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KKU Science Journal

การหาคาความนาจะเปนในการสงผานสำหรบพลงงานศกยแบบตางๆ โดยวธการประมาณแบบดบเบลยเคบ

Kunlapat Sansuk, Petarpa Boonserm, Tritos Ngampitipan

Published Dec 28, 2020• 0 citations
OpenAlex

Abstract

Quantum mechanics is a theory that can explain phenomena of atom-sized matter or smaller, which classical mechanics cannot do. The central equation of quantum mechanics is the Schrödinger equation. Mathematically, the Schrödinger equation is a second order partial differential equation which tells us how a wave function changes with time. For most problems in quantum mechanics, the Schrödinger equation cannot present exact solutions, except for a few idealized systems in which the Schrödinger equation can be solved exactly. Therefore, approximation methods are needed to find solutions to the Schrödinger equation. In this paper, systems in stationary states are of special interest, in which the Schrödinger equation is reduced to the time-independent Schrödinger equation. One of the approximation methods that work for systems in stationary states is the WKB approximation. In this paper, transmission probabilities for various potentials are calculated using the WKB approximation. The potentials studied in this paper are composed of the asymmetric square-well potential, the hyperbolic tangent potential, the squared hyperbolic secant potential, and the double squared hyperbolic secant potential.

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วธการทางคณตศาสตรสำหรบสมการชเรอดงเงอร และวธประมาณคาแบบดบเบลยเคบ

เพชรอาภา บุญเสริม

The Schrödinger equation is a fundamental equation in physics for describing quantum mechanical behavior. It is a partial differential equation that shows how a wave function of a physical system evolves over time. In addition, it is associated with the kinetic energy and the potential energy, both of which contribute towards the total energy. This paper explains mathematical aspect of the Schrödinger equation. The concept of the WKB approximation has also been reviewed, which is an important method in finding the approximate solutions for the wave function. In addition, the paper also shows how to derive the transmission and reflection probabilities in relation to a one-dimensional quantum problem, called a “tunneling problem”, which occurs when a classical particle passes through a region where total energy is less than its potential energy. The WKB approximation can be used as a basis for formally writing down the exact solutions. Furthermore, it gives a high accuracy for the propagation of waves with high frequency or short wavelength.

KKU Science Journal•Mar 30, 2013•0 citations

- Approximation Methods I: Time-Independent Perturbation Theory

S. Rajasekar, R. Velusamy

The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.

CRC Press eBooks•Dec 11, 2014•0 citations

การหาขอบเขตลางของความนาจะเปนของการสงผานในปญหาการกระเจงทางควอนตมใน 1 มต

ไตรทศ งามปิติพันธ์, เพชรอาภา บุญเสริม

Quantum mechanics is the theory that describes dynamics of small objects such as atom and molecule. In this paper, Schrodinger’s wave mechanics, a part of quantum mechanics, is studied. The central equation of this wave mechanics is the Schrodinger’s equation. Solving this equation, quantum system dynamics can be described. In this work, the quantum scattering problem in one dimension is studied. Wave functions are obtained by exactly solving the Schrodinger’s equation in case of the delta function potential and the rectangular potential. The transmission and reflection probabilities are calculated from the obtained wave functions. Lower bounds on the transmission probabilities are presented. Finally, the lower bounds on the transmission probabilities are applied to the delta function potential and the rectangular potential problems. The results show that the exact transmission probabilities satisfy the lower bounds on the transmission probabilities.

KKU Science Journal•Dec 30, 2015•0 citations

Approximate methods of finding quantum states

Vladimir Mitin, Dmitry I. Sementsov, Nizami Z. Vagidov

The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method , using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics. Stationary perturbation theory for a system with non-degenerate states This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.

Cambridge University Press eBooks•May 20, 2010•0 citations

Reflection and transmission probabilities from various potentials using quasilinearization and WKB approximation techniques

Kittinut Singhanat

Quantum tunneling is a phenomenon in quantum mechanics. In this phenomenon, Quantum particle can go through potential barrier that potential energy is higher than total energy of quantum particle. Physicists are interested in transmitted and reflected value of Quantum particle that is transmission and reflection probabilities. Then, Schrödinger equation is used to find transmission and reflection probabilities by directly solving. However, some potential barriers are too complex to solve directly. Then, we use another method to solve more complex problem. That method is called "quasilinearization''. Quasilinearization method (QLM) is a numerical method to solve nonlinear differential equations. In this thesis, we compare the quasilinearization method with the well-known WKB approximation method. The results show that the transmission and reflection probabilities calculated by the QLM and WKB approximation have the same trend as the transmission and reflection probabilities calculated from the exact solution.

•Jan 1, 2025•0 citations

Approximation Methods III: WKB and Asymptotic Methods

S. Rajasekar, R. Velusamy

The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.

