Wave equation electronic

The miderstanding of the quantum mechanics of atoms was pioneered by Bohr, in his theory of the hydrogen atom. This combined the classical ideas on planetary motion—applicable to the atom because of the fomial similarity of tlie gravitational potential to tlie Coulomb potential between an electron and nucleus—with the quantum ideas that had recently been introduced by Planck and Einstein. This led eventually to the fomial theory of quaiitum mechanics, first discovered by Heisenberg, and most conveniently expressed by Schrodinger in the wave equation that bears his name. 

The earliest appearance of the nonrelativistic continuity equation is due to Schrodinger himself [2,319], obtained from his time-dependent wave equation. A relativistic continuity equation (appropriate to a scalar field and formulated in terms of the field amplitudes) was found by Gordon [320]. The continuity equation for an electron in the relativistic Dirac theory [134,321] has the well-known form [322] ... 

How are the electrons distributed in an atom You might recall from your general chemistry course that, according to the quantum mechanical model, the behavior of a specific electron in an atom can be described by a mathematical expression called a wave equation—the same sort of expression used to describe the motion of waves in a fluid. The solution to a wave equation is called a wave function, or orbital, and is denoted by the Greek letter psi, i/y. 

An atom consists of a positively charged nucleus surrounded by one or more negatively charged electrons. The electronic structure of an atom can be described by a quantum mechanical wave equation, in which electrons are considered to occupy orbitals around the nucleus. Different orbitals have different energy levels and different shapes. For example, s orbitals are spherical and p orbitals are dumbbell-shaped. The ground-state electron configuration of an... 

Wave equation (Section 1.2) A mathematical expression that defines the behavior of an electron in an atom. 

In 1926 Erwin Schrodinger (1887-1961), an Austrian physicist, made a major contribution to quantum mechanics. He wrote down a rather complex differential equation to express the wave properties of an electron in an atom. This equation can be solved, at least in principle, to find the amplitude (height) of the electron wave at various points in space. The quantity ip (psi) is known as the wave function. Although we will not use the Schrodinger wave equation in any calculations, you should realize that much of our discussion of electronic structure is based on solutions to that equation for the electron in the hydrogen atom. 

The system to be considered consists of two nuclei and one electron. For generality let the nuclear charges be ZAe and ZBe. From Born and Oppenheimer s results it is seen that the first step in the determination of the stationary states of the system is the evaluation of the electronic energy with the nuclei fixed an arbitrary distance apart. The wave equation is... 

The form of the functions may be closely similar to that of the molecular orbitals used in the simple theory of metals. If there are M interatomic positions in the crystal which might be occupied by any one of the N electron-pair bonds, then the M functions linear aggregates that approximate the solutions of the wave equation with inclusion of the interaction terms representing resonance. This combination can be effected with use of Bloch factors ... 

Wave mechanics is based on the fundamental principle that electrons behave as waves (e.g., they can be diffracted) and that consequently a wave equation can be written for them, in the same sense that light waves, soimd waves, and so on, can be described by wave equations. The equation that serves as a mathematical model for electrons is known as the Schrodinger equation, which for a one-electron system is... 

In the VB method, a wave equation is written for each of various possible electronic structures that a molecule may have (each of these is called a canonical form), and the total )/ is obtained by summation of as many of these as seem plausible, each with its weighting factor ... 

The free electron resides in a quantized energy well, defined by k (in wave-numbers). This result Ccm be derived from the Schroedinger wave-equation. However, in the presence of a periodic array of electromagnetic potentials arising from the atoms confined in a crystalline lattice, the energies of the electrons from all of the atoms are severely limited in orbit and are restricted to specific allowed energy bands. This potential originates from attraction and repulsion of the electron clouds from the periodic array of atoms in the structure. Solutions to this problem were... 

Quantum numbers The four quantum numbers—principal, angular momentum, magnetic, and spin—arise from solutions to the wave equation and govern the electron configuration of atoms. 

Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. 

An expression for e(k) in the case of a Fermi gas of free electrons can be obtained by considering the effect of an introduced point charge potential, small enough so the arguments of perturbation theory are valid. In the absence of this potential, the electronic wave functions are plane waves V 1/2exp(ik r), where V is the volume of the system, and the electron density is uniform. The point charge potential is screened by the electrons, so that the potential felt by an electron, O, is due to the point charge and to the other electrons, whose wave functions are distorted from plane waves. The electron density and the potential are related by the Poisson equation,... 

ABBA molecules, 631-633 HCCS radical, 633-640 perturbative handling, 641-646 theoretical principles, 625-633 Hamiltonian equation, 626-628 vibronic problem, 628-631 Thouless determinantal wave function, electron nuclear dynamics (END) ... 

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