Irreversibilities in Quantum Mechanics (eBook)

CONTENTS 6
FOREWORD 8
INTRODUCTION 9
CHAPTER I QUANTUM- THEORETICAL BASIS. DENSITY MATRIX 11
1.1 Basic concepts 11
1.2 Various representations of physical quantities. 13
1.3 Quantum state and statistical ensembles 15
1.4 The wave function 16
1.5 Entropyofquantum ensembles 20
1.6 Pure and mixed states. Proper mixtures. 21
1.7 Transition probability per unit time 27
1.8 Continuous spectrum of energies and irreversibility 32
CHAPTER II QUANTUM THEORY OF RELAXATION PROCESSES 34
2.1 Exact equations describing temporal behavior of interacting dissipative and dynamic systems 35
2.2 Relaxation of the dissipative system. Markovian approximation 38
2.3 Equations for density matrix of dynamic systems 43
2.4 Generalized master equations 48
2.5 Time convolutionless equations. Argyres and Kelley projection operators and reduced dynamics. 51
2.6 Semigroup theory of irreversible processes 57
2.7 Master equations for dynamic systems 62
CHAPTER III INTERACTION WITH PHONONS AND MOLECULAR VIBRATIONS 74
3.1 Description of time- dependent electron- nuclear system in the Born- Oppenheimer approximation 74
3.2 Phonons. Phonon-phonon interaction and relaxation 76
3.3 Two-state electronic systems 80
3.4 Radiationless transitions. Basic models of electron and energy transfer 85
3.5 Tunneling in the condensed media 88
3.6 Equations ofmotion of the two-state electronic (nuclear) system interacting with vibrations of the medium 92
3.7 Calculation of rate coefficients 95
3.8 The energy gap law 102
CHAPTER IV INTERACTION WITH PHOTONS 116
4.1 Interaction of matter with the electromagnetic field 116
4.2 A system of two-level molecules interacting with the electromagnetic field 118
4.3 Quantum theory of spontaneous and stimulated emission in a system of two- level molecules 120
4.4 Spontaneous emission vs. vacuum fluctuations 123
4.5 Superradiance ( small volumes L ³ < <
4.6 Large sample superradiance 127
4.7 Time- development of the superradiance ( small volumes) 132
4.8 Time-development of the superradiance (large volumes) 138
CHAPTER V MEMORY EFFECTS IN RELAXATION PROCESSES, EXACT SOLUTIONS. 144
5.1 Time development of quantum systems: general relations 144
5.2 General criteria for emerging of dissipationless regimes 147
5.3 The configuration interaction between one discrete state and the continuum of the states 150
5.4 Spontaneous emission of bosons (phonons) and tunneling in the rotating wave approximation 152
5.5 Weak coupling case ( beyond RWA) 160
5.6 Semiquantitive analysis of the dissipationless regime 166
5.7 Rotating wave, Markovian, and weak coupling approximations 172
5.8 Impossibility of exponential relaxation 180
CHAPTER VI QUANTUM MEASUREMENT AND IRREVERSIBILITY 182
6.1 Another view on quantum mechanics. EPR. Bell’s theorems. Nonlocality. 182
6.2 Reduction of wave-packet 191
6.3 Reduction of the wave packet as a result of interaction with the dissipative system 195
6.4 Measurement of spins in the Stern-Gerlach experiment 199
6.5 Realization of quantum measurement by the irreversible relaxation process 205
6.6 Gedanken experiment: measurement of the z-component of spin ½ 207
CONCLUDING REMARKS 211
REFERENCES 214
INDEX 220

CHAPTER 1 QUANTUM-THEORETICAL BASIS. DENSITY MATRIX (p. 1-2)

The present chapter gives a short account of the basic concepts of quantum theory. Since our main purpose is quantum theory of irreversible processes, the use of the density matrix concept is indispensable. A conventional presentation of quantum theory principles uses the wave function as the basic characteristic of a quantum system. In this case the density matrix is, in a sense, a derivative of the wave function. In this chapter we consider the density matrix as a basic, primary characteristic of the quantum system [1], while the description by the wave function is a specific case of the description with aid of the density matrix. We realize that such an approach is a generalization of the conventional procedure. However, this generalization seems to be quite natural, and only due to historical reasons quantum theory text books use the wave function presentation. In this chapter we follow conventional (Copenhagen) interpretation of quantum mechanics. A different approach will be considered in the last chapter (Chapter VI).

1.1 Basic concepts

The nature of the phenomena occurring at the atomic level is very different from the nature of the phenomena of the macrocosm. For this reason the basic concepts of the classical theory proved to be invalid in describing the microcosm. The concept of the state of a physical system underwent a most radical re-examination. In classical physics it is assumed that the physical quantities (or properties of a system) found from various measurements made on a system are characteristics of the particular state of the system, that they are always present in a given system in a definite form and that this does not depend on the observational methods and equipment. In quantum physics they are at the same time characteristics of the methods and equipment used for the observations. In the microcosm we cannot ignore the effect of the measuring apparatus on the measured object. Therefore the concept of the quantum state takes into account both the object which is in this state and possible experimental devices used to make the measurement. Accordingly, the quantum theoretical description of quantum objects differs essentially from the classical description. Quantum theory, unlike the classical theory, is a statistical theory in principle. The laws of quantum theory do not govern the actual behavior of a particular object, but give the probabilities of the various ways in which the object may behave as a result of an interaction with its surroundings. The following postulates form the basis of the quantum description of physical phenomena.