Elsevier S & T

Quantum Mechanics of Non-Hamiltonian and Dissipative Systems

Tarasov, Vasily

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Table of contents
  • Cover
  • Prefacev
  • Contentsvii
  • A Very Few Preliminaries1
  • 1. Potential and conservative systems1
  • 2. Hamiltonian and non-Hamiltonian classical systems2
  • 3. Examples of non-Hamiltonian systems2
  • 4. Non-Hamiltonian and dissipative classical systems3
  • 5. Non-Hamiltonian and dissipative quantum systems4
  • 6. Quantization of non-Hamiltonian and dissipative systems5
  • Part I: Quantum Kinematics9
  • Chapter 1. Quantum Kinematics of Bounded Observables11
  • 1.1. Observables and states11
  • 1.2. Pre-Hilbert and Hilbert spaces11
  • 1.3. Separable Hilbert space16
  • 1.4. Definition and examples of operators18
  • 1.5. Quantum kinematical postulates20
  • 1.6. Dual Hilbert space21
  • 1.7. Dirac's notations23
  • 1.8. Matrix representation of operator24
  • Chapter 2. Quantum Kinematics of Unbounded Observables27
  • 2.1. Deficiencies of Hilbert spaces27
  • 2.2. Spaces of test functions28
  • 2.3. Spaces of generalized functions31
  • 2.4. Rigged Hilbert space33
  • 2.5. Linear operators on a rigged Hilbert space36
  • 2.6. Coordinate representation40
  • 2.7. X-representation43
  • Chapter 3. Mathematical Structures in Quantum Kinematics47
  • 3.1. Mathematical structures47
  • 3.2. Order structures49
  • 3.3. Topological structures50
  • 3.4. Algebraic structures52
  • 3.5. Examples of algebraic structures57
  • 3.6. Mathematical structures in kinematics66
  • Chapter 4. Spaces of Quantum Observables69
  • 4.1. Space of bounded operators69
  • 4.2. Space of finite-rank operators71
  • 4.3. Space of compact operators72
  • 4.4. Space of trace-class operators74
  • 4.5. Space of Hilbert-Schmidt operators78
  • 4.6. Properties of operators from K1 (H) and K2 (H)79
  • 4.7. Set of density operators81
  • 4.8. Operator Hilbert space and Liouville space83
  • 4.9. Correlation functions87
  • 4.10. Basis for Liouville space88
  • 4.11. Rigged Liouville space91
  • Chapter 5. Algebras of Quantum Observables95
  • 5.1. Linear algebra95
  • 5.2. Associative algebra96
  • 5.3. Lie algebra96
  • 5.4. Jordan algebra98
  • 5.5. Involutive, normed and Banach algebras100
  • 5.6. C*-algebra103
  • 5.7. W*-algebra106
  • 5.8. JB-algebra111
  • 5.9. Hilbert algebra112
  • Chapter 6. Mathematical Structures on State Sets115
  • 6.1. State as functional on operator algebra115
  • 6.2. State on C*-algebra117
  • 6.3. Representations C*-algebra and states123
  • 6.4. Gelfand-Naimark-Segal construction124
  • 6.5. State on W*-algebra128
  • Chapter 7. Mathematical Structures in Classical Kinematics129
  • 7.1. Symplectic structure129
  • 7.2. Poisson manifold and Lie-Jordan algebra130
  • 7.3. Classical states133
  • 7.4. Classical observables and C*-algebra136
  • Chapter 8. Quantization in Kinematics139
  • 8.1. Quantization and its properties139
  • 8.2. Heisenberg algebra147
  • 8.3. Weyl system and Weyl algebra149
  • 8.4. Weyl and Wigner operator bases152
  • 8.5. Differential operators and symbols157
  • 8.6. Weyl quantization mapping159
  • 8.7. Kernel and symbol of Weyl ordered operator161
  • 8.8. Weyl symbols and Wigner representation162
  • 8.9. Inverse of quantization map165
  • 8.10. Symbols of operators and Weyl quantization166
  • 8.11. Generalization of Weyl quantization174
  • Chapter 9. Spectral Representation of Observable181
  • 9.1. Spectrum of quantum observable181
  • 9.2. Algebra of operator functions187
  • 9.3. Spectral projection and spectral decomposition189
