Particle waves

The time-independent Schrbdinger equation describes the particle-wave duality, the square of the wave function giving the probability of finding the particle at a given position. 

Direct substitution of Eq. (8-83) leads to the proof of these, when we recall that the one-particle wave functions satisfy the relation... 

This is a vector having magnitude equal to the Schrddinger iV-particle wave function at the coordinates X. 

The appropriate expression for the operator H in the above equations is that appearing in Eq. (8-160). Eor a first example, consider an ideal gas without interactions. Assuming that the one-particle wave functions used in the population density operators are the energy eigenfunctions, then the matrix H0llA is diagonal, and we can write... 

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

The next step is the decomposition of the total density into single particle densities which are related to single particle wave functions by... 

In the next step, which is numerically the most demanding, the differential equations (3) are solved. Two possible strategies using a variational expansion of the single particle wave functions, /., are described below. After the eigenvalues and eigenfunctions have been found, a new ("output") charge density can be... 

Both photons and electrons are particle-waves, but different equations describe their properties. Table 7-1 summarizes the properties of photons and free electrons, and Example shows how to use these equations. 

This problem has two parts, one dealing with photons and the other with electrons. We are asked to relate the wavelengths of the particle-waves to their corresponding energies. Table 7J, emphasizes that photons and electrons have different relationships between energy and wavelength. Thus, we use different equations for the two calculations. 

This problem deals with particle-waves that have mass. Equation, the de Broglie equation, relates the mass and speed of an object to its wavelength. 

Mathematically, the position and momentum of a wave-particle are linked. Werner Heisenberg, a German physicist, found in the 1920s that the momentum and position of a particle-wave cannot be simultaneously pinned... 

Wave-like properties cause electrons to be smeared out rather than localized at an exact position. This smeared-out distribution can be described using the notion of electron density Where electrons are most likely to be found, there is high electron density. Low electron density correlates with regions where electrons are least likely to be found. Each electron, rather than being a point charge, is a three-dimensional particle-wave that is distributed over space in... 

We need ways to visualize electrons as particle-waves delocalized in three-dimensional space. Orbital pictures provide maps of how an electron wave Is distributed In space. There are several ways to represent these three-dimensional maps. Each one shows some important orbital features, but none shows all of them. We use three different representations plots of electron density, pictures of electron density, and pictures of electron contour surfaces. 

C07-0118. Neutrons, like electrons and photons, are particle-waves whose diffraction patterns can be used to determine the structures of molecules. Calculate the kinetic energy of a neutron with a wavelength of 75 pm. 

The essential features of the particle-wave duality are clearly illustrated by Young s double-slit experiment. In order to explain all of the observations of this experiment, light must be regarded as having both wave-like and particlelike properties. Similar experiments on electrons indicate that they too possess both particle-like and wave-like characteristics. The consideration of the experimental results leads directly to a physical interpretation of Schrodinger s wave function, which is presented in Section 1.8. 

In this section we state the postulates of quantum mechanics in terms of the properties of linear operators. By way of an introduction to quantum theory, the basic principles have already been presented in Chapters 1 and 2. The purpose of that introduction is to provide a rationale for the quantum concepts by showing how the particle-wave duality leads to the postulate of a wave function based on the properties of a wave packet. Although this approach, based in part on historical development, helps to explain why certain quantum concepts were proposed, the basic principles of quantum mechanics cannot be obtained by any process of deduction. They must be stated as postulates to be accepted because the conclusions drawn from them agree with experiment without exception. 

Equation (8.43) is the completeness relation for a complete set of symmetric (antisymmetric) multi-particle wave funetions. 

We may express the single-particle wave function tpniqd fhe product of a spatial wave function 0n(r,) and a spin function % i). For a fermion with spin such as an electron, there are just two spin states, which we designate by a(i) for m = and f i) for Therefore, for two particles there are three... 

It has been suggested that quasi-particle wave functions do not deviate much from LDA wave functions [26], Furthermore, in the evaluation of momentum densities shown in Figure 9, the characteristics of the quasi-particle states dominantly reflect on the occupation number densities which should be evaluated by using the general quasi-particle Green s function. In GWA, however, the corresponding occupation number densities are... 

The requirement that electrons (and fermions in general) have antisymmetric many-particle wave functions is called the Pauli principle, which can be stated as follows ... 

Although Einstein made use of the assumption that light behaves as a particle, there is no denying the validity of the experiments that show that light behaves as a wave. Actually, light has characteristics of both waves and particles, the so-called particle-wave duality. Whether it behaves as a wave or a particle depends on the type of experiment to which it is being subjected. In the study of atomic and molecular structure, it necessary to use both concepts to explain the results of experiments. 

Having now demonstrated that a moving electron can be considered as a wave, it remained for an equation to be developed to incorporate this revolutionary idea. Such an equation was obtained and solved by Erwin Schrodinger in 1926 when he made use of the particle-wave duality ideas of de Broglie even before experimental verification had been made. We will describe this new branch of science, wave mechanics, in Chapter 2. 

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