Wave-particle duality equations

Erwin Schrodinger (1887-1961 Nobel Prize for physics 1932) transferred the concept of wave-particle duality of matter developed by L. V. de Broglie for electrons to the whole atom and thus developed wave mechanics. The Schrodinger equation allows a description of orbitals as the probability of the location of the electrons. Wave mechanics represented a significant development, but were subsequently shown to be insufficient. 

Schrodinger equation valence electrons wave function wavelength, X wave mechanical model wave-particle duality of nature... 

The electron, discovered by J. J. Thomson in 1895, was first considered as a corpuscule, a piece of matter with a mass and a charge. Nowadays things are viewed differently. We rather speak of a wave-particle duality whereby electrons exhibit a wavelike behavior. But, in Levine s own words [45], quanmm mechanics does not say that an electron is distributed over a large region of space as a wave is distributed it is the probability patterns (wavefunctions) used to describe the electron s motion that behave like waves and satisfy a wave equation. 

In classical mechanics, Newton s laws of motion determine the path or time evolution of a particle of mass, m. In quantum mechanics what is the corresponding equation that governs the time evolution of the wave function, F(r, t) Obviously this equation cannot be obtained from classical physics. However, it can be derived using a plausibility argument that is centred on the principle of wave-particle duality. Consider first the case of a free particle travelling in one dimension on which no forces act, that is, it moves in a region of constant potential, V. Then by the conservation of energy... 

But by de Broglie s principle of wave-particle duality, this free particle has associated with it a wave vector, = p/h, and angular frequency, = E/h (cf eqs (2.13) and (2.14)), which substituting into the above equation gives... 

The de Broglie concept of wave-particle duality enables us to calculate the wavelength of an electron. According to the de Broglie equation,... 

This is an example of the de Broglie wave-particle duality. The resulting wave equation is... 

For the time being, let us consider the conventional view wave-particle duality. Then, propogation of photon is the same as propagation of electromagnetic field E, B. In free space the charge density is null everywhere, except possibly at the source. The photon is chargeless hence, if Maxwell s equations are applicable to a photon in vacuum, pe = 0 everywhere. This leads to some contradiction. 

Considerations of this sort led us to suggest that, in order to avoid violations of causality within the wave-particle duality, there are two possible interpretations of Maxwell s equations [76] ... 

As photon momentum p = E/c, the quantum assumption E = hu implies that p = hu/c = h/X. This relationship between mechanical momentum and wavelength is an example of electromagnetic wave-particle duality. It reduces the Compton equation into ... 

Equation (1.5) establishes a bridge between a description of fight as an (electromagnetic) wave of frequency v and as a beam of -q energy particles. If phenomena related to time averages, such as diffraction and interference, can be easily interpreted in terms of waves, other phenomena, involving a one-to-one relation such as the photoelectric and the Compton effects, require a description based on corpuscular attributes. This wave-particle duality reflects the use of one or the other description depending on the experiment performed, while no experiment exists which exhibits both aspects of the duality simultaneously. 

Quantum mechanics represents one of the cornerstones of modem physics. Though there were a variety of different clues (such as the ultraviolet catastrophe associated with blackbody radiation, the low-temperature specific heats of solids, the photoelectric effect and the existence of discrete spectral lines) which each pointed towards quantum mechanics in its own way, we will focus on one of these threads, the so-called wave-particle duality, since this duality can at least point us in the direction of the Schrodinger equation. 

Quantum chemistry is the appfication of quantum mechanical principles and equations to the study of molecules. In order to nnderstand matter at its most fundamental level, we must use quantum mechanical models and methods. There are two aspects of quantum mechanics that make it different from previous models of matter. The first is the concept of wave-particle duality that is, the notion that we need to think of very small objects (such as electrons) as having characteristics of both particles and waves. Second, quantum mechanical models correctly predict that the energy of atoms and molecules is always quantized, meaning that they may have only specific amounts of energy. Quantum chemical theories allow us to explain the structure of the periodic table, and quantum chemical calculations allow us to accurately predict the structures of molecules and the spectroscopic behavior of atoms and molecules. 

Electronic structure calculations as we will consider them here concentrate on solving the Schrddinger equation. This equation takes into account one of the main postulates of quantum mechanics, namely that of the wave-particle duality. All particles, in particular the very light ones such as the electrons, must be described in terms of a wave function, E. The square of this wave function gives the probability of finding the particle at a given position. In the time-independent Schrddinger equation, which we will consider here, E depends only on the location of all particles r, and is solved from ... 

The formulation of the rules has gone through various phases, the most comprehensive statement being that based upon the wave equation. This expresses the possible energy levels for any kind of system, and provides information about their multiplicity. The equation is not itself based upon any explicit theory of the nature of things, except in so far as it contains a general implication of the wave-particle duality in systems of minute enough dimensions. 

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