Density Matrix Theory and ApplicationsSpringer Science & Business Media, 31 oct. 1996 - 327 pages Quantum mechanics has been mostly concerned with those states of systems that are represented by state vectors. In many cases, however, the system of interest is incompletely determined; for example, it may have no more than a certain probability of being in the precisely defined dynamical state characterized by a state vector. Because of this incomplete knowledge, a need for statistical averaging arises in the same sense as in classical physics. The density matrix was introduced by J. von Neumann in 1927 to describe statistical concepts in quantum mechanics. The main virtue of the density matrix is its analytical power in the construction of general formulas and in the proof of general theorems. The evaluation of averages and probabilities of the physical quantities characterizing a given system is extremely cumbersome without the use of density matrix techniques. The representation of quantum mechanical states by density matrices enables the maximum information available on the system to be expressed in a compact manner and hence avoids the introduction of unnecessary variables. The use of density matrix methods also has the advan tage of providing a uniform treatment of all quantum mechanical states, whether they are completely or incompletely known. Until recently the use of the density matrix method has been mainly restricted to statistical physics. In recent years, however, the application of the density matrix has been gaining more and more importance in many other fields of physics. |
Table des matières
Basic Concepts | 1 |
112 The Polarization Vector | 4 |
113 Mixed Spin States | 8 |
114 Pure versus Mixed States | 10 |
115 The SpinDensity Matrix and Its Basic Properties | 12 |
116 The Algebra of the Fault Matrices | 19 |
117 Summary | 22 |
122 Pure and Mixed Polarization States of Photons | 25 |
542 Quantum Beats Produced by Symmetry Breaking | 154 |
55 Time Integration over Quantum Beats | 156 |
552 Depolarization Effects Caused by Fine and Hyperfine Interactions | 158 |
Some Applications | 161 |
612 Influence of Fine and Hyperfine Interactions on the Emitted Radiation | 166 |
62 SteadyState Excitation | 167 |
622 Threshold and Pseudothreshold Excitations | 170 |
63 Effect of a Weak Magnetic Field | 172 |
123 The Quantum Mechanical Concept of Photon Spin | 27 |
124 The Polarization Density Matrix | 29 |
125 Stokes Parameter Description | 32 |
General Density Matrix Theory | 39 |
22 The Density Matrix and Its Basic Properties | 43 |
23 Coherence versus Incoherence | 47 |
232 The Concept of Coherent Superposition | 49 |
24 Time Evolution of Statistical Mixtures | 52 |
242 The Liouville Equation | 55 |
243 The Interaction Picture | 57 |
25 Spin Precession in a Magnetic Field | 62 |
26 Systems in Thermal Equilibrium | 63 |
Coupled Systems | 67 |
32 Interaction with an Unobserved System The Reduced Density Matrix | 69 |
33 Analysis of Light Emitted by Atoms Nuclei | 73 |
332 Description of the Emitted Photon | 76 |
34 Some Further Consequences of the Principle of Nonseparability | 77 |
342 Complete Coherence in Atomic Excitation | 79 |
35 Excitation of Atoms by Electron Impact I | 81 |
352 Restrictions due to Symmetry Requirements | 86 |
4 Irreducible Components of the Density Matrix | 91 |
42 The Definition of Tensor Operators | 92 |
422 Transformation Properties under Rotations The Rotation Matrix | 94 |
423 Examples | 97 |
424 Some Important Properties of the Tensor Operators | 99 |
43 State Multipoles Statistical Tensors | 101 |
432 Basic Properties of State Multipoles | 103 |
433 Physical Interpretation of State Multipoles The Orientation Vector and Alignment Tensor | 104 |
Spin Tensors | 106 |
442 Description of Spin1 Particles | 107 |
45 Symmetry Properties Relation between Symmetry and Coherence | 111 |
452 Classification of Axially Symmetric Systems | 112 |
Photoabsorption by Atoms Nuclei | 117 |
462 General Consequences of Reflection Invariance | 120 |
463 Axially Symmetric Atomic Systems | 122 |
464 Symmetry Relations in the Natural System | 123 |
465 Coordinate Representation of the Density Matrix Shape and Spatial Orientation of Atomic Charge Clouds | 125 |
47 Time Evolution of State Multipoles in the Presence of an External Perturbation | 131 |
