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Introduction to quantum relaxation processes

Abraham Nitzan

Published Jun 14, 2024• 0 citations
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Abstract

Abstract This chapter introduces the theory and practice of quantum relaxation and quantum steady state processes. (1) A simple quantum-mechanical model for relaxation. (2) Relaxation in the time domain. (3) The origin of irreversibility. (4) Preparation of the initial state. (5) The effect of relaxation on absorption lineshapes. (6) Relaxation in a two-level system model. (7) Relaxation of a quantum harmonic oscillator. (8) Quantum mechanics of steady states. (9) Resonance tunnelling

DOI
10.1093/9780191947971.003.0009

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•Jan 26, 1995•42 citations
Abstract This chapter introduces the theory and practice of quantum relaxation and quantum steady state processes. (1) A simple quantum-mechanical model for relaxation. (2) Relaxation in the time domain. (3) The origin of irreversibility. (4) Preparation of the initial state. (5) The effect of relaxation on absorption lineshapes. (6) Relaxation in a two-level system model. (7) Relaxation of a quantum harmonic oscillator. (8) Quantum mechanics of steady states. (9) Resonance tunnelling
The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.
Abstract This chapter the reviews essential concepts and methods of time-dependent quantum mechanics. (1) Formal solutions of the time-dependent Schrödinger equation. (2) The two-level system. (3) Interacting systems: The time-dependent self-consistent field approximation and the Ehrenfest approximation. (4) The Born–Oppenheimer approximation and nuclear potential surfaces. (5) Time-dependent Hamiltonians: A two-level system in a time-dependent field. (6) Non-adiabatic dynamics: The surface-hopping method. (7) Expressing the time evolution in terms of the Green operator. (8) Representations (the Schrödinger and Heisenberg representations, the interaction representation). (9) Time-dependent perturbation theory. (10) Quantum dynamics of the free particles. (11) Quantum dynamics of the harmonic oscillator. (12) The raising/lowering operators formalism. (13) Tunneling.
In this paper, the non-exponential decay is analyzed with the help of simple computer experiments performed by T. Petrosky, simulating classical radiation damping. The non-exponential decay is studied and shown to depend on the preparation of the system. However, whatever the initial preparation, the system reaches the decay predicted by classical radiation theory after a short time we call the Zeno's time. The similitude of Petrosky's results with computer experiments for the approach to equilibrium in many-body systems is emphasized. However, while there one deals with times which are multiple of the relaxation time, the irreversibility manifest in radiation theory occurs always over a much shorter time scale, the Zeno's time. In atomic systems, this would be a time order of 10/sup /minus/18/ seconds. These results are of great interest for the understanding of the microscopic mechanism of radiation. Let us consider a charged oscillator. In a first stage, this oscillator has to produce the field oscillators to which it may transfer energy through the usual resonance mechanism. Radiation appears therefore as a kind of non linear autocatalytic process, involving a self-organization mechanism. The behavior during the Zeno period can be explained easily in terms of subdynamics as introduced by the Brussel's group. We see that there is no transition from reversibility to irreversibility. Irreversible processes start at the very moment at which the system is prepared. It is important to stress that an unstable particle is itself the result of irreversible processes. As a result, an unstable particle (or an excited atomic state) can no more be described in terms of wave functions, as irreversible processes are not included in Schroedinger's equation. 14 refs. 3 figs.
The work is devoted to the investigation of the influence of a heat bath on the physical processes in a quantum system. We use the density matrix theory as one of the most powerfool tool for investigation of quantum relaxation. In the beginning of the work (chapter 2) we mention and recall the most important steps of derivation of the equation of motion for the reduced density matrix (master equation) for an arbitrary quantum system in diabatic representation interacting with the environment modeled by a set of independent harmonic oscillators. Chapter 3 deals with the question of the border between classical and quantum effects and reports on a study of the environmental influence on the time evolution of a coherent state or the superposition of two coherent states of a harmonic oscillator as a simple system displaying the peculiarities of the transition from quantum to classical regime. Chapters 4 and 5 concern the electron transfer (ET) problem, namely the mathematical description of the ET in molecular zinc-porphyrin-quinone complexes modeling artificial photosynthesis (chapter 4) and photoinduced processes in the porphyrin triad (chapter 5). Each chapter starts with an introduction and ends with a brief summary. The main achievements of the present work are summarized in the Conclusions.