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AIP conference proceedings

The new finite temperature Schrödinger equations with strong or weak interaction

Heling Li, Bin Yang, Hongjun Shen

Published Jan 1, 2017• 1 citation
Publisher

Abstract

Implanting the thoughtway of thermostatistics into quantum mechanics, we formulate new Schrödinger equations of multi-particle and single-particle respectively at finite temperature. To get it, the pure-state free energies and the microscopic entropy operators are introduced and meantime the pure-state free energies take the places of mechanical energies at finite temperature. The definition of microscopic entropy introduced by Wu was also revised, and the strong or weak interactions dependent on temperature are considered in multi-particle Schrödinger Equations. Based on the new Schrödinger equation at finite temperature, two simple cases were analyzed. The first one is concerning some identical harmonic oscillators in N lattice points and the other one is about N unrelated particles in three dimensional in finite potential well. From the results gotten, we conclude that the finite temperature Schrödinger equation is particularly important for mesoscopic systems.

DOI
10.1063/1.4992279

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We study the behavior of two weakly coupled quantum systems, described by a separable density operator. One of them is a single oscillator, representing a microscopic system, while the other is a set of oscillators, which perform the role of a reservoir in thermal equilibrium. From the Liouville-Von Neumann equation for the reduced density operator, we devise the master equation that governs the evolution of the microscopic system, incorporating the effects of temperature via Thermofield Dynamics formalism and suitably redefining the vacuum of the macroscopic system. As applications, we study an oscillator interacting with a set of boson oscillators at finite temperature, usually employed in quantum mechanics to describe a Brownian particle; and investigate the behavior of an oscillator in the presence of a heat bath consisting of a set of fermion oscillators.
In this paper, we investigate the Dunkl–Schrödinger equation in the presence of energy-dependent potentials. A consistent formulation is developed by deriving modified expressions for the probability density, normalization condition, and scalar product, thereby ensuring the conservation of probability. Exact analytical solutions are obtained for both one- and three-dimensional systems. In the one-dimensional case, we study an energy-dependent harmonic oscillator and analyze the resulting energy spectrum for different forms of energy dependence. The analysis is then extended to three dimensions, where central potentials are considered, including the harmonic oscillator and Coulomb interactions. We show that the interplay between the Dunkl deformation and the energy dependence leads to significant modifications of the spectral structure, including nonlinear energy behavior and, in some cases, a finite number of bound states. Concrete physical realizations of such energy-dependent potentials are discussed, including semiconductors with an energy-dependent effective mass, two-dimensional electron gases in strong magnetic fields, and cold atoms in optically engineered traps.
In this paper, we have proposed the non-Hermitian theory of quantum thermodynamics, and given the quantum thermodynamics equations, including non-relativistic and relativistic quantum theory with temperature, i.e., the temperature-dependent schrodinger equation, Dirac equation and photon equation. We give the solution of wave function and energy level with temperature, and introduced the temperature constant T0, which should be determined by the atom or molecule spectrum experiments. PACS: 03.65.-w, 05.70.Ce, 05.30.Rt.
We discuss formal analogies between a nonlinear Schrödinger equation derived by the author from the theory of scale relativity and the equations of Brownian theory. By using the Madelung transformation, the nonlinear Schrödinger equation takes the form of hydrodynamic equations involving a friction force, an effective thermal pressure, a pressure due to the self-interaction, and a quantum potential. These hydrodynamic equations have a form similar to the damped Euler equations obtained for self-interacting Brownian particles in the theory of simple liquids. In that case, the temperature is due to thermal motion and the pressure arises from spatial correlations between the particles. More generally, the correlations can be accounted for by using the dynamical density functional theory. We determine the excess free energy of Brownian particles that reproduces the standard quantum potential. We then consider a more general form of excess free energy functionals and propose a new class of generalized Schrödinger equations. For a certain form of excess free energy, we recover the generalized Schrödinger equation associated with the Tsallis entropy considered in a previous paper.