Solutions of the one-dimensional time-independent Schrödinger equation

H. T. Williams

I present a method in the spirit of Green’s functions for generating bound state solutions to the Schrödinger equation. The method is demonstrated by generating ground state and excited state solutions to the one-dimensional, time-independent Schrödinger equation. Its applicability to the radial equation for two and three dimensions is also discussed.

S. Brandt, Hans Dieter Dahmen

Juan C. Morales, Carlos A. Arango

An alternative method for obtaining analytical solutions of the time-independent 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.

S. Brandt, Hans Dieter Dahmen

Rohit Gupta, Rahul Gupta

This paper adds up how the matrix method can be used for solving the one-dimensional time-independent Schrodinger's equation for some specific potential energy variation like half-infinite potential well.The matrix method is illustrated to obtain solution of the time-independent Schrodinger's equation for half-infinite potential well, which is generally done by ordinary algebraic and analytical methods.The transcendental equation determining the discrete eigenvalues for bound state and the corresponding eigenwave functions are obtained by the time-independent Schrodinger's equation for half-infinite potential well via the application of matrix method.