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Journal of Physics B Atomic Molecular and Optical Physics

On the rules for filling electron shells and the properties of the atomic Hamiltonian

A. A. Danshin, M. I. Gurevich, A. A. Kovalishin

Published Jun 2, 2021• 0 citations
Publisher

Abstract

Abstract The aim of this work is to determine the dependence of the total energy of the atomic system on the mutual orientation of the orbital angular moments of electrons. It is shown that the minimum value of the energy is attained for those configurations that are an eigenfunction of the orbital angular momentum operator L ̂ 2 . Thus, at least for individual atoms of the periodic table it is shown that Dirac’s postulate on the necessity of a wave function to be an eigenfunction of L ̂ 2 is a consequence that follows from the properties of the Hamiltonian, and it is not a complementary condition. Additionally, it is shown that the second Hund’s rule is incorrect with respect to the mutual orientation of the electron orbital angular momentum projections.

DOI
10.1088/1361-6455/ac0768

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Abstract The aim of this work is to determine the dependence of the total energy of the atomic system on the mutual orientation of the orbital angular moments of electrons. It is shown that the minimum value of the energy is attained for those configurations that are an eigenfunction of the orbital angular momentum operator L ̂ 2 . Thus, at least for individual atoms of the periodic table it is shown that Dirac’s postulate on the necessity of a wave function to be an eigenfunction of L ̂ 2 is a consequence that follows from the properties of the Hamiltonian, and it is not a complementary condition. Additionally, it is shown that the second Hund’s rule is incorrect with respect to the mutual orientation of the electron orbital angular momentum projections.
This chapter analyses the orbital angular momentum to three dimensions and introduces the spin angular momentum of the electron. The description of a localized particle in orbit requires a superposition of eigenfunctions analogous to the packet state that describes motion in one dimension. If the distance between molecules is fixed, all the energy changes are rotational kinetic energy changes, and the quantization of total angular momentum gives rise to discrete rotational energy levels. The analysis of orbital angular momentum is a powerful tool for describing molecular rotation. The Stern-Gerlach experiment and many other lines of evidence show that the measured magnetic moment associated with an electron spin is almost exactly one Bohr magneton. When orbital angular momentum and spin angular momentum both exist in the same atom, the magnetic moments that result from orbital and spin angular momenta interact to cause a splitting in the corresponding energy level.
Angular momentum plays a similarly important role in atomic systems, such as an electron subject to the central force provided by the Coulomb field of a nucleus. This chapter introduces the properties of angular momentum, both orbital and spin, in the quantum-mechanical scheme of things. The angular momentum of atomic systems is inseparable from the magnetic properties of atoms. The various permitted values of the quantized angular momentum play a central role in determining the number and variety of possible quantum states in atomic systems. The Stern-Gerlach experiment was deliberately designed to test whether or not the angular momentum of individual neutral atoms is quantized in a magnetic field. As in the development of the Schrodinger equation, classical physics gives some hints of the correct forms of these angular momentum operators. The chapter discusses rough estimate using the Neils Bohr theory and attributing the magnetic moment to the orbital motion of the electron.
Angular momentum is a very important and revealing property of many physical systems. In classical mechanics the principle of conservation of angular momentum is a powerful aid to the solution of such problems as the orbits of planets and satellites and the behaviour of gyroscopes and tops. The role of angular momentum in quantum mechanics is probably even more important and this will be the subject of the present chapter and the next. We shall find that the operators representing the components of angular momentum do not commute with each other, although they all commute with the operator representing the total angular momentum. It follows that no pair of these components can be measured compatibly and we shall therefore look for a set of eigenfunctions that are common to the operators representing the total angular momentum and one of its components. We shall find that the angular momentum eigenvalues always form a discrete set and that, in the case of a central field, the operators commute with the Hamiltonian, implying that the total angular momentum and one component can be measured compatibly with the energy in this case, so that their values, once measured, remain constant in time. Consideration of angular momentum will also enable us to make predictions about the behaviour of atoms in magnetic fields, and we shall find that these are in agreement with experiment only if we assume that the electron (in common with many other fundamental particles) has an intrinsic “spin” angular momentum in addition to the “orbital” angular momentum associated with its motion. The interaction between spin and orbital angular momenta leads to detailed features of the atomic spectra which are known as “fine structure.” We shall also use the measurement of angular momentum to illustrate the quantum theory of measurement discussed in the previous chapter.
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.