Dualism of the Heisenberg and Schrödinger Approaches to the Quantum States Entering a One-Dimensional Electron Gas

• 作者: Stanis?aw Olszewski
• 来源期刊: 现代物理(英文)
• 学科: 物理学
• 关键词: Fundamentals of the Modern Quantum Theory HEISENBERG PICTURE ITS Momentum RESULTS and the ENERGY Matrix Schr?dinger PICTURE and ITS ENERGY RESULTS
• 年，卷（期）: 2020,（3）
• 页码范围: 475-485
• 页数: 11页
• 分类号: O41

摘要

According to quantum mechanics, the commutation property of the energy Hamiltonian with the momentum operator should give the definite values not only for energy but also for the momentum quantum levels. A difficulty provided by the standing-like boundary conditions of the electron gas is that the Hamiltonian eigenfunctions are different than eigenfunctions of the momentum operator. In results the electron momenta are obtained from the correspondence rule between the classical and quantum mechanics given by Landau and Lifshits. As a consequence the statistics of solutions representing not only the energy values but also the electron momenta should be taken into account. In the Heisenberg picture of quantum mechanics, the momenta are easily obtained because the electron oscillators are there directly considered. In fact, the Hamiltonian entering the Heisenberg method can be defined in two different ways each giving the set of the electron energies known from the Schr?dinger’s approach.