  • 9.4. Symmetrical and self-adjoint operators192
  • 9.5. Resolution of the identity194
  • 9.6. Spectral theorem196
  • 9.7. Spectral operator through ket-bra operator199
  • 9.8. Function of self-adjoint operator201
  • 9.9. Commutative and permutable operators203
  • 9.10. Spectral representation205
  • 9.11. Complete system of commuting observables208
  • Part II: Quantum Dynamics211
  • Chapter 10. Superoperators and its Properties213
  • 10.1. Mathematical structures in quantum dynamics213
  • 10.2. Definition of superoperator217
  • 10.3. Left and right superoperators220
  • 10.4. Superoperator kernel223
  • 10.5. Closed and resolvent superoperators226
  • 10.6. Superoperator of derivation227
  • 10.7. Hamiltonian superoperator231
  • 10.8. Integration of quantum observables233
  • Chapter 11. Superoperator Algebras and Spaces237
  • 11.1. Linear spaces and algebras of superoperators237
  • 11.2. Superoperator algebra for Lie operator algebra240
  • 11.3. Superoperator algebra for Jordan operator algebra241
  • 11.4. Superoperator algebra for Lie-Jordan operator algebra243
  • 11.5. Superoperator C*-algebra and double centralisers244
  • 11.6. Superoperator W*-algebra247
  • Chapter 12. Superoperator Functions251
  • 12.1. Function of left and right superoperators251
  • 12.2. Inverse superoperator function253
  • 12.3. Superoperator function and Fourier transform254
  • 12.4 Exponential superoperator function255
  • 12.5. Superoperator Heisenberg algebra257
  • 12.6. Superoperator Weyl system258
  • 12.7. Algebra of Weyl superoperators259
  • 12.8. Superoperator functions and ordering261
  • 12.9. Weyl ordered superoperator263
  • Chapter 13. Semi-Groups of Superoperators267
  • 13.1. Groups of superoperators267
  • 13.2. Semi-groups of superoperators269
  • 13.3. Generating superoperators of semi-groups273
  • 13.4. Contractive semi-groups and its generators275
  • 13.5. Positive semi-groups279
  • Chapter 14. Differential Equations for Quantum Observables285
  • 14.1. Quantum dynamics and operator differential equations285
  • 14.2. Definition of operator differential equations286
  • 14.3. Equations with constant bounded superoperators288
  • 14.4. Chronological multiplication289
  • 14.5. Equations with variable bounded superoperators291
  • 14.6. Operator equations with constant unbounded superoperators294
  • 14.7. Generating superoperator and its resolvent295
  • 14.8. Equations in operator Hilbert spaces298
  • 14.9. Equations in coordinate representation301
  • 14.10. Example of operator differential equation302
  • Chapter 15. Quantum Dynamical Semi-Group305
  • 15.1. Dynamical semi-groups305
  • 15.2. Semi-scalar product and dynamical semi-groups307
  • 15.3. Dynamical semi-groups and orthogonal projections309
  • 15.4. Dynamical semi-groups for observables311
  • 15.5. Quantum dynamical semi-groups on W*-algebras313
  • 15.6. Completely positive superoperators315
  • 15.7. Bipositive superoperators319
  • 15.8. Completely dissipative superoperators320
  • 15.9. Lindblad equation323
  • 15.10. Example of Lindblad equation328
  • 15.11. Gorini-Kossakowski-Sudarshan equation331
  • 15.12. Two-level non-Hamiltonian quantum system333
  • Chapter 16. Classical Non-Hamiltonian Dynamics337
  • 16.1. Introduction to classical dynamics337
  • 16.2. Systems on symplectic manifold340
  • 16.3. Systems on Poisson manifold346
  • 16.4. Properties of locally Hamiltonian systems349
  • 16.5. Quantum Hamiltonian and non-Hamiltonian systems352
  • 16.6. Hamiltonian and Liouvillian pictures354
  • Chapter 17. Quantization of Dynamical Structure361
