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JURNAL PENDIDIKAN FISIKA DAN KEILMUAN (JPFK)

Penggunaan Teori Perturbasi untuk Menentukan Tingkat Energi Partikel pada Kotak Potensial Tiga Dimensi

Indah Kharismawati

Published Sep 7, 2020• 1 citation
Publisher

Abstract

Hamiltonian a known system in many matters does not guarantee that the equation can be resolved, e.g. due to minor disturbances such as electric fields or magnetic fields that can lead to slight changes in energy and its particle functions, for such issues it should be used disorder theory (perturbation theory). The perturbation theory can determine how much the result of the presence of disruption to energy levels and Eigen's functions. The energy possessed by particles in addition to being influenced by quantum numbers (nx, ny and nz) and the size of the potential box are also influenced by interference in the form of magnetic energy produced due to the motion of particles (electrons) in the core intrinsic magnetic field (proton) which affects the energy value does not appear due to the value of this magnetic energy disorder is very small, while for interference that is an extrinsic magnetic energy has shown a 10 (Tesla). The wider the size of the box then the energy that the particles have is smaller so that the energy level and the particle energy spectrum in the potential box will appear continuous. It can be concluded that the particle energy level relies on the length of the potential box, the quantum number of particles, and also the interference that particles have.

DOI
10.25273/jpfk.v6i2.7237

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•Jan 1, 1996•0 citations
Hamiltonian a known system in many matters does not guarantee that the equation can be resolved, e.g. due to minor disturbances such as electric fields or magnetic fields that can lead to slight changes in energy and its particle functions, for such issues it should be used disorder theory (perturbation theory). The perturbation theory can determine how much the result of the presence of disruption to energy levels and Eigen's functions. The energy possessed by particles in addition to being influenced by quantum numbers (nx, ny and nz) and the size of the potential box are also influenced by interference in the form of magnetic energy produced due to the motion of particles (electrons) in the core intrinsic magnetic field (proton) which affects the energy value does not appear due to the value of this magnetic energy disorder is very small, while for interference that is an extrinsic magnetic energy has shown a 10 (Tesla). The wider the size of the box then the energy that the particles have is smaller so that the energy level and the particle energy spectrum in the potential box will appear continuous. It can be concluded that the particle energy level relies on the length of the potential box, the quantum number of particles, and also the interference that particles have.
Telah dilakukan review jurnal dengan judul ‘Dimensionless equation of non-relativistic quantum mechanics’ yang membahas tentang persamaan tak berdimensi dalam mekanika kuantum non-relativistik pada model potensial satu dimensi, model atom hidrogen dan teori perturbasi tak bergantung waktu. Tujuan dari penelitian ini untuk menyajikan persamaan tak berdimensi secara jelas dan terperinci pada model potensial satu dimensi, atom hidrogen serta teori perturbasi tak bergantung waktu dengan menggunakan operator Hamiltonian. Dalam menentukan persamaan tak berdimensi dilakukan terlebih dahulu menurunkan persamaan Hamiltonian tak bergantung waktu. Setelah menentukan persamaan Hamiltonian tak bergantung waktu, ditinjau setiap kasus yang terdapat pada model potensial satu dimensi, model atom hidrogen dan teori perturbasi tak bergantung waktu, kemudian ditransformasikan beberapa solusi dan parameter yang sederhana serta skala yang tepat sehingga menghasilkan persamaan dalam bentuk tak berdimensi. Hasil penelitian menunjukan bahwa dengan menggunakan persamaan Hamiltonian dapat menentukan persamaan tak berdimensi di berbagai kasus yang terdapat pada model potensial satu dimensi, atom hidrogen dan teori perturbasi tak bergantung waktu.
Schrodinger equations are widely used in solving a variety of problems, one of them is particles in a box in especially of potential wells bound state. Particle in a box is a discrete circumstances, so we must examined whether or not the degeneration of these particles. The Schrodinger equation analysed to get a form of modeling of energy levels and wave functions in potential wells of the circumstance used numerical method. We also can show energy level of each state from the model. The numerical approaches is finnite difference method using  Matlab R2010 software. Numerical calculation shows k value. k 1 and k 3 gets from solution of cos equation, whereas k 2 and k 4 gets from solution of sin equation. The k value is used to know the energy level in potential wells. energy level model show wave fucntion. In potential well bound state, Energy level and wave functions be smaller when potential energy is high and it is high when potential when potential energy is small. When wide of potential well is smaller, energy level and wave function be smaller too. But there isn’t show energy level and wave function when wide of potential well is high. The shape of energy level and and wave function of particle in box.
Suatu partikel Bosonik dan Dirac bermassa nol yang beraksi pada potensial penghalang satu dimensi telah dikaji secara mekanika kuantum relativistik dengan menggunakan persamaan Klein-Gordon dan persamaan Dirac. Persamaan ini selanjutnya mengalami modifikasi akibat pengaruh medan elektromagnetik yang dihadirkan dan pendekatan matriks telah diaplikasikan untuk mendapatkan representasi energi dan spinor eigennya. Terkhusus partikel Dirac, Hamiltonian Dirac memiliki bentuk yang identik dengan osilator harmonik sehingga representasi energi merupakan perpanjangan dari energi osilator harmonik. Selain itu fungsi eigennya melibatkan energi positif dan energi negatif yang bergerak secara eksponensial ketika melewati penghalang. DOI: http://dx.doi.org/10.17977/um024v4i12019p008
Abstract In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H0, H1, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H0 + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H0 + H2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H0, H1, H2 (t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H0, H1, H2 (t). To obtain Born’s theory we take H1 = 0, and for Oppenheimer’s we take H2 (t) = H2 (T) and T = 0. In Dirac’s theory we have H1 = 0, H2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.
There is a physical problem found in the Quantum Mechanics that is usually started with a Schr6dinger equation. The problem appears when a certain system with nondegenerate and time independent characteristic, such as hydrogen atom at the ground state, receives outside pertubation that causes the system ( in the Schr6dinger equation ) to change. The problem solving of the pertubation may be done by calculating it directly, but it has a high level of mathematic difficulty. Another way can be done in the cases of nondegenerate and time independent system that is by using pertubation theory. The result of the calculation using pertubation theory in the case of Hydrogen atom at ground state which receives outside electric field pertubatoin = t k, shows that the polarizability value a = 0.67 (A°)3 resembles the measured value in the amount of 0.68 (A°)3. Thus, it is proven that pertubation theory is available to calculate any pertubation found in a certain system.