Equations Schrodinger wave equation

Schrodinger wave equation The fundamental equation of wave mechanics which relates energy to field. The equation which gives the most probable positions of any particle, when it is behaving in a wave form, in terms of the field. 

In 1926 Erwin Schrodinger (1887-1961), an Austrian physicist, made a major contribution to quantum mechanics. He wrote down a rather complex differential equation to express the wave properties of an electron in an atom. This equation can be solved, at least in principle, to find the amplitude (height) of the electron wave at various points in space. The quantity ip (psi) is known as the wave function. Although we will not use the Schrodinger wave equation in any calculations, you should realize that much of our discussion of electronic structure is based on solutions to that equation for the electron in the hydrogen atom. 

The quantum number ms was introduced to make theory consistent with experiment. In that sense, it differs from the first three quantum numbers, which came from the solution to the Schrodinger wave equation for the hydrogen atom. This quantum number is not related to n, , or mi. It can have either of two possible values ... 

After the discovery of quantum mechanics in 1925 it became evident that the quantum mechanical equations constitute a reliable basis for the theory of molecular structure. It also soon became evident that these equations, such as the Schrodinger wave equation, cannot be solved rigorously for any but the simplest molecules. The development of the theory of molecular structure and the nature of the chemical bond during the past twenty-five years has been in considerable part empirical — based upon the facts of chemistry — but with the interpretation of these facts greatly influenced by quantum mechanical principles and concepts. 

The physical interpretation of the quantum mechanics and its generalization to include aperiodic phenomena have been the subject of papers by Dirac, Jordan, Heisenberg, and other authors. For our purpose, the calculation of the properties of molecules in stationary states and particularly in the normal state, the consideration of the Schrodinger wave equation alone suffices, and it will not be necessary to discuss the extended theory. 

Considering only forward scattering by a crystal, the one-body Schrodinger wave equation may be transformed into a first order eigenequation [44, 51]... 

Just like any spectroscopic event EPR is a quantum-mechanical phenomenon, therefore its description requires formalisms from quantum mechanics. The energy levels of a static molecular system (e.g., a metalloprotein in a static magnetic field) are described by the time-independent Schrodinger wave equation,... 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

The coefficients of the various p orbitals of complex molecules containing n systems are obtained by computer programming based on approximate solutions of the Schrodinger wave equation. 

Erwin Schrodinger developed an equation to describe the electron in the hydrogen atom as having both wavelike and particle-like behaviour. Solution of the Schrodinger wave equation by application of the so-called quantum mechanics or wave mechanics shows that electronic energy levels within atoms are quantised that is, only certain specific electronic energy levels are allowed. 

Solving the Schrodinger wave equation yields a series of mathematical functions called wavefunctions, represented by R (Greek letter psi), and their corresponding energies. 

Atomic orbitals are actually graphical representations for mathematical solutions to the Schrodinger wave equation. The equation provides not one, but a series of solutions termed wave functions t[ . The square of the wave function, is proportional to the electron density and thus provides us with the probability of finding an electron within a given space. Calculations have allowed us to appreciate the shape of atomic orbitals for the simplest atom, i.e. hydrogen, and we make the assumption that these shapes also apply for the heavier atoms, like carbon. 

A newly discovered, highly organized state of matter in which clusters of 20-30 component atoms are magnetically contained and adiabatically cooled to within 2-3 X 10 K of absolute zero. At this point, the motions of the contained atoms are overcome by very weak cohesive forces of the Bose-Einstein condensate. While of no apparent relevance to biochemical kinetics, the Bose-Einstein condensate represents one of the most perfect forms of self-assembly, inasmuch as aU atoms within the condensate share identical Schrodinger wave equations. 

Substitution of the potential energy for this harmonic oscillator into the Schrodinger wave equation gives the allowed vibrational energy levels, which are quantified and have energies Ev given by... 

The Schrodinger wave equation for a particle in the potential energy regions I and [ can be written as... 

A rigorous quantum-mechanical treatment of directed valence bonds has not been given, for the reason that the Schrodinger wave equation has not been rigorously solved for any complicated molecule. Several approximate treatments have, however, been carried out, leading in a... 

Scientific theories themselves can be distinguished as deductive or inductive in nature, according to the underlying character of their premises. In a deductive theory, the fundamental premises are axioms or postulates that are neither questionable nor explainable within the theory itself. Outstanding examples of deductive theories include Euclidean geometry (based on Euclid s five axioms) and quantum mechanics (based on Schrodinger s prescription for converting classical trajectory equations into wave equations). An inductive theory, on the other hand, is based on universal laws of experience that express what has always been found to be true in the past, and may therefore be reasonably expected to hold in the future. Thermodynamics is the pre-eminent example of an inductive theory. 