  • 17.1. Quantization in kinematics and dynamics361
  • 17.2. Quantization map for equations of motion363
  • 17.3. Quantization of Lorenz-type systems370
  • 17.4. Quantization of Poisson bracket371
  • 17.5. Discontinuous functions and nonassociative operators377
  • Chapter 18. Quantum Dynamics of States381
  • 18.1. Evolution equation for normalized operator381
  • 18.2. Quantization for Hamiltonian picture383
  • 18.3. Expectation values for non-Hamiltonian systems384
  • 18.4. Adjoint and inverse superoperators389
  • 18.5. Adjoint Lie-Jordan superoperator functions392
  • 18.6. Weyl multiplication and Weyl scalar product397
  • 18.7. Weyl expectation value and Weyl correlators400
  • 18.8. Evolution of state in the Schrödinger picture404
  • Chapter 19. Dynamical Deformation of Algebras of Observables409
  • 19.1. Evolution as a map409
  • 19.2. Rule of term-by-term differentiation412
  • 19.3. Time evolution of binary operations414
  • 19.4. Bilinear superoperators417
  • 19.5. Cohomology groups of bilinear superoperators419
  • 19.6. Deformation of operator algebras422
  • 19.7. Phase-space metric for classical non-Hamiltonian system427
  • Chapter 20. Fractional Quantum Dynamics433
  • 20.1. Fractional power of superoperator433
  • 20.2. Fractional Lindblad equation and fractional semi-group435
  • 20.3. Quantization of fractional derivatives444
  • 20.4. Quantization of Weierstrass nondifferentiable function448
  • Chapter 21. Stationary States of Non-Hamiltonian Systems453
  • 21.1. Pure stationary states453
  • 21.2. Stationary states of non-Hamiltonian systems455
  • 21.3. Non-Hamiltonian systems with oscillator stationary states456
  • 21.4. Dynamical bifurcations and catastrophes459
  • 21.5. Fold catastrophe461
  • Chapter 22. Quantum Dynamical Methods463
  • 22.1. Resolvent method for non-Hamiltonian systems463
  • 22.2. Wigner function method for non-Hamiltonian systems466
  • 22.3. Integrals of motion of non-Hamiltonian systems473
  • Chapter 23. Path Integral for Non-Hamiltonian Systems475
  • 23.1. Non-Hamiltonian evolution of mixed states475
  • 23.2. Path integral for quantum operations477
  • 23.3. Path integral for completely positive quantum operations480
  • Chapter 24. Non-Hamiltonian Systems as Quantum Computers487
  • 24.1. Quantum state and qubit487
  • 24.2. Finite-dimensional Liouville space and superoperators489
  • 24.3. Generalized computational basis and ququats491
  • 24.4. Quantum four-valued logic gates495
  • 24.5. Classical four-valued logic gates509
  • 24.6. To universal set of quantum four-valued logic gates512
  • Bibliography521
  • Subject Index533
Book details
  • Vendor Elsevier S & T
  • SKU 9780444530912
  • ISBN-13 9780080559711
  • Author Tarasov, Vasily
  • Category Mathematics
  • Subject Applied

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Quantum Mechanics of Non-Hamiltonian and Dissipative Systems is self-contained and can be used by students without a previous course in modern mathematics and physics. The book describes the modern structure of the theory, and covers the fundamental results of last 15 years. The book has been recommended by Russian Ministry of Education as the textbook for graduate students and has been used for graduate student lectures from 1998 to 2006.

• Requires no preliminary knowledge of graduate and advanced mathematics
• Discusses the fundamental results of last 15 years in this theory
• Suitable for courses for undergraduate students as well as graduate students and specialists in physics mathematics and other sciences