472 Perturbation Coefficients for the Fine and Hyperfine Interactions | 133 |
473 An Explicit Example | 138 |
474 Influence of an External Magnetic Field | 139 |
48 Notations Used b Other Authors | 141 |
Radiation from Polarized Atoms Quantum Beats | 143 |
Separation of Dynamical and Geometrical Factors | 147 |
53 Discussion of the General Formulas | 149 |
532 Manifestations of Coherence Quantum Beats | 151 |
54 Perturbed Angular Distribution and Polarization | 153 |
632 Magnetic Depolarization Theory of the Hanle Effect | 175 |
633 Physical Interpretation of Zeeman Quantum Beats Rotation of the Atomic Charge Cloud | 180 |
64 Influence of Electric Fields OrientationAlignment Conversion | 182 |
642 Variation of Shape and Spatial Direction of Atomic Charge Clouds | 184 |
643 Creation of Orientation out of Alignment | 187 |
The Role of Orientation and Alignment in Molecular Processes | 189 |
Distribution Functions of Angular Momentum Vectors | 190 |
73 Axis Distributions of Linear Rotors | 195 |
732 General Equations Examples and Experimental Studies | 196 |
General Description of Axis Orientation and Alignment | 200 |
742 Relation between Angular Momenta and Axis Distributions for Linear Rotors Pendulum States | 204 |
75 Angular Momenta and Axis Distributions of Rotors after Photoabsorption Quantum Mechanical and Classical Theory | 206 |
752 Absorption of Circularly Polarized and Unpolarized Light | 209 |
76 Distribution Functions for Nonlinear Molecules and for Diatomics with Electronic Angular Momentum | 212 |
762 Angular Momentum and Axis Distributions of Symmetric Tops | 213 |
763 Theory of Oriented SymmetricTop Molecules Semiclassical Interpretation | 216 |
764 Order Parameters for Nonlinear Molecules | 218 |
77 Electronic Orbital Orientation and Alignment | 220 |
Spatial Orientation and Selective Population | 225 |
773 Combined Description of Rotational Polarization and Orbital Anisotropies | 229 |
774 Vector Correlations Analysis of Emitted Light | 233 |
775 Photoabsorption and Photofragmentation | 238 |
78 Dynamical Stereochemistry | 241 |
782 Discussion and Examples | 248 |
783 Product Rotational Polarization Quantum Mechanical Theory and Semiclassical Approximation | 253 |
784 AlignmentInduced Chemical Reactions | 257 |
Quantum Theory of Relaxation | 261 |
812 Time Correlation Functions Discussion of the Markoff Approximation | 264 |
813 The Relaxation Equation The Secular Approximation | 267 |
82 Rate Master Equations | 270 |
83 Kinetics of Stimulated Emission and Absorption | 275 |
84 The Bloch Equations | 282 |
842 Longitudinal and Transverse Relaxation Spin Echoes | 286 |
843 The Optical Bloch Equations | 290 |
85 Some Properties of the Relaxation Matrix | 291 |
852 Relaxation of State Multipoles | 293 |
86 The Liouville Formalism | 295 |
87 Linear Response of a Quantum System to an External Perturbation | 299 |
Appendixes | 303 |
State Multipoles for Coupled Systems | 306 |
Formulas from Angular Momentum Theory | 308 |
The Efficiency of a Measuring Device | 312 |
The Scattering and Transition Operators | 314 |
317 | |
321 | |
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Expressions et termes fréquents
3j symbol amplitudes angular momentum applying Eq assumed axes axially symmetric axis distribution beam characterized charge cloud coherence collision system components condition consider corresponding cos² defined denotes density matrix density operator described diagonal elements dipole direction discussed in Section eigenstates electric field electrons emitted energy ensemble Equation Euler angles evolution example excited atoms expression Figure filter follows given by Eq gives Hamiltonian Hence initial interaction J₁ light linear Liouville Liouville equation M₁ M₂ magnetic field molecular molecules momenta multipoles obtain from Eq orbital order parameters orientation and alignment P₁ parallel particles Pauli matrices perpendicular photons polarization density polarization vector probability pure quantization axis quantum beats quantum mechanical quantum numbers relation relaxation relevant representation respect rotation Schrödinger picture semiclassical spin spin tensors Stokes parameters superposition symmetry properties tensor operators theory tion transformation transition values vanish W(Ba W₁ zero