Eigenvalues of λ<i>x2m</i> anharmonic oscillators

Biswas, S. N.; Datta, Kamal; Saxena, Reena; Srivastava, Pramod K.; Varma, Vijayalakshmi

doi:10.1063/1.1666462

Cited by 239 publications

(62 citation statements)

References 14 publications

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“…As a quantum system, the onedimensional Schroedinger equation involving the potential x 2 /(1 + gx 2 ) was considered in [5] as an example of anharmonic oscillator, and later on studied in [6]- [10]; the three-dimensional quantum problem was considered in [11], [12] (e.g., in [11] the Bohr-Sommerfeld quantization procedure was applied in relation with some previous studies [13], [14], in nonpolynomial quantum mechanical models). We observe that this system can also be considered as an oscillator with a position-dependent effective mass (see [15] and references therein).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A non-linear oscillator with quasi-harmonic behaviour: two- andn-dimensional oscillators

et al. 2004

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A nonlinear two-dimensional system is studied by making use of both the Lagrangian and the Hamiltonian formalisms. The present model is obtained as a twodimensional version of a one-dimensional oscillator previously studied at the classical and also at the quantum level. First, it is proved that it is a super-integrable system, and then the nonlinear equations are solved and the solutions are explicitly obtained. All the bounded motions are quasiperiodic oscillations and the unbounded (scattering) motions are represented by hyperbolic functions. In the second part the system is generalized to the case of n degrees of freedom. Finally, the relation of this nonlinear system with the harmonic oscillator on spaces of constant curvature, two-dimensional sphere S 2 and hyperbolic plane H 2 , is discussed.MSC Classification: 37J35, 34A34, 34C15, 70H06 a)

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A non-linear oscillator with quasi-harmonic behaviour: two- andn-dimensional oscillators

et al. 2004

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“…We follow the works of others [5][6][7] and set the following: ϭ 2, m ϭ 0.5, and ϭ {0.1, 0.2, 0.3, 1, 2, 3}.…”

Section: Anharmonic Oscillatormentioning

confidence: 99%

An analytic iterative approach to solving the time‐independent Schrödinger equation

Junkermeier

Transtrum

Berrondo

2008

Int J of Quantum Chemistry

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ABSTRACT:In this article, we introduce a simple analytic method for obtaining approximate solutions of the Schrö dinger equation for a wide range of potentials in one-and two-dimensions. We define an operator, called the iteration operator, which will be used to solve for the lowest order state(s) of a system. The method is simple in that it does not require the computation of any integrals in order to obtain a solution. We use this method on several potentials which are well understood or even exactly solvable in order to demonstrate the strengths and weaknesses of this method.

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“…Em 1971, Biswas e cols. [4] apresentaram um método para a obtenção do espectro de energia para osciladores anarmônicos com termos do tipo λx 2m . Falco e cols.…”

Section: Introductionunclassified

Quantização de sistemas hamiltonianos via método de diferenças finitas

Monerat

Filho

Silva

et al. 2010

Rev. Bras. Ensino Fís.

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Propomos a introdução do método de diferenças finitas como tópico a ser abordado na disciplina de mecânica quântica de um curso de graduação em física. O métodoé suficientemente simples para ser introduzido e exemplificado em seis horas de aula, permitindo a obtenção de resultados tanto qualitativamente corretos como de alta precisão quantitativa. Devido a sua grande aplicabilidade, seu entendimento por parte dos alunos de graduação em física, futuros pesquisadores,é essencial para a formação acadêmica destes. No presente trabalho, apresentamos o método em detalhes. Sua precisãoé verificada calculando o espectro de energia para o oscilador harmônico e comparando-os com os resultados analíticos conhecidos. Em seguida aplicamos o método a dois outros sistemas, o oscilador anarmônico quártico e o potencial linear. Para cada um destes sistemas, calculamos os dez níveis de energia mais baixos, bem como seus respectivos auto-estados. Palavras-chave: quantização de sistemas hamiltonianos, diferenças finitas.We propose the introduction of the finite differences method as one topic to be inserted in the discipline of quantum mechanics in a physics undergraduate course. The method is simple enough to be introduced and exemplified in about six hours of class allowing both qualitatively correct and high-precision quantitative results. Due to the great applicability of the method, it is essential that the undergraduate students, future researchers, learn it. In the present work, we show the method in detail and verify its precision initially by calculating the energy spectrum of the harmonic oscillator and comparing it to its well-known analytical results. Then we apply it to two other systems, the anharmonic oscillator and the one with linear potential. For each of those systems We compute, for both systems, the ten lowest energy eigenvalues are calculated, as well as their corresponding eigenfunctions.

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Eigenvalues of λx2m anharmonic oscillators

Cited by 239 publications

References 14 publications

A non-linear oscillator with quasi-harmonic behaviour: two- andn-dimensional oscillators

A non-linear oscillator with quasi-harmonic behaviour: two- andn-dimensional oscillators

An analytic iterative approach to solving the time‐independent Schrödinger equation

Quantização de sistemas hamiltonianos via método de diferenças finitas

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