Schroedinger wave equation

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... 

The relativistic (DSW) version incorporates the same approximations but starts from the Dirac rather than the Schroedinger wave equation,(11)... 

The foregoing discussion of valence is. of course, a simplified one. From ihe development of the quantum theory and its application to the structure of the atom, there has ensued a quantum theory of valence and of the structure of the molecule, discussed in this hook under Molecule. Topics thal are basically important to modem views of molecular structure include, in addition to those already indicated the Schroedinger wave equation the molecular orbital method (introduced in the article on Molecule) as well as directed valence bonds bond energies, hybrid orbitals, the effect of Van der Waals forces and electron-dcticiem molecules. Some of these subjects are clearly beyond the space available in this book and its scope of treatment. Even more so is their use in interpretation of molecular structure. [However, sec Crystal Field Theory and Ligand.)... 

Any description of the free atom begins with the Schroedinger wave equation for a single electron in the field of a positive point charge +Ze ... 

As in the case of atoms, the starting point is the Schroedinger wave equation corresponding to equation 1 ... 

In order to understand these and related differences between cyclopropane and other carbocycles, it is helpful first to consider the results of theoretical calculations. It is now possible to obtain high quality wave functions for cyclopropane and related molecules via ab initio molecular orbital calculations. The most satisfactory way in which to examine these wave functions is by calculating the charge densities. It must be remembered that the molecular orbitals themselves are just convenient functions for solving the Schroedinger wave equation ... 

The reason a single equation = ( can describe all real or hypothetical mechanical systems is that the Hamiltonian operator H takes a different form for each new system. There is a limitation that accompanies the generality of the Hamiltonian and the Schroedinger equation We cannot find the exact location of any election, even in simple systems like the hydrogen atom. We must be satisfied with a probability distribution for the electron s whereabouts, governed by a function (1/ called the wave function. 

We cannot solve the Schroedinger equation in closed fomi for most systems. We have exact solutions for the energy E and the wave function (1/ for only a few of the simplest systems. In the general case, we must accept approximate solutions. The picture is not bleak, however, because approximate solutions are getting systematically better under the impact of contemporary advances in computer hardware and software. We may anticipate an exciting future in this fast-paced field. 

In the few two- and three-dimensional cases that pemiit exact solution of the Schroedinger equation, the complete equation is separated into one equation in each dimension and the energy of the system is obtained by solving the separated equations and summing the eigenvalues. The wave function of the system is the product of the wave functions obtained for the separated equations. 

By this time, we have introduced so many approximations and restrictions on our wave function and energy spectrum that is no longer quite legitimate to call it a Schroedinger equation (Schroedinger s initial paper treated the hydrogen atom only.) We now write... 

Electron-electron repulsion integrals, 28 Electrons bonding, 14, 18-19 electron-electron repulsion, 8 inner-shell core, 4 ionization energy of, 10 localization of, 16 polarization of, 75 Schroedinger equation for, 2 triplet spin states, 15-16 valence, core-valence separation, 4 wave functions of, 4,15-16 Electrostatic fields, of proteins, 122 Electrostatic interactions, 13, 87 in enzymatic reactions, 209-211,225-228 in lysozyme, 158-161,167-169 in metalloenzymes, 200-207 in proteins ... 

Auxiliary Formulae— The first of the equations (7) represents the wave motion in the atom not subjected to an external field. This problem has been exhaustively treated by Schroedinger and its solution is known to be... 

Finally, U(q, Q) is the total coulomb potential energy, and V(Q) is the potential energy of the nuclei. The electronic wave function at a fixed nuclear configuration n(q, Q) is chosen to satisfy the partial Schroedinger equation... 

Abstract. Calculations of the non-linear wave functions of electrons in single wall carbon nanotubes have been carried out by the quantum field theory method namely the second quantization method. Hubbard model of electron states in carbon nanotubes has been used. Based on Heisenberg equation for second quantization operators and the continual approximation the non-linear equations like non-linear Schroedinger equations have been obtained. Runge-Kutt method of the solution of non-linear equations has been used. Numerical results of the equation solutions have been represented as function graphics and phase portraits. The main conclusions and possible applications of non-linear wave functions have been discussed. 

A molecular system consists of electrons and nuclei. Their position vectors are denoted hereafter as rel and qa, respectively. The potential energy function of the whole system is V(rel, qa). For simplicity, we skip the dependence of the interactions on the spins of the particles. The nuclei, due to their larger mass, are usually treated as classical point-like objects. This is the basis for the so called Bom-Oppenheimer approximation to the Schroedinger equation. From the mathematical point of view, the qnuc variables of the Schroedinger equation for the electrons become the parameters. The quantum subsystem is described by the many-dimensional electron wave function rel q ). 

We may think of a free-electron gas as having a vanishing potential (or equivalently, a constant potential, since wc can measure energies from that potential level). The Hamiltonian becomes simply -h V Ilm, and the solutions of the time-independent Schroedinger equation, Eq. (1-5), can be written as plane waves, e h Wc must apply suitable boundary conditions, and this is most conveniently done by imagining the crystal to be a rectangular parallelepiped, as shown in Fig. 15-1. Then wc apply periodic boundary conditions on the surface, as wc did following F.q. (2-2). The normalized plane-wave stales may be written as... 

The /bnu of the wave function outside depends upon the value of the constant potential chosen outside the sphere. Andersen recognized that the results were rather insensitive to that choice, so he selected the simplest value, a value equal to the energy of the state being considered, so that the kinetic energy of the state is zero in that region. Such a potential is shown in Fig. 20-1.3,b. The consequences of this assumption will turn out, in this section and the next, to be extraordinary. For this case, the form of the general solution of the Schroedinger equation is simply... 

The solution of the Schroedinger equation for a Morse potential is well known (see e.g. [5]), so that the suppression-enhancement amplimde can be calculated using the available information on the solution of the respective equation (see, e.g. [17]). Let us introduce two dimensionless quantities, the reduced wave vector k, associated with the incoming wave, and the wave-vector-like quantity ko, associated with the potential well depth ... 

At ultra-low temperatures, where we expect quantum dynamics to govern the translational motion of atoms, this magnetic potential should be included in some sort of Schroedinger equation for the translational motion. However, this inclusion is not trivial, since even the simplest description of such an atom must also include its internal angular-momentum degrees of freedom. For example, the wave function of an atom with angular momentum A obeys the following equation ... 

Despite its strong resemblance to the Schroedinger equation, this equation encompasses veiy different physics the wave function P(x) does not describe a single-particle state in the usual sense, but is rather a building block for the (approximate) ground-state A -particle wave function... 

As a first step in obtaining an approximate solution to the molecular Schroedinger equation, we agree to regard the many electron wave function P(r) as having been broken up into molecular orbitals y/i... 

Because the product (3.8) is a probability, it must integrate to 1.0 when all possible outcomes are taken into account. Consequently, the wave functions are multiplied by arbitrary constants (n ),/2 chosen to make this integral come out to 1.0 over the complete range of motion. These are called normalization constants. It is legitimate to multiply solutions to the Schroedinger equation by an arbitrary constant because they are elements of a closed binary vector space. Multiplication of a solution by any scalar yields another element in the space, hence the product of the normalization constant and the wave function (or any other state vector in Hilbert space) also is a solution. 

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