<i>Quantum Mechanics</i>

Powell, John L.; Crasemann, Bernd; Guth, Eugene

doi:10.1063/1.3058277

Cited by 46 publications

(69 citation statements)

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“…Podemos destacar o potencial degrau e o potencial barreira retangular apresentado em diversos livros-textos para introdução do assunto. 11,12 Exemplos de potenciais para os quais se tem soluções analíticas podem ser encontrados na referência Razavy. 13 Dentre os métodos numéricos destaca-se o método da matriz de transferência que é amplamente utilizado para cálculos de coeficientes de transmissão/reflexão.…”

Section: Introductionunclassified

Transmission and Reflection Coefficients by the Variable Amplitude Method

Costa¹,

Cordeiro²,

Lemes³

et al. 2016

Química Nova

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Recebido em 22/10/2015; aceito em 28/01/2016; publicado na web em 12/04/2016 TRANSMISSION AND REFLECTION COEFFICIENTS BY THE VARIABLE AMPLITUDE METHOD. In this work, a simple derivation of the variable amplitude method using the variation of parameters to solve a differential equation is presented. The variable amplitude method was originally devised by Tikochinsky in 1977, using the quantum theory of scattering. The method is applied to two model potentials, the rectangular potential barrier and the Eckart potential, both with analytical solutions for the reflection coefficient. Numerical results will be compared with the exact values for several energies. The problem of calculating the reflection coefficient, usually involving extensive algebra as described in several textbooks, is reduced to solving a first order differential equation with initial condition. The method is very simple to apply, representing an attractive tool for teaching introductory quantum mechanics. A simple computer code is available from which reflection coefficients for the Eckart potential can be calculated.Keywords: transmission coefficient; quantum tunneling; variation of parameters. INTRODUÇÃONo estudo do movimento unidimensional de uma partícula de massa m e energia E, existem duas perspectivas: tratar o problema do ponto de vista da mecânica clássica ou de acordo com a mecânica quântica. No tratamento clássico, quando não ocorre perdas de energia de qualquer forma, a partícula fica presa a uma região do espaço x, tal que sua energia potencial V(x) é sempre menor ou igual a E. Em outras palavras, a energia total é conservada, V(x)+mv 2 /2=E. Este é um dos aspectos mais fascinantes que distingue a mecânica quântica da mecânica clássica. Na teoria quântica o estado da partícula é representado por uma função y (x) obtida pela solução da equação de Schrödinger, cujo módulo ao quadrado |y (x)| 2 dx é interpretado como a probabilidade de encontrar a partícula na região x+dx. Neste caso, como a função é espalhada além do limite x 0 , em que x 0 satisfaz a relação V(x 0 )=E, existe uma probabilidade finita de que uma partícula com energia total E penetre em uma região do espaço onde a energia total é menor que a energia potencial V(x).O processo, conhecido como tunelamento quântico, é de extrema relevância para explicação de várias fenômenos físicos e químicos, como o decaimento alfa, 1,2 a emissão de elétrons por metais frios sob a influência de um campo forte, 3 reações de transferência de hidrogênio, fundamentais para a compreensão do efeito isotópico cinético H/D, 4,5 mecanismos de transferência de elétrons 6,7 e inversão espectral. 8,9 Além disso, o tunelamento quântico é de importância para a fabricação de novos semicondutores e dispositivos eletrônicos. 10A compreensão das propriedades físicas dos sistemas requer em muitos casos a avaliação da equação unidimensional de Schrödinger para um potencial específico. No entanto, soluções analíticas são limitadas a potenciais modelos, que embora sejam úteis para a análise da situação física...

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Section: Introductionunclassified

Transmission and Reflection Coefficients by the Variable Amplitude Method

Costa¹,

Cordeiro²,

Lemes³

et al. 2016

Química Nova

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“…The fractional equations of motion which are obtained from application of SR, were applied directly by Hermann [14], in terms of a large number of explicit numerically simulated trajectories for a free particle in an infinite one-dimensional box [15] [16] [17]. Similarly, Al-Rashid [18] [19] [20], applied SR to the finite one-dimensional square well potential and special case in a double oscillator problems.…”

Section: Introductionmentioning

confidence: 99%

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

Al-Rashid¹,

Habeeb²,

Ahmed³

2017

JQIS

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In the present work, Scale Relativity (SR) is applied to a particle in a simple harmonic oscillator (SHO) potential. This is done by utilizing a novel mathematical connection between SR approach to quantum mechanics and the well-known Riccati equation. Then, computer programs were written using the standard MATLAB 7 code to numerically simulate the behavior of the quantum particle utilizing the solutions of the fractal equations of motion obtained from SR method. Comparison of the results with the conventional quantum mechanics probability density is shown to be in very precise agreement. This agreement was improved further for some cases by utilizing the idea of thermalization of the initial particle state and by optimizing the parameters used in the numerical simulations such as the time step and number of coordinate divisions. It is concluded from the present work that SR method can be used as a basis for description the quantum behavior without reference to conventional formulation of quantum mechanics. Hence, it can also be concluded that the fractal nature of space-time implied by SR, is at the origin of the quantum behavior observed in these problems. The novel mathematical connection between SR and the Riccati equation, which was previously used in quantum mechanics without reference to SR, needs further investigation in future work.

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“…It provides a better foundation for Physics, improves the accuracy and completeness of Physics theories, and answers fundamental Physics questions. The Unified Field Theory can derive the Planck [4][5][6][7][8][9][10][11][12][13], equation and Newton's gravity [14][15][16][17][18] equation and concludes that the electron [19][20][21][22][23][24][25] is stable. Finally, the paper unifies the four fundamental fields.…”

Section: Introductionmentioning

confidence: 99%

Unified Field Theory

Cao¹

2013

AJMP

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Abstract:The paper "SR Equations without Constant One-Way Speed of Light" [1] brings us a century back by discussing contradictory Relativity postulates. Instead of dealing with the aether theory, this paper proposes the Unified Field Theory (UFT) that replaces both the Theory of Relativity and Theory of Aether. In this new theory, the universe exists in form of Torque Grids in which Space, Time and Energy are unified. The ideal Torque Grids have same size and String movement cycle is the same as the Grid size in all directions. When Torque Grids have different sizes, the Torque Grids have size distortions. When Torque String movement speeds are different between two opposite directions, the Torque Grids have charge distortion. Both size and charge distortion can be measured by energy. The charge distortion can be measured by electronic charge in addition to the energy. The Gravity interactions and Strong interactions are related to Torque Grids' size distortions. The electron-magnetic interactions and weak interactions are related to charge distortions. The Unified Field Theory derives the Planck equation and Newton's gravity equation and explains why the electron is stable. The electron has a single Torque Grid distortion on its shell and is in resonance with its wavelength. Therefore, electron is stable and electronic field is formed. Strong force and weak force are result of two additional main resonance wave forms on the shell of the electron. Finally, the paper unifies the four fundamental fields.

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Quantum Mechanics

Cited by 46 publications

References 0 publications

Transmission and Reflection Coefficients by the Variable Amplitude Method

Transmission and Reflection Coefficients by the Variable Amplitude Method

Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential

Unified Field Theory

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