TITLE:
Dualism of the Heisenberg and Schrödinger Approaches to the Quantum States Entering a One-Dimensional Electron Gas
AUTHORS:
Stanisław Olszewski
KEYWORDS:
Fundamentals of the Modern Quantum Theory, Heisenberg Picture, Its Momentum Results and the Energy Matrix, Schrödinger Picture and Its Energy Results
JOURNAL NAME:
Journal of Modern Physics,
Vol.11 No.3,
March
25,
2020
ABSTRACT: 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.