Wave equation, matrix formulation

Earlier studies of positron-molecule elastic scattering did not involve such detailed descriptions of the scattering process as do the variational and R-matrix formulations. Instead, the interaction between the positron and the molecule was represented by a relatively simple model potential, and the positron wave function F(r 1) was assumed to satisfy the equivalent single-particle Schrodinger equation... [Pg.128]

Schrodinger discovered the equation that bears his name in 1926, and it has provided the foundation for the wave-mechanical formulation of quantum mechanics. Heisenberg had independently, and somewhat earlier, proposed a matrix formulation of the problem, which Schrodinger later showed was an equivalent alternative to his approach. We choose to present Schrodinger s version because its physical interpretation is much easier to understand. [Pg.141]

It is commonly accepted that the old quantum theory era spans from the birth of Planck s quantum hypothesis to the formulation of Schrodinger s equation. This section describes the old quantum theory in three parts the failure of classical mechanics, the birth of the quantum theory, and the completion of wave mechanics.5 8) This century obviously began with the birth of quantum theory. Many researchers appeared on the scene of quantum theory at the time, but we remember mostly the contributions of four researchers Max Planck (1901), Albert Einstein (1905), Niels Bohr (1913), and de Broglie (1923). Then Schrodinger proposed the new wave equation to conclude the age of the old quantum theory. Heisenberg established matrix mechanics and formulated the uncertainty principle. [Pg.21]

Since at the voltages with which we are concerned there are neither abrupt changes in the indices of refraction (except at the boundaries) nor any periodicity in the structure, we can safely neglect the reflected waves and solve Eq. (A.13) for transmitted waves only. This transforms Eq. (A.13) into a 2x2 differential matrix equation. The 4x4 matrix formulation is necessary only when considering the reflections at surfaces. Neglecting the reflection eigenmodes also eliminates the effects of interference as the surface reflections can be treated independently. [Pg.141]

We could numerically solve the Pople-Nesbet equations for minimal basis STO-3G H2, just as we have solved them for CH3, N2, and O2. An appropriate unrestricted initial guess would be required if the iterations were to lead to an unrestricted solution rather than to the restricted solution. The transition from a restricted to an unrestricted wave function will be more transparent, however, if, rather than obtain a numerical solution to the Pople-Nesbet matrix equations, we formulate the problem in an analytical fashion. [Pg.222]

We have used a 4 x 4 matrix formulation of the electromagnetic wave equations in stratified media to compute the reflectance and transmittance of single-domain cholesteric liquid crystal films. Our technique is basically equivalent to the 4x4 matrix technique first described by Teitler and Henvis, ) applied by them to... [Pg.39]

In this chapter we present the time-dependent quantum wave packet approaches that can be used to compute rate constants for both nonadiabatic and adiabatic chemical reactions. The emphasis is placed on our recently developed time-dependent quantum wave packet methods for dealing with nonadiabatic processes in tri-atomic and tetra-atomic reaction systems. Quantum wave packet studies and rate constants computations of nonadiabatic reaction processes have been dynamically achieved by implementing nuclear wave packet propagation on multiple electronic states, in combination with the coupled diabatic PESs constructed from ab initio calculations. To this end, newly developed propagators are incorporated into the solution of the time-dependent Schrodinger equation in matrix formulism. Applications of the nonadiabatic time-dependent wave packet approaches and the adiabatic ones to the rate constant computations of the nonadiabatic tri-atomic F (P3/2, P1/2) + D2 (v = 0,... [Pg.228]

However, similar substitutions cannot be made in the case of the system of equations for state-specific wave operators, eqs. (4.59). Specifically, we are not able to replace this system of equations by a single equation in operator form, whilst simultaneously retaining its Brillouin-Wigner character. To some extent, this replacement can be achieved within a matrix formulation. By using the multi-root wave operator (4.60), the system of eqs. (4.59) can be shown to be equivalent to... [Pg.146]

A matrix formulation of these equations was developed in Section 14.1.2, using an orthonormal basis for the expansion of the perturbed wave functions (14.1.28). In this exercise, this matrix formulation is generalized to the expansion of the perturbed states ... [Pg.286]

Just as the variational condition for an HF wave function can be formulated either as a matrix equation or in terms of orbital rotations (Sections 3.5 and 3.6), the CPFIF may also be viewed as a rotation of the molecular orbitals. In the absence of a perturbation the molecular orbitals make the energy stationary, i.e. the derivatives of the energy with respect to a change in the MOs are zero. This is equivalent to the statement that the off-diagonal elements of the Fock matrix between the occupied and virtual MOs are zero. [Pg.245]

The solution of single-particle quantum problems, formulated with the help of a matrix Hamiltonian, is possible along the usual line of finding the wave-functions on a lattice, solving the Schrodinger equation (6). The other method, namely matrix Green functions, considered in this section, was found to be more convenient for transport calculations, especially when interactions are included. [Pg.223]

Here, the potential energy and the wavefunction depend on the three space coordinates x, y, z, which we write for brevity as r. We have thus arrived at the time-dependent Schriidinger equation for the amplitude I (r, t) of the matter waves associated with the particle. Its formulation in 1926 represents the starting point of modem quantum mechanics. (Heisenberg in 1925 proposed another version known as matrix mechanics.)... [Pg.19]

The theory of the saturable absorption effect in single-wall carbon nanotubes has been elaborated. The kinetic equations for density matrix of n-electrons in a single-wall carbon nanotube have been formulated and solved analytically within the rotating wave approximation. The dependence of the carbon nanotube absorption coefficient on the driving field intensity has been shown to be different from the absorption coefficient behavior predicted forthe case of two level systems. [Pg.108]

Just as the variational condition for an HE wave function can be formulated either as a matrix equation or in terms of orbital rotations (Sections 3.5 and 3.6), the CPHF may... [Pg.245]

Kirtman formulated a density matrix treatment using a separated pair an-satz [86], He derived variational equations in the local space approximation to determine fragment wave functions and discussed the role of strong orthogonality in the localization of the wave function. [Pg.69]

A formal similarity arises because both the scattering matrix in quantum electrodynamics and the wave operator in a full coupled-cluster method [cf. Equations (1) and (4)] are exponential operators. In Bogo-liubov s axiomatic formulation of the scattering matrix,29... [Pg.223]

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