scholarly journals An Alternative Approach to Solutions of the MGECSC Potential in Presence of External Electric Field

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
M. K. Bahar
Keyword(s):  
Schrödinger Equation ◽  
Electric Field ◽  
External Electric Field ◽  
Schrodinger Equation ◽  
Mathematical Structure ◽  
Wave Functions ◽  
Electric Field Effect ◽  
Electric Field Effects ◽  
Alternative Approach ◽  
First Time

For the first time the Schrödinger equation with more general exponential cosine screened Coulomb potential in the presence of external electric field is solved approximately and analytically by applying an ansatz to eigenfunction of corresponding Hamiltonian and then energy values and wave functions are obtained. Since this potential turns into four different potential cases when considering different cases of the parameters in the potential, energies and eigenfunctions for these four different potentials are already to be found by solving Schrödinger equation with MGECSC potential. Energy values and wave functions obtained by using different values of potential parameters for each of these four different potential are compared with the results of other studies. Since the obtained general solutions in this study have been found in the presence of external electric field, the external electric field effects on systems with the mentioned four different potentials are also easily investigated. One of advantages of the present results and method is that if external electric field is equal to zero, general mathematical structure of corresponding equations does not change and then electric field effect can be eliminated. The presence or absence of electric field does not prevent solving the Schrödinger equation analytically.

Quantum Boxes, Stringed Instruments

2017 ◽  
Author(s):  
Frank S. Levin
Keyword(s):  
Energy Level ◽  
Schrödinger Equation ◽  
Quantum System ◽  
Schrodinger Equation ◽  
Probability Distributions ◽  
Wave Functions ◽  
Energy Level Diagram ◽  
One Dimensional ◽  
Stringed Instruments ◽  
Symmetry Properties

Chapter 7 illustrates the results obtained by applying the Schrödinger equation to a simple pedagogical quantum system, the particle in a one-dimensional box. The wave functions are seen to be sine waves; their wavelengths are evaluated and used to calculate the quantized energies via the de Broglie relation. An energy-level diagram of some of the energies is constructed; on it are illustrations of the corresponding wave functions and probability distributions. The wave functions are seen to be either symmetric or antisymmetric about the midpoint of the line representing the box, thereby providing a lead-in to the later exploration of certain symmetry properties of multi-electron atoms. It is next pointed out that the Schrödinger equation for this system is identical to Newton’s equation describing the vibrations of a stretched musical string. The different meaning of the two solutions is discussed, as is the concept and structure of linear superpositions of them.

Directionality and the Role of Polarization in Electric Field Effects on Radical Stability

2017 ◽  
Vol 70 (4) ◽  
pp. 367 ◽  
Author(s):  
Ganna Gryn'ova ◽  
Michelle L. Coote
Keyword(s):  
Electric Field ◽  
External Electric Field ◽  
Radical Reactions ◽  
Electric Field Effects ◽  
Dissociation Energies ◽  
Barrier Heights ◽  
Diol Dehydratase ◽  
Group A ◽  
Activity Of Enzymes ◽  
The Stability

Accurate quantum-chemical calculations are used to analyze the effects of charges on the kinetics and thermodynamics of radical reactions, with specific attention given to the origin and directionality of the effects. Conventionally, large effects of the charges are expected to occur in systems with pronounced charge-separated resonance contributors. The nature (stabilization or destabilization) and magnitude of these effects thus depend on the orientation of the interacting multipoles. However, we show that a significant component of the stabilizing effects of the external electric field is largely independent of the orientation of external electric field (e.g. a charged functional group, a point charge, or an electrode) and occurs even in the absence of any pre-existing charge separation. This effect arises from polarization of the electron density of the molecule induced by the electric field. This polarization effect is greater for highly delocalized species such as resonance-stabilized radicals and transition states of radical reactions. We show that this effect on the stability of such species is preserved in chemical reaction energies, leading to lower bond-dissociation energies and barrier heights. Finally, our simplified modelling of the diol dehydratase-catalyzed 1,2-hydroxyl shift indicates that such stabilizing polarization is likely to contribute to the catalytic activity of enzymes.

A Student's Guide to the Schrödinger Equation

2020 ◽  
Author(s):  
Daniel A. Fleisch
Keyword(s):  
Quantum Mechanics ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Wave Functions ◽  
Plain Language ◽  
Science And Engineering ◽  
Popular Approach ◽  
Matlab Code ◽  
Mathematical Techniques

Quantum mechanics is a hugely important topic in science and engineering, but many students struggle to understand the abstract mathematical techniques used to solve the Schrödinger equation and to analyze the resulting wave functions. Retaining the popular approach used in Fleisch's other Student's Guides, this friendly resource uses plain language to provide detailed explanations of the fundamental concepts and mathematical techniques underlying the Schrödinger equation in quantum mechanics. It addresses in a clear and intuitive way the problems students find most troublesome. Each chapter includes several homework problems with fully worked solutions. A companion website hosts additional resources, including a helpful glossary, Matlab code for creating key simulations, revision quizzes and a series of videos in which the author explains the most important concepts from each section of the book.

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