Schrodinger equation basic principles

Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle. Several experiments, which lead to a physical interpretation of the wave function, are also described. In Chapter 2, the Schrodinger differential wave equation is introduced and the wave function concept is extended to include particles in an external potential field. The formal mathematical postulates of quantum theory are presented in Chapter 3. 

But there is a more basic difficulty in the Hohenberg-Kohn formulation [19-21], which has to do with the fact that the functional iV-representability condition on the energy is not properly incorporated. This condition arises when the many-body problem is presented in terms of the reduced second-order density matrix in that case it takes the form of the JV-representability problem for the reduced 2-matrix [19, 22-24] (a problem that has not yet been solved). When this condition is not met, an energy functional is not in one-to-one correspondence with either the Schrodinger equation or its equivalent variational principle therefore, it can lead to energy values lower than the experimental ones. 

Following the development of quantum theory by Heisenberg [1] and Schrodinger [2] and a few further discoveries, the basic principles of the structure of atoms and molecules were described around 1930. Unfortunately, the complexity of the Schrodinger equation increases dramatically with the number of electrons involved in a system, and thus for a long time the hydrogen and helium atoms and simple molecules as H2 were the only species whose properties could really be calculated from these first principles. In 1929, Dirac [3] wrote ... 

The Schrodinger equation cannot be solved exactly for any molecule with more than one ( ) electron. Thus approximations are used the less serious these are, the higher the level of the ab initio calculation is said to be. Regardless of its level, an ab initio calculation is based only on basic physical theory (quantum mechanics) and is in this sense from first principles . 

Because the Schrodinger wave equation is an expression of some of the most basic principles of quantum mechanics, it cannot be derived from a more fundamental equation. However, it is possible to arrive at the equation by considering a general wave equation and applying the de Broglie relation to it. This is done below. 

One can almost marvel at the ability and way of those who worked on molecular orbitals to make sense of the molecular spectral data. And, yet, the apparently simplistic approach to "duplicate" Bohr s aufbau program for molecules turned out to be a great success. Furthermore, it adhered to a basic chemical characteristic that of being visualizable. The molecule conceived as "united atoms" helped in visualizing the new bonding mechanisms. Notably, one did not need to make use of the Schrodinger equation, even though the Pauli principle was absolutely crucial. 

On the other hand, if a model is simple, empirical, and parametric, that does not necessarily means that it has no firm, apparently hidden link to generally accepted fundamental laws, such as those of quantum chemistry. Just as the opposite may be the case, a model that is thought to be based on basic axioms may turn out not to reflect this deep connection with quantum chemistry. It thus was found, mostly through the work of D. J. Kleln, that the conjugated circuits model has a quite firm foundation in quantum chemical principles, while as we all know, the Hiickel molecular orbital (MO) model that started as a quantum chemical model turned out to be a consequence of molecular topology, rather than an intricate interaction of r-electrons governed by the Schrodinger equation. 

The principle of quantum mechanical methods consist basically to evaluate the total energy of a molecule by solving an JV-electron Schrodinger equation... 

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