 219
11.3.2 Potential of the Form x , n 2 222
12. Partially and Completely Periodic Potentials 225
12.1 Potential Periodic in a Finite Region of Space 225
12.1.1 Periodic Potential inside Infinitely Deep Well 229
12.1.2 Array of Square Wells and Barriers 230
12.1.3 Array of Delta Function Potentials 232
12.1.4 Scattering States 233
12.1.5 Limit of Completely Periodic Potential 234
Contents
xi
12.2 Completely Periodic Potential 237
12.2.1 Eigenvalue Spectrum 239
12.2.2 Kronig-Penney Model 240
12.2.3 Dirac Comb 242
13. Harmonic Oscillator 243
13.1 Hilbert Space Operators Method 243
13.2 Ladder Operators in Position Representation 247
13.2.1 Finding Expectation Values of H.O. Operators 250
13.3 Time Evolution 252
13.4 Schrodinger’s Classical H.O. States 252
13.5 Coherent and Squeezed States 255
14. Three-Dimensional Central Potentials 259
14.1 Spherical Polar Coordinates 259
14.2 Radial Momentum 262
14.2.1 Classical Radial and Angular Momenta 262
14.2.2 Quantum Radial Momentum 262
14.3 Solution of Schrodinger Equation for Central Potentials 264
14.4 Spherical Harmonics 265
14.5 Angular Momentum States of Free Particle 269
14.6 Three-Dimensional Symmetric Potential Well 270
14.6.1 Solution in Interior Region 271
14.6.2 Solution in Exterior Region 271
14.6.3 Energy Eigenvalues 272
14.6.4 Infinite Potential Well 272
14.7 Isotropic Harmonic Oscillator 272
14.8 Hydrogen Atom 274
15. Symmetry in Quantum Mechanics 281
15.1 Symmetry Transformation 281
15.1.1 Wigner’s Theorem 282
15.2 Symmetries, Conservation Laws and Degeneracies 283
15.3 Continuous Transformations 284
15.3.1 Special Unitary Group SU(ri) 284
15.3.2 Real Orthogonal Transformation 0(n) 285
15.4 Symmetry Transformations in Position Space 288
15 A1 Translational Symmetry 288
15.4.2 Rotational Symmetry 289
15.4.3 Space Inversion Symmetry: Parity 291
15.5 Time-Reversal Symmetry 294
15.6 Dynamical Symmetry 298
15.6.1 Hydrogen Atom 299
15.6.2 Isotropic Harmonic Oscillator 300
xii Contents
16. Quantum Theory of Angular Momentum 305
16.1 Hilbert Space of Angular Momentum 305
16.2 Rotation Operator 312
16.2.1 In Terms of Euler Angles 315
16.2.2 Relationship between Angular Momentum Eigenstates in Two Directions 316
16.3 Minimum Uncertainty States 318
16.3.1 Spin Coherent States 319
16.3.2 Squeezed Spin States 320
16.4 Time Evolution 322
16.5 Ladder Operators in Position Representation 323
16.6 Comparison of Orbital Angular Momentum and Spin 326
16.7 Addition of Angular Momenta 327
16.7.1 C-G Coefficients without Recursion Relations 331
16.7.2 Addition of jx = j2 = 1/2 332
16.7.3 Addition of jx = /, j2 = 1/2 333
16.7.4 Addition of j - /, j2 = 1 334
17. Approximation Methods 337
17.1 WKB Approximation 337
17.1.1 Connection Formulas 339
17.1.2 Potential Well 341
17.1.3 Tunnelling 343
17.1.4 Infinite Potential on One or Both Ends 344
17.2 Variational Method 345
17.3 Stationary Perturbation 346
17.3.1 Non-Degenerate Perturbation Theory 348
17.3.2 Degenerate Perturbation Theory 349
17.3.3 Fine Structure of Hydrogen 353
17.3.4 Zeeman Effect 357
17.4 Time-Dependent Perturbation Theory 358
17.4.1 Sinusoidal Perturbation 360
17.5 Adiabatic Approximation 364
17.5.1 Berry’s Phase 366
17.5.2 Geometric Phase 367
17.6 Sudden Approximation 374
18. Entanglement and Local Hidden Variable Theory 377
18.1 Entanglement 377
18.1.1 Separability of a Pure State 378
18.1.2 Separability of a Mixed State: Peres Criterion 378
18.1.3 Measure of Entanglement 380
18.2 Local Hidden Variable Theory 383
18.2.1 Bell’s Inequality 384
18.3 Joint Quasiprobabilities 387
Contents xiii
18.4 Joint Quasiprobability-Based Criterion for Classification of States of Spin-l/2s 392
18.5 Quantumness of Mutual Information 395
18.5.1 Classical Information 395
18.5.2 Quantum Information 398
18.5.3 Quantum Discord 401
18.5.4 Measurement Protocols-Based Quantumness 402
