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Ebook Fundamentals of quantum mechanics - For solid state electronics and optics: Part 2

Part 2 book "Fundamentals of quantum mechanics - For solid state electronics and optics" includes content: Multi-electron ions and the periodic table, interaction of atoms with electromagnetic radiation, simple molecular orbitals and crystalline structures, electronic properties of semiconductors and the p-n junction, the density matrix and the quantum mechanic Boltzmann equation. | 7 Multi-electron ions and the periodic table An electron in a hydrogenic atom or ion can occupy any of the jn sjmj i eigen states of the Hamiltonian of the atom or ion. In ions or atoms with more than one electron the solutions of the time independent Schrodinger equations become complicated because the electrons interact not only with the positively charged nucleus but also with each other. Particles with half-integer spin angular momentum such as electrons must also satisfy Pauli s exclusion principle which forbids two such particles to occupy the same quantum state. Furthermore the electrons in the multi-electron ion or atom are indistinguishable from one another. Taking these considerations into account the electrons will systematically fill all the available single-electron states of successively higher energies in multi-electron ions or atoms. Because of the nature of the quantum states occupied by the electrons the physical and chemical properties of the elements exhibit certain patterns and trends which form the basis of the periodic table. 7.1 Hamiltonian of the multi-electron ions and atoms Consider an ion with N electrons and Z protons in the nucleus for a neutral multi- electron atom Z N. Again because the nucleus is much heavier than the electrons we assume it to be stationary at the origin of a spherical coordinate system as shown in Figure 7.1. Including only the kinetic energy of the electrons and the potential energy due to the electrostatic interactions among the electrons and between the electrons and the nucleus the Hamiltonian of the electrons in the ion for the orbital part of the motion only is X N quot 2 2 Ze2 h X e2 N H À ri À þ 7 1 i 1 2m ri r i gt j 1 i j The form of the summation sign in the last term is to ensure that the electrostatic interaction between each pair of electrons is counted only once. The factor 110 7.1 Hamiltonian of the multi-electron ions and atoms 111 z rij rj ri y x Figure 7.1. Coordinate system used for the model .