Wave Model of the Electron

The ripple that occurs in undistorted space should be isotropic and of a type described with a general wave equation 

This solution represents two spherical waves (T 5.5.6), one travelling toward the origin, the other from the origin. The factor 1/r, without which U would not be a solution of (2) and therefore not a wave, accounts for attenuation of a spherical wave as it moves from its source. By suitable choice of f and f2 diverse wave complexes can be formed, of which standing waves, defined with a condition U(r,t) = F(r) G(t) in which F and G represent new functions, are perhaps the simplest. Allowing for time-inversion symmetry a more general solution of (2) is 

This result is consistent with the Lorentz transformations 

Waves in the moving cavity therefore lock together in a relativistically invariant way. 

In order to reconcile this relativistic invariance with De Broglie s wavelength hypothesis, for waves at rest and in a moving frame, 

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Because quantum theory is supposed only to deal with observables it may be, and is, argued as meaningless to enquire into the internal structure of an electron, until it has been observed directly. To treat an electron as a point particle is therefore considered mathematically sufficient. However, an electron has experimentally observed properties such as the Compton wavelength and spin, which can hardly be ascribed to a point particle. The only reasonable account of such properties has, to date, been provided by wave models of the electron. 

Interatomic distance is calculated by mathematical modelling of the electron exchange that constitutes a covalent bond. Such a calculation was first performed by Heitler and London using Is atomic wave functions to simulate the bonding in H2. To model the more general case of homonuclear diatomic molecules the interacting atoms in their valence states are described by monopositive atomic cores and two valence electrons with constant wave functions (3.36). 

The effects of the crystallographic face and the difference between metals are evidence of the incorrectness of the classical representations of the interface with all the potential decay within the solution (Fig. 3.13a). In fact a discontinuity is physically improbable and experimental evidence mentioned above confirms that it is incorrect, the schematic representation of Fig. 3.136 being more correct. This corresponds to the chemical models (Section 3.3) and reflects the fact that the electrons from the solid penetrate a tiny distance into the solution (due to wave properties of the electron). In this treatment the Galvani (or inner electric) potential, (p, (associated with EF) and the Volta (or outer electric) potential, ip, that is the potential outside the electrode s electronic distribution (approximately at the IHP, 10 5cm from the surface) are distinguished from each other. The difference between these potentials is the surface potential x (see Fig. 3.14 and Section 4.6). 

Massa, L., Goldberg, M., Frishberg, C., Boehme, R. F. and LaPlaca, S. J. Wave functions derived by quantum modeling of the electron density from coherent X-ray diffraction beryllium metal. Phys. Rev. Lett. 55, 622-625 (1985). 

BohrPT Bohr Model of the Atom - Four Quantum Numbers -Electron Configuration of the Atom - Electron Shells - Shapes of Orbitals - Wave Nature of the Electron - Wave Functions, Radial and... 

De Broglie s hypothesis and Heisenberg s uncertainty principle set the stage for a new and more broadly applicable theory of atomic structure. In this approach, any attempt to define precisely the instantaneous location and momentum of the electron is abandoned. The wave nature of the electron is recognized, and its behavior is described in terms appropriate to waves. The result is a model that precisely describes the energy of the electron while describing its location not precisely but rather in terms of probabilities. 

The change in the electric potential within the double layer is illustrated in Fig. 1.1.3. It is assumed that the electrode is charged negatively. The electric potential, (f)M, is virtually constant throughout the metallic phase except for the layers of metal atoms located next to the solution, where a discontinuity in the metal structure takes place and the wave properties of the electron are exposed (the jellium model [1, 3]). This effect is much stronger in semiconductor electrodes, where the accessible electronic levels are more restricted [5]. 

Open problems in writing the basic organic chonistry textbook include the selection of concepts for the representation of the material, but also the level of the explanation of the complex phenomena such as reaction mechanisms or the electron structure. Here I propose the compromises. First compromise is related to the mode of the systematization of the contents, which can traditionally be based either on the classes of compounds, or on the classes of reactions. Here, the main chapter titles contain the reaction types, but the subtitles involve the compound classes. The electronic effects as well as the nature of the chemical bond is described by using the quasi-classical approach starting with the wave nature of the electron, and building the molecular orbitals from the linear combination of the atomic orbitals on the principle of the qualitative MO model. Hybridization is avoided because all the phenomena on this level can be simply explained by non-hybridized molecular orbitals. 

In the kinetic field there is another way in which the small mass of the proton may be important. It is well known that the behaviour of electrons cannot be accounted for in terms of a particulate model but that it is necessary to take into account the wave nature of the electron on the other hand, it is usually supposed that the motion of nuclei can be described with sufficient accuracy by the laws of classical mechanics. This is undoubtedly true for most nuclei, but calculation shows that the proton may, on account of its small mass, show considerable deviations from classical behaviour. This phenomenon is often described as the tunnel effect and should be detectable experimentally, especially by a detailed analysis of kinetic isotope effects. At present the experimental evidence is meagre, but the problem is an interesting one and will be treated in some detail. 

The polaron model is properly an extension of the primitive cavity model. In both, the electron is considered to be solvated by a number of ammonia molecules. However, in the cavity model one considers the localization or solvation of the electron as described by a cavity of some shape whose boundaries act as limiting points for the potential or the electronic wave function. In the polaron model, on the other hand, the electron is considered to polarize the surrounding ammonia molecules in such a way as to provide a trapping potential for itself. The potential is derived from the laws of electrostatics adapted to the quantum mechanical description of the electron density in terms of the electronic wave function. In the final development of the theory one would of course, require self-consistency between the wave function of the electron and the potential in which it moves. It is possible that the end result may indicate that the electronic wave function is in fact almost localised within a definite volume of certain shape. However, no such assumption is made a priori as in the cavity model. 

For the cavity model, a careful analysis of the two-electron polaron is necessary to test the correctness of the conclusion that it is unstable. Also, for the one-electron polaron, the orthogonality of the unpaired electron wave function to the wave functions of the electrons on neighboring ammonia molecules has to be considered carefully for possible contributions to the observed and proton Knight shifts in N.M.R. measurements. 

In the quantum mechanical model of the hydrogen atom, the wave behaviour of the electron is described by a mathematical function known as the wave function, /. Each allowed wave function has a precisely known energy, but the location of the electron cannot be determined exacdy, only the probability. 

Unlike the neutron paramagnetic form factor, the de Haas van Alphen effect does not give direct information on the wave function of the electrons. A model of the Fermi surface, such as that obtained from a band calculation, is needed to sort out the various pieces. On the other hand, the experiment usually provides such a large amqpnt of specific information that only a highly realistic model can have any chance to fit the observed frequencies and their dependences on the field direction. 

The concept of spin tends to emphasize the particie nature of the electrons, yet the name wave mechanical model is meant to emphasize the wave nature of the electrons. This is an unresolved dilemma in quantum physics. With no analog of electron spin in our everyday macroscopic world, any attempt we make to try to explain spin will be inadequate. 

The surprising implication is that Dirac s equation does not allow of a self-consistent single-particle interpretation, although it has been used to calculate approximate relativistic corrections to the Schrodinger energy spectrum of hydrogen. The obvious reason is that a 4D point particle is without duration and hence undefined. An alternative description of elementary units of matter becomes unavoidable. Prompted by such observation, Dirac [3] re-examined the classical point model of the electron only to find that it has three-dimensional size, with an interior that allows superluminal signals. It all points at a wave structure with phase velocity > c. 

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