Abstract
Evolution of a self-consistent joint system (JS), i.e., a quantum system (QS) + thermal bath (TB), is considered within the framework of the Langevin–Schrödinger (L-Sch) type equation. As a tested QS, we considered two linearly coupled quantum oscillators that interact with TB. The influence of TB on QS is described by the white noise type autocorrelation function. Using the reference differential equation, the original L-Sch equation is reduced to an autonomous form on a random space–time continuum, which reflects the fact of the existence of a hidden symmetry of JS. It is proven that, as a result of JS relaxation, a two-dimensional quantized small environment is formed, which is an integral part of QS. The possibility of constructing quantum thermodynamics from the first principles of non-Hermitian quantum mechanics without using any additional axioms has been proven. A numerical algorithm has been developed for modeling various properties and parameters of the QS and its environment.
Highlights
When we try to approach the study of a quantum system (QS) strictly and consistently, it becomes obvious that its isolation from the environment is an almost non-realizable task
Issues related to taking into account the influence of the environment on the properties of a quantum system have become the subject of increased interest in connection with the importance of solving a number of fundamental and applied problems of science and technology
We postulated an equation of the Langevin–Schrödinger type as the basic equation for describing the joint system (JS), for which the Schrödinger equation plays the role of the principle of local correspondence
Summary
When we try to approach the study of a quantum system (QS) strictly and consistently, it becomes obvious that its isolation from the environment is an almost non-realizable task. The manuscript is organized as follows: In Section 2, the problem of coupled quantum oscillators is formulated taking into account the presence of a random environment. The explicit form of the wave function of the joint system (JS) “two coupled quantum oscillators + TB ” is obtained by solving the L-Sch equation in the form of an orthonormal probability processes. It contains the basic definitions with the help of which different statistical distributions and the average values of corresponding parameters of the QS and its small environment are constructed. L=1 where {ξ} = (ξ1, ξ2) denotes some complex stochastic process, which will be defined below, and Ys(tlc)(ql, t, {ξl}) is the time-dependent solution—the wave function of the 1D quantum harmonic oscillator for an arbitrary frequency Ωl(t; {f}).
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