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Energy and Quantum Dynamics

Gennaro Auletta, Shang-Yung Wang

Published Jun 20, 2014• 0 citations
Publisher

Abstract

In this chapter we shall deal with quantum dynamics and in particular with the basic equation that governs quantum dynamics, the Schro¨dinger equation. We shall consider two of the most important model systems in quantum mechanics: the free particle and the harmonic oscillator. Moreover, we shall deal with the density matrix formalism and composite systems. Another crucial concept that will be introduced here is that of entanglement, a particular manifestation of quantum features.

DOI
10.1201/b17083-13

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In this chapter we shall deal with quantum dynamics and in particular with the basic equation that governs quantum dynamics, the Schro¨dinger equation. We shall consider two of the most important model systems in quantum mechanics: the free particle and the harmonic oscillator. Moreover, we shall deal with the density matrix formalism and composite systems. Another crucial concept that will be introduced here is that of entanglement, a particular manifestation of quantum features.
This chapter begins by developing the relation between the quantum theory and the linear algebra. It discusses the most fundamental postulates of the theory and provides a phenomenological development of the Schrödinger wave equation (SWE) although the results might be gleaned from the classical Lagrangian and Hamiltonian with the Poisson bracket relations. Afterward some simple examples for the infinitely and finitely deep wells help clarify the basic postulates. The simple harmonic oscillator has perhaps the most sweeping implications and applications for the quantum theory. For this reason, we present the most applicable formalism for the Harmonic oscillator that uses the operator approach rather than the classical method of partial differential equations. The angular momentum and spin are presented for the study of the atom and for applications for nanodevices, and quantum computing. Next the representation formalism clarifies the distinction between the dynamical operators and the wave functions and their interrelations. The chapter discusses both timeindependent and time-dependent perturbation theories. The density operator combines the quantum theory with the classical theory for finding the behavior of a system. The density operator represents one of the basic concepts in the quantum theory. The remainder of the chapter introduces the more advanced material meant to introduce the quantum field theory starting with the second quantization and the propagator.
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.
Chapter 2 begins with a discussion of the de Broglie wave concept that is the foundation of quantum mechanics. The idea that particles can exhibit wave-like behavior is an important part of the explanation of why atoms have discrete energy levels and make transitions in which electromagnetic radiation is absorbed or emitted at specific frequencies. Important aspects of how quantum systems may be observed and how they develop in time are presented. The Schrödinger equation is introduced and the importance of the wavefunction covered. Mathematical simplifications that arise from considering symmetry are covered. Several examples of simple quantum systems are presented in which the calculation of electronic energy structure based on boundary conditions is illustrated. Sections on wave–particle duality and quantum observables are followed by others on the dynamics of two-level systems, mathematically complete representations of arbitrary states, and standard approaches for handling the temporal development of wavefunctions.
In this chapter we shall deal with the quantum dynamics of an open system. By open quantum system we mean here a system which interacts with an environment (see Sec. 9.4): since we are not interested in the dynamics of the environment, we shall have to describe the evolution of the system in some “effective” way. In particular, if we consider only the evolution of the system, it will be non-unitary, and this will represent the subject of this chapter. As we know, Hamiltonian quantum dynamics is unitary and changes pure states into pure states. On the other hand, non-unitary dynamics changes an initially pure state into a mixture, which must be described by a density matrix (see Ch. 5). In the case of macroscopic systems, the coupling with the environment may be arbitrarily reduced and therefore its influence can be made correspondingly small (see Sec. 1.1). Microscopic systems, however, always couple to the environment and this coupling cannot be considered negligible. This is the reason why the quantum theory of open systems is one of the most important and fundamental chapters of quantum mechanics that, though born in quantum optics, has many implications in almost all fields of physics. The present chapter can be seen as a further development of the measurement theory (see Ch. 9), as open systems manifest a decoherent dynamics (see in particular Sec. 9.4).
In this chapter we introduce the basic concept of a quantum harmonic oscillator, and how such oscillators couple to their environment. Our purpose is not to exhaustively cover all aspects of such systems, but rather to provide the foundation material required to treat the quantum mechanics of mechanical resonators and optical cavities – and their interactions – in later chapters. The quantum fluctuation-dissipation theorem is derived relating the power spectral density of environmental forcing of an oscillator to the rate of energy decay from the oscillator into the environment. This is in direct analogy to the familiar fluctuation-dissipation theorem in classical physics. However, in contrast to the classical case, the noncommutation of position and momentum introduces an asymmetry between the positive and negative frequency components of the power spectrum. This provides a natural definition for the effective temperature of the oscillator. The quantum Langevin approach is introduced as a method to model the open dynamics of quantum systems, both with and without recourse to the rotating wave approximation.