An alternative method to obtain analytical solutions of the time-independent Schrödinger equation
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Juan C. Morales [1] ; Carlos A. Arango [1]
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[1] Universidad Icesi
Universidad Icesi
Colombia
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- Localización: International journal of mathematical education in science and technology, ISSN 0020-739X, Vol. 54, Nº. 1, 2023, págs. 151-158
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
An alternative method for obtaining analytical solutions of the timeindependent Schrödinger equation (TISE) is developed and applied to 1-dimensional model quantum systems of interest in atomic and molecular physics. The method is based on proposing a wave function that is a product of two functions, one of them an integrating factor. The differential equation resulting from operating the TISE on the product wave function is compared with a differential equation with a known solution. This comparison produces a system of differential equations for the integrating factor. This system of equations is easily solved for model quantum systems such as the harmonic oscillator, the hydrogen atom and the Morse oscillator. The method presented simplifies some approaches shown in textbooks, based on asymptotic analyses of the TISE, and power series methods with recurrence relations. In addition, the method presented here facilitates the understanding of the relationship between the ordinary differential equations of the mathematical physics and the TISE of physical models as the harmonic oscillator, the Hydrogen atom, and the Morse oscillator.