Pauling showed that the quantum mechanical wave functions for s and p atomic orbitals derived from the Schrodinger wave equation (Section 5.7) can be mathematically combined to form a new set of equivalent wave functions called hybrid atomic orbitals. When one s orbital combines with three p orbitals, as occurs in an excited-state carbon atom, four equivalent hybrid orbitals, called sp3 hybrids, result. (The superscript 3 in the name sp3 tells how many p atomic orbitals are combined to construct the hybrid orbitals, not how many electrons occupy each orbital.)... 

Unlike molecular mechanics, the quantum mechanical approach to molecular modelling does not require the use of parameters similar to those used in molecular mechanics. It is based on the realization that electrons and all material particles exhibit wavelike properties. This allows the well defined, parameter free, mathematics of wave motions to be applied to electrons, atomic and molecular structure. The basis of these calculations is the Schrodinger wave equation, which in its simplest form may be stated as ... 

I think that the theory of resonance is independent of the valence-bond method of approximate solution of the Schrodinger wave equation for molecules. I think that it was an accident in the development of the sciences of physics and chemistry that resonance theory was not completely formulated before quantum mechanics. It was, of course, partially formulated before quantum mechanics was discovered and the aspects of resonance theory that were introduced after quantum mechanics, and as a result of quantum mechanical argument, might well have been induced from chemical facts a number of years earlier. [25]... 

Is resonance a real phenomenon The answer is quite definitely no. We cannot say that the molecule has either one or the other structure or even that it oscillates between them. .. Putting it in mathematical terms, there is just one full, complete and proper solution of the Schrodinger wave equation which describes the motion of the electrons. Resonance is merely a way of dissecting this solution or, indeed, since the full solution is too complicated to work out in detail, resonance is one way - and then not the only way - of describing the approximate solution. It is a calculus , if by calculus we mean a method of calculation but it has no physical reality. It has grown up because chemists have become used to the idea of localized electron pair bonds that they are loath to abandon it, and prefer to speak of a superposition of definite structures, each of which contains familiar single or double bonds and can be easily visualizable. [30]... 

The position and energy of each electron surrounding the nucleus of an atom are described by a wave function, which represents a solution to the Schrodinger wave equation. These wave functions express the spatial distribution of electron density about the nucleus, and are thus related to the probability of finding the electron at a particular point at an instant of time. The wave function for each electron, F(r,6,), may be written as the product of four separate functions, three of which depend on the polar coordinates of the electron... 

Surfaces may be drawn to enclose the amplitude of the angular wave function. These boundary surfaces are the atomic orbitals, and lobes of each orbital have either positive or negative signs resulting as mathematical solutions to the Schrodinger wave equation. 

The quantum number, m , originating from the 0(6) and Schrodinger wave equation, indicates how the orbital angular momentum is oriented relative to some fixed direction, particularly in a magnetic field. Thus, ml roughly characterizes the directions of maximum extension of the electron... 

Orbitals. Atomic orbitals represent the angular distribution of electron density about a nucleus. The shapes and energies of these amplitude probability functions are obtained as solutions to the Schrodinger wave equation. Corresponding to a given principal quantum number, for example n = 3, there are one 3s, three 3p and five 3d orbitals. The s orbitals are spherical, the p orbitals are dumb-bell shaped and the d orbitals crossed dumb-bell shaped. Each orbital can accomodate two electrons spinning in opposite directions, so that the d orbitals may contain up to ten electrons. 

Although the Schrodinger wave equation is difficult to solve for increasingly complicated atoms and molecules, we could, if we had a large enough computer, deduce the properties of all known chemicals from this equation. Quantum mechanics, however, is even more powerful than an ordinary cookbook because it also allows us to calculate the properties of chemicals that we have yet to see in nature. 

Molecular mechanics simulations are useful methods when dealing with large molecules or when limited information is required. When more sophisticated analysis is desired, such as thermodynamic data, it is usually necessary to switch to ab initio methods that seek to solve, or approximate a solution to, the Schrodinger wave equation for the entire molecule. Programs are rapidly improving both in terms of time taken to generate a solution to the molecular orbital and in the size of molecule that can be analyzed by these methods. Despite these advances only the simplest of supramolecular systems can usefully be investigated at this level of... 

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