18.5.5 LHV Theoretic Measure of Quantumness of Mutual Information 403
Appendices
A. Delta Function 409
B. Second-Order Ordinary Differential Equations 413
В. 1 Ordinary and Singular Points 415
В .2 Power Series Solution Near Ordinary Point 415
B.3 Frobenius Series Solution Near a Regular Singularity 416
В .4 Three Regular Singularities 417
B.5 A Regular and a Rank One Singularity 421
C. Riccati Equation 425
D. Some Mathematical Formulas 427
References
Bibliography
Index
429
433
435
Non-Relativistic Quantum Mechanics
Quantum mechanics has unraveled several mysteries of nature, on the scale from the
subatomic to astronomical, and in diverse areas ranging from device physics to biology.
This book introduces the reader to non-relativistic quantum mechanics and its mathematical
methods. It begins by recounting the main historical developments leading to the
formulation of the quantum postulates. Starting with postulates, a wide range of
fundamental topics and topics of current research interests such as quantum information
are discussed in detail. Relevant topics in mathematics such as linear algebra, problem
of disentanglement of exponential of sum of operators, and the theory of second-order
ordinary linear differential equations are discussed at length.
The theory of second-order ordinary linear differential equations is presented by classifying
them according to their singularity structure. This approach is used to identify and solve
time-independent exactly solvable continuous one-dimensional potentials. A number of
exercises are given in each chapter, with hints wherever necessary.
Ravinder R. Puri is visiting professor in the discipline of Physics, Indian Institute of
Technology Gandhinagar, India. During 1974-2012 he served in various positions in
Bhabha Atomic Research Centre, Mumbai, India, and during 2004—2012 served also as
Dean, Homi Bhabha National Institute. He also served as a distinguished guest professor
at the Indian Institute of Technology Bombay, as a visiting professor at the University
of Manchester Institute of Science and Technology (UMIST), United Kingdom and was
a Humboldt Fellow in the University of Essen, Germany. His research interests include
theoretical quantum optics, foundations of quantum mechanics and quantum information
processing.
|
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| dewey-ones | 530 - Physics |
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| dewey-search | 530.12 |
| dewey-sort | 3530.12 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
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| spelling | Puri, Ravinder Rupchand 1950- Verfasser (DE-588)122132084 aut Non-relativistic quantum mechanics Ravinder R. Puri Cambridge Cambridge University Press 2017 xvii, 438 Seiten txt rdacontent n rdamedia nc rdacarrier Quantenmechanik (DE-588)4047989-4 gnd rswk-swf Nonrelativistic quantum mechanics 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=029775846&sequence=000001&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=029775846&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Puri, Ravinder Rupchand 1950- Non-relativistic quantum mechanics Quantenmechanik (DE-588)4047989-4 gnd |
| subject_GND | (DE-588)4047989-4 |
| title | Non-relativistic quantum mechanics |
| title_auth | Non-relativistic quantum mechanics |
| title_exact_search | Non-relativistic quantum mechanics |
| title_full | Non-relativistic quantum mechanics Ravinder R. Puri |
| title_fullStr | Non-relativistic quantum mechanics Ravinder R. Puri |
| title_full_unstemmed | Non-relativistic quantum mechanics Ravinder R. Puri |
| title_short | Non-relativistic quantum mechanics |
| title_sort | non relativistic quantum mechanics |
| topic | Quantenmechanik (DE-588)4047989-4 gnd |
| topic_facet | Quantenmechanik |
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