Table of Contents
Preface ix
Author biography xi
1 Probability operator and statistical averages 1-1
1.1 Expectation, density operator and averages 1-2
1.1.1 Expectation value 1-2
1.1.2 Density operator 1-3
1.1.3 Statistical average 1-3
1.2 Uniform weight density of wave space 1-6
1.2.1 Probability flux and trajectory uniformity 1-7
1.2.2 Time average on the hypersurface 1-10
1.3 Canonical equilibrium system 1-12
1.3.1 Entropy of energy states 1-13
1.3.2 Wave function entanglement 1-14
1.3.3 Expectation values and wave function collapse 1-36
1.3.4 Statistical average and probability operator 1-18
1.4 Environmental selection 1-22
1.5 Wave function collapse and the classical universe 1-26
1.5.1 Mechanism of statistical collapse 1-26
1.5.2 Probabilistic nature of the wave function 1-28
1.5.3 Quantum interference is uniquely non-classical 1-30
1.5.4 Classical phase space and Hamilton's equations 1-32
References 1-36
2 Examples and applications: equilibrium 2-1
2.1 Bosons, fermions and wave function symmetry 2-1
2.2 Ideal quantum gas 2-3
2.2.1 Leading classical term 2-3
2.2.2 First quantum correction 2-5
2.2.3 Quantum correction as a potential of mean force 2-6
2.3 State occupancy by ideal particles 2-8
2.3.1 Bosons 2-8
2.3.2 Fermions 2-10
2.3.3 Classical particles 2-11
2.4 Thermodynamics and statistical mechanics of ideal particles 2-12
2.5 Classical ideal gas 2-19
2.6 Ideal Bose gas 2-20
2.6.1 Black body radiation 2-24
2.6.2 Heat capacity of solids 2-25
2.7 Ideal Fermi gas 2-27
2.8 Simple harmonic oscillator 2-33
References 2-37
3 Probability in quantum systems 3-1
3.1 Formulation of probability 3-1
3.1.1 States 3-1
3.1.2 Weight 3-3
3.1.3 Probability 3-6
3.1.4 Entropy 3-8
3.1.5 Averages 3-12
3.2 Transitions 3-12
3.2.1 Transition weight operator 3-12
3.2.2 Time correlation function 3-15
3.2.3 First reduction condition 3-17
3.2.4 Second reduction condition 3-18
3.2.5 Parity and reversibility 3-19
3.3 Non-equilibrium probability 3-24
References 3-26
4 Time propagator for an open quantum system 4-1
4.1 Adiabatic time propagator 4-1
4.1.1 Constant Hamiltonian operator 4-1
4.1.2 Time-varying Hamiltonian operator 4-2
4.1.3 Adiabatic Heisenberg picture 4-7
4.1.4 Adiabatic Liouville operator 4-9
4.1.5 Adiabatic Pauli master equation 4-11
4.2 Stochastic time propagator 4-13
4.2.1 General features 4-13
4.2.2 Variance of the stochastic operator 4-17
4.2.3 A quantum fluctuation-dissipation theorem 4-18
4.2.4 Properties of the stochastic time propagator 4-19
4.2.5 Heisenberg picture 4-22
4.2.6 Stochastic Liouville super-operator 4-24
4.2.7 Kubo cumulant expansion 4-26
4.2.8 Liouville equation 4-28
4.3 Kraus representation and Lindblad equation 4-29
4.3.1 Kraus operators for a sub-system and reservoir 4-30
4.3.2 Lindblad equation 4-31
4.4 Caldeira-Leggett model 4-33
4.4.1 Fundamentals 4-33
4.4.2 Generalised Langevin equation 4-35
4.5 Time correlation function 4-38
4.5.1 Equilibrium time correlation function 4-40
4.5.2 Unitary and evolution conditions 4-41
4.6 Transition probability 4-43
4.6.1 Unconditional transition probability operator 4-43
4.6.2 Conditional transition probability operator 4-45
4.7 Microscopic reversibility 4-46
References 4-49
5 Evolution of the canonical equilibrium system 5-1
5.1 Transitions between entropy states 5-1
5.1.1 Random phase for transitions 5-1
5.1.2 Mean and stochastic parts of the state transition 5-3
5.1.3 Form of the time propagator. 1 5-5
5.1.4 Time correlation function 5-7
5.1.5 Stochastic and dissipative operators 5-8
5.2 Second entropy for transitions 5-10
5.2.1 Fluctuation form 5-10
5.2.2 Small time expansion for most likely transition 5-13
5.2.3 Stochastic, dissipative equation of motion 5-16
5.2.4 Stationarity of the state probability 5-17
5.2.5 Form of the time propagator. 2 5-19
5.3 Trajectory in wave space 5-22
5.3.1 Stochastic dissipative Schrouml;dinger equation 5-22
5.3.2 Average on the trajectory 5-23
5.4 Time derivative of entropy operator 5-24
References 5-27
6 Probability operator for non-equilibrium systems 6-1
6.1 Entropy operator for a trajectory 6-2
6.1.1 Wave space formulation 6-2
6.1.2 Propagator formulation 6-7
6.2 Point entropy operator 6-9
6.2.1 Reduction of the trajectory entropy 6-10
6.2.2 Form and interpretation of the point entropy 6-12
6.2.3 Propagator formulation 6-14
6.3 Non-equilibrium probability operator 6-16
6.3.1 Operator and average 6-16
6.3.2 Time derivative 6-17
6.4 Approximations for the dynamic entropy operator 6-17
6.4.1 Odd projection of the dynamic entropy operator 6-18
6.4.2 Adiabatic approximation for steady state thermodynamic systems 6-19
6.5 Perturbation of the non-equilibrium probability operator 6-21
6.6 Linear response theory 6-24
6.6.1 Unitary transformations 6-25
6.6.2 Interaction picture 6-25
6.6.3 Probability operator 6-26
6.6.4 Susceptibility 6-28
6.6.5 Dissipation 6-32
References 6-32
Appendices
A Probability densities and the statistical average 7-1
B Stochastic state transitions for a non-equilibrium system 8-1
C Entropy eigenfunctions, state transitions, and phase space 9-1