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College Physics

On the general properties of time-dependent invariants

Qiong-Gui Lin

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Several general properties of time-dependent invariants in quantum mechanics are discussed, some of which were not mentioned in the literature.

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In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.

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In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
In the first part of the lectures dynamical invariants in classical mechanics and conventional quantum mechanics will be considered. In particular, we will begin with some remarks on classical mechanics and on quantization in order to establish the theory in the form that will be used later on. Starting from the time‐dependent Schrödinger equation, the dynamics of Gaussian wave packets and Ermakov invariants, the time‐dependent Green function/Feynman kernel, quantum‐classical connections, energetics and Lagrange—Hamilton formalism for quantum uncertainties, momentum space representation and the relation between the Wigner function and the Ermakov invariant will be discussed. The representation of canonical transformations in time‐independent and time‐dependent quantum mechanics, factorization of the Ermakov invariant and generalized creation/annihilation operators will be studied. Subsequently, the time‐independent Schrödinger equation, leading to nonlinear quantum mechanics related to Riccati/Ermakov systems as well as the occurrence of Riccati/Ermakov systems in the treatment of Bose—Einstein condensates via the so‐called moment method will be analyzed.In part two, irreversible dynamics of dissipative systems, classical and quantum mechanical descriptions and corresponding invariants will be treated. After some general remarks on classical and quantum mechanics with unitary time‐evolution and energy conservation, phenomenological Langevin and Fokker—Planck equations, master equations in classical and quantum mechanics and the system‐plus‐reservoir approach will be mentioned briefly. Then follows a more detailed discussion of modified Schrödinger equations and, particularly, of a nonlinear Schrödinger equation with complex logarithmic nonlinearity; its properties, solutions, invariants and energetics will be studied. Finally, a comparison with a classical description in expanding coordinates will lead to a non‐unitary connection between the logarithmic nonlinear Schrödinger equation, the Caldirola‐Kanai approach and the expanding coordinate approach.
In Chapter 3, we discussed charging effects in nanostructures: the most important manifestation of electron–electron interaction in quantum transport. In this chapter, we concentrate on another aspect of interactions which we have so far mentioned very briefly. It concerns interaction with slow modes . In most cases these slow modes are electromagnetic excitations in the nanostructure and nearby circuit that form an electromagnetic environment of the nanostructure. The effect of this interaction is threefold. First, it may affect and alter transport properties of the nanostructure. Secondly, it provides energy relaxation : transporting electrons and qubits may exchange their energy with the electromagnetic environment. Thirdly, the environment provides decoherence , inducing time-dependent phase shifts to wave functions of propagating electrons and qubits, thereby destroying the quantum coherence of corresponding states. The physics discussed in this chapter is sometimes involved and various. It requires effort to see a “common denominator” in all effects mentioned. We choose to present material starting from the ideas of dissipative quantum mechanics : a branch of quantum mechanics developed in the 1970s and 1980s. For several concrete phenomena this presentation manner deviates from that commonly accepted in the literature. Although this may be inconvenient for the reader, we did this for the sake of the “big picture,” which allows us to see links and analogies between formally different phenomena. The structure of the chapter is as follows. Sections 6.1 and 6.2 are introductory. In Section 6.1, we discuss electromagnetic excitations in linear circuits and the way to treat them quantum-mechanically. In Section 6.2 we review general ideas of dissipative quantum mechanics, which are not specific for quantum transport: the orthogonality catastrophe, shake-up, classification of environments.