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Вестник Российского университета дружбы народов. Серия: Математика, информатика, физика

Исследование решений квазистационарных состояний для квазипотенциального уравнения

Амирханов Илькизар Валиевич, Саркар Нил Ратан, Сархадов Иброхим, Тухлиев Зафар Камаридинович, Шарипов Зариф Алимжонович

Published Jan 1, 2014• 0 citations

Abstract

The excited states of quantum systems are nonstationary, and they break up. These states are called unstable or quasi-stationary. Such states are already observed in the study of scattering problems, the accumulation of particles in the lens (the particle prefers to live inside the lens) is accompanied by a large delay τ (the lifetime of the quasi-level). Here the lifetime of the quasi-level τ = γ−1, width of the quasi-level Г = ~γ and complex energy level E = E1 − iE2,E2 = Γ/2. Investigation of the quasi-stationary states is carried out for the quasi-potential equation with piecewise-constant potentials at various values of the parameter of the equation e and the potential parameters. A comparative analysis of solutions of the quasi-potential equation for the different values of e with the solutions of the Schredinger equation is performed. Found that at e → 0 the solutions of quasi-potential equation tend to the solutions of the Schredinger equation.

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The excited states of quantum systems are nonstationary, and they break up. These states are called unstable or quasi-stationary. Such states are already observed in the study of scattering problems, the accumulation of particles in the lens (the particle prefers to live inside the lens) is accompanied by a large delay τ (the lifetime of the quasi-level). Here the lifetime of the quasi-level τ = γ−1, width of the quasi-level Г = ~γ and complex energy level E = E1 − iE2,E2 = Γ/2. Investigation of the quasi-stationary states is carried out for the quasi-potential equation with piecewise-constant potentials at various values of the parameter of the equation e and the potential parameters. A comparative analysis of solutions of the quasi-potential equation for the different values of e with the solutions of the Schredinger equation is performed. Found that at e → 0 the solutions of quasi-potential equation tend to the solutions of the Schredinger equation.

Figure 2.1 shows a schematic diagram of a two-level system, called a two-level atom, with the ground state |1〉 and the upper (excited) state |2〉. In fact, the electron can make transitions between many energy states, but we can limit the transitions to only between two states. In practice, it is done by a suitable choice of the frequency of an external ﬁeld that will force the electron to oscillate only between these two selected states [3, 4]. Multi-level transitions involving more than two energy states are much more complex and we are not intend to consider them here, but the formalism presented here can be extended to multi-level cases.a The energy of the electron in the selected states |2〉 and |1〉 is determined by the stationary Schro¨dinger equation HˆA |1〉 = E1 |1〉 , HˆA |2〉 = E2 |2〉 , (2.10) where HˆA is the Hamiltonian of the atom whose explicit form is to be determine.

The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.

Based on the method of V-function, a continuation of the optical-mechanical analogy is attained. In contrast to classic quantum mechanics, a trajectory-wave motion of the particle is explored. We highlight the presence of energy quantization of the particle and the availability of solution without a particle in the case of rectilinear uniform motion at constant speed. A solution to the direct and inverse problems of dynamics is searched for in a new statement for a hydrogen-like atom. When solving a direct problem, we find a stationary wave function of the electron in a hydrogen-like atom, with its properties investigated. When searching for a final solution to the stationary wave equation, we take into account a solution to the inverse problem of dynamics for the electron. A linear dependence between two particular solutions is shown. The second linearly independent solution is found, decaying exponentially to zero. We present charts of the stationary solution for a wave of the particle (electron) for three lower stationary states. Energy levels of a hydrogen-like atom are determined as a solution to the inverse problem of dynamics, which fully coincide with the classical results by Schrödinger and Bohr. A wave function is regarded as a physical reality, which makes it possible to open up new possibilities in order to study the structure of the microcosm