Electron wave-like properties

A convenient orbital method for describing eleetron motion in moleeules is the method of molecular orbitals. Molecular orbitals are defined and calculated in the same way as atomic orbitals and they display similar wave-like properties. The main difference between molecular and atomic orbitals is that molecular orbitals are not confined to a single atom. The crests and troughs in an atomic orbital are confined to a region close to the atomic nucleus (typieally within 1-2 A). The electrons in a molecule, on the other hand, do not stick to a single atom, and are free to move all around the molecule. Consequendy, the crests and troughs in a molecular orbital are usually spread over several atoms. 

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... 

Because electrons have wave-like properties, orbital interactions involve similar addition or subtraction of wave functions. When two orbitals are superimposed, one result is a new orbital that is a composite of the originals, as shown for molecular hydrogen in Figure 10-2. This interaction is called orbital overlap, and it is the foundation of the bonding models described in this chapter. 

Because electrons have wave-like properties, orbital Interactions Involve addition or subtraction of amplitudes, as we describe in Section 10-1. So far, we have described only additive orbital interactions. The wave amplitudes add in the overlap region, generating a new bonding orbital with larger amplitude between the nuclei. However, a complete mathematical treatment of orbital overlap requires that orbitals be conserved. In other words, whenever several orbitals interact, they must generate an equal number of new orbitals. 

De Broglie s hypothesis of matter waves received experimental support in 1927. Researchers observed that streams of moving electrons produced diffraction patterns similar to those that are produced hy waves of electromagnetic radiation. Since diffraction involves the transmission of waves through a material, the observation seemed to support the idea that electrons had wave-like properties. 

In this section, you saw how the ideas of quantum mechanics led to a new, revolutionary atomic model—the quantum mechanical model of the atom. According to this model, electrons have both matter-like and wave-like properties. Their position and momentum cannot both be determined with certainty, so they must be described in terms of probabilities. An orbital represents a mathematical description of the volume of space in which an electron has a high probability of being found. You learned the first three quantum numbers that describe the size, energy, shape, and orientation of an orbital. In the next section, you will use quantum numbers to describe the total number of electrons in an atom and the energy levels in which they are most likely to be found in their ground state. You will also discover how the ideas of quantum mechanics explain the structure and organization of the periodic table. 

Because electrons have wave-like properties, atomic orbitals can be described with wave functions and can overlap in different ways ... 

In the late nineteenth century, a whole set of experiments progressively lead to the conclusion that classical physics, namely, Newtonian mechanics, thermodynamics, and nascent electromagnetism, were unable to explain empirical evidence gathered by experimentalists. Scientists of that time were unable to conciliate two apparent contradictory aspects exhibited by radiation and matter. Some experiments demonstrated that light behaved like a wave, while others showed a rather corpuscular nature. On the other hand, electrons, protons, and the other massive particles would manifest wave-like properties in certain experimental conditions. 

I wave-like properties, and allow de Broglie s formula (eqn 2.21) to be verified quantitatively. No-one who has used an LEED apparatus can be left in any doubt that this theory is correct. Simply turning a knob on the control panel changes the accelerating voltage for electrons, and hence their kinetic energy and momentum. The diffraction spots then move in exactly the way predicted by eqn 2.21, in conjunction with simple diffraction theory (see Problem 2 below). 

In 1924, considering the nature of the light and matter, Lois de Broglie proposed that small particles sometimes show wave-like properties. In 1927, his hypothesis was proved by the deviation of electron beams like X-rays by a crystal. 

The observation that the wavelength of light is linked to the particle-like momentum of a photon prompted de Broglie to postulate the likelihood of an inverse situation whereby particulate objects may exhibit wave-like properties. Hence, an electron with linear momentum p could under appropriate conditions exhibit a wavelength A = h/p. The demonstration that an electron beam was diffracted by periodic crystals in exactly the same way as X-radiation confirmed de Broglie s postulate and provided an alternative description of the electronic stationary states on an atom. Instead of an accelerated particle the orbiting electron could be described as a standing wave. To avoid self-destruction by wave interference it is necessary to assume an... 

Another popular approach to wave-particle duality, which originated with Schrodinger, was to view the quantum particle as a wave structure or wave packet. This model goes a long way towards the rationalization of particle-like and wave-like properties in a single construct. However, the simplified textbook discussion, which is unsuitable for the definition of quantum wave packets, relies on the superposition of many waves with a continuous spread of wavelengths, defines a dispersive wave packet, and therefore fails in modelling an electron as a stable particle. 

Shortly after Bohr s proposal, Louis de Broglie made an important proposal. He said that if waves have matter-like properties, then matter should have wave-like properties. It is not necessary for you to know the de Broglie equation, but you should understand that it predicted that matter of normal mass would create infinitesimally small waves. It is only matter with an extremely small mass, like an electron, and traveling at high speed that will emit appreciable wavelengths. 

Quantum theory was developed primarily to find an explanation for the stability of atomic matter, specifically the planetary model of the hydrogen atom. In the Schrodinger formulation the correct equation was obtained by recognizing the wave-like properties of an electron. The first derivation by Schrodinger [30] was done by analogy with the relationship that was known to exist between wave optics and geometrical optics in the limit where the index of refraction, n does not change appreciably over distances of order A. This condition leads to the eikonal equation (T3.15)... 

Diffraction is an interference phenomenon occurring when waves are scattered by objects in different positions. Electron diffraction depends on the wave-like properties of electrons and can be used in various ways. One application in inorganic chemistry is the determination of bond lengths and angles of molecules in the gas phase. Its scope is limited as only volatile substances may be studied, and a full interpretation is only possible for molecules containing rather few atoms. 

When dealing with the interactions of crystals with particles that can display wave-like properties, like photons, phonons or electrons, it is useful to introduce a reciprocal lattice associated with the real (or direct) crystal lattice. Let us consider a set of vectors R constituting a given 3D BL and a plane wave elk r. For special choices of k, it can be shown that k can also display the periodicity of a BL, known as the reciprocal lattice of the direct BL. For all R of the direct BL, the set of all wave vectors G belonging to the reciprocal lattice verify the relation... 

If an electron has wave-like properties, there is an important and difficult consequence it becomes impossible to know exactly both the momentum and position of the electron at the same instant in time. This is a statement of Heisenberg s uncertainty principle. In order to get around this problem, rather than trying to define its exact position and momentum, we use the probability of finding the electron in a given volume of space. The probability of finding an electron at a given point in space is determined from the function ij where is a mathematical function which describes the behaviour of an electron-wave ip is the wavefunction. 

The size-evolution of the physical properties from atom to bulk might also be related in part to the variation of the surface-to-volume ratio. In addition to these classical effects, however, the quantum mechanical properties of the electrons play an equally important role. These so-called quantum-size effects can be understood most simply by realizing that a conduction electron in a metal has both particle-and wave-like properties, according to the famous particle-wave duality of quantum mechanics. Treated as a wave-phenomenon, the electron in a metal has a wavelength of one to a few nanometers. The wave-character of the electron will... 

Schrodinger and de Broglie suggested a "wave-particle duality" for small particles—that is, if electromagnetic radiation showed some particle-like properties, then perhaps small particles might exhibit some wave-like properties. Explain. How does the wave mechanical picture of the atom fundamentally differ from the Bohr model How do wave mechanical orbitals differ from Bohr s orbits What does it mean to say that an orbital represents a probability map for an electron ... 

As we discussed earlier, the electron has both particle-like and wave-like properties. Thus, the picture of an electron as a spinning charged sphere is, strictly speaking, just a useful pictorial representation that helps us understand the two directions of magnetic field that an electron can j... 

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