CRC Press eBooks•Sep 29, 2022•0 citations
Quantum mechanics is a theory that can explain phenomena of atom-sized matter or smaller, which classical mechanics cannot do. The central equation of quantum mechanics is the Schrödinger equation. Mathematically, the Schrödinger equation is a second order partial differential equation which tells us how a wave function changes with time. For most problems in quantum mechanics, the Schrödinger equation cannot present exact solutions, except for a few idealized systems in which the Schrödinger equation can be solved exactly. Therefore, approximation methods are needed to find solutions to the Schrödinger equation. In this paper, systems in stationary states are of special interest, in which the Schrödinger equation is reduced to the time-independent Schrödinger equation. One of the approximation methods that work for systems in stationary states is the WKB approximation. In this paper, transmission probabilities for various potentials are calculated using the WKB approximation. The potentials studied in this paper are composed of the asymmetric square-well potential, the hyperbolic tangent potential, the squared hyperbolic secant potential, and the double squared hyperbolic secant potential.
The Schrödinger equation is a fundamental equation in physics for describing quantum mechanical behavior. It is a partial differential equation that shows how a wave function of a physical system evolves over time. In addition, it is associated with the kinetic energy and the potential energy, both of which contribute towards the total energy. This paper explains mathematical aspect of the Schrödinger equation. The concept of the WKB approximation has also been reviewed, which is an important method in finding the approximate solutions for the wave function. In addition, the paper also shows how to derive the transmission and reflection probabilities in relation to a one-dimensional quantum problem, called a “tunneling problem”, which occurs when a classical particle passes through a region where total energy is less than its potential energy. The WKB approximation can be used as a basis for formally writing down the exact solutions. Furthermore, it gives a high accuracy for the propagation of waves with high frequency or short wavelength.
The starting point in a quantum mechanical problem is the construction of a Schrödinger equation. We try to solve it exactly to obtain wave function. A Schrödinger equation is exactly solvable only for a certain number of simple potentials. In most of the cases the Schrödinger equation cannot be solved exactly. The solution of the Schrödinger equation for the real systems is usually complicated and often exact solutions are very difficult to find or solutions not exist. Examples include as anharmonic oscillator, hydrogen atom in an electric field, a plane rotator in the presence of an applied electric field and so on. Therefore, we need to rely on approximation methods. The simple and most interesting and powerful approximate method is the perturbation theory developed by Schrödinger in 1926 [1]. Earlier Lord Rayleigh [2] analysed harmonic vibrations of a string perturbed by small inhomogeneities. The work of Rayleigh was pointed out by Schrödinger. Consequently, the method is also called Rayleigh-Schrödinger perturbation theory. This method is powerful for determining the changes in the discrete energy values and the associated eigenfunctions of a system due to a weak perturbation, provided the eigenpairs of the undisturbed system are available. The approximations obtained from the perturbation theory are used to calculate energy level diagrams and to understand the splitting and assignments of atomic and molecular spectra [35]. The perturbation theory has been proven to be useful in several fields of theoretical physics and also in chemistry. It provides considerable insight into various phenomena. The main disadvantage of this method is that in certain problems the perturbation series converges slowly or diverges.
Quantum mechanics is the theory that describes dynamics of small objects such as atom and molecule. In this paper, Schrodinger’s wave mechanics, a part of quantum mechanics, is studied. The central equation of this wave mechanics is the Schrodinger’s equation. Solving this equation, quantum system dynamics can be described. In this work, the quantum scattering problem in one dimension is studied. Wave functions are obtained by exactly solving the Schrodinger’s equation in case of the delta function potential and the rectangular potential. The transmission and reflection probabilities are calculated from the obtained wave functions. Lower bounds on the transmission probabilities are presented. Finally, the lower bounds on the transmission probabilities are applied to the delta function potential and the rectangular potential problems. The results show that the exact transmission probabilities satisfy the lower bounds on the transmission probabilities.
The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method , using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics. Stationary perturbation theory for a system with non-degenerate states This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.
Quantum tunneling is a phenomenon in quantum mechanics. In this phenomenon, Quantum particle can go through potential barrier that potential energy is higher than total energy of quantum particle. Physicists are interested in transmitted and reflected value of Quantum particle that is transmission and reflection probabilities. Then, Schrödinger equation is used to find transmission and reflection probabilities by directly solving. However, some potential barriers are too complex to solve directly. Then, we use another method to solve more complex problem. That method is called "quasilinearization''. Quasilinearization method (QLM) is a numerical method to solve nonlinear differential equations. In this thesis, we compare the quasilinearization method with the well-known WKB approximation method. The results show that the transmission and reflection probabilities calculated by the QLM and WKB approximation have the same trend as the transmission and reflection probabilities calculated from the exact solution.
The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The WKB method is due to Gregor Wentzel, Hendrik Anthony Kramers and Leon Nicolas Brillouin. The method is applicable for slowly varying potentials, that is, the potentials that change not more than a few wavelengths. The WKB approximation is a physically intuitive method and makes many connections with intuition about waves and classical mechanics as well as with results from the early days of quantum theory as it leads to the Bohr quantization condition. The Schrödinger equation is solved to find the penetration of a square potential barrier. The WKB method is applied to obtain a transmission coefficient for penetration through a barrier.