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International Journal of Quantum Chemistry

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

Chad E. Junkermeier, Mark K. Transtrum, Manuel Berrondo

Published Nov 14, 2008• 0 citations
Publisher

Abstract

Abstract In this article, we introduce a simple analytic method for obtaining approximate solutions of the Schrödinger 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. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

DOI
10.1002/qua.21917

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Abstract In this article, we introduce a simple analytic method for obtaining approximate solutions of the Schrödinger 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. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2009
We found a simple procedure for the solution of the time - independent Schrodinger equation in one dimension without making any approximation. The wave functions are always periodic. Two difficulties may be encountered: one is to solve the equation E=U(x), where E and U(x) are the total and potential energies, respectively, and the other is to calculate the integral of the square root of U(x). If these calculations cannot be made analytically, it should then be performed by numerical methods. To find the energy and the wave function of the ground state, there is no need to calculate this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(x).
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.
A simple procedure has been found for the general solution of the time-independent Schrödinger Equation (SE) with the help of quantization of potential area in one dimension without making any approximation.Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and anti symmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well, the relativistic theory of Dirac as examples.
We solve the Schrodinger equation analytically for systems whose potentials have a certain time-dependence (which is monotonously decreasing) and general coordinate-dependences.Only a few time-dependent systems have been reported to be analytically solved whose potentials are constant, linear, and quadratic functions of coordinate with arbitrary time-dependences.From a different perspective, we focus on the time-dependent systems whose potentials are monotonously decreasing functions of time with arbitrary coordinate dependences.Time-dependent potential of any coordinate-dependence can be handled analytically by transforming it to a time-independent potential of known solutions if its time-dependence is monotonously decreasing.We do this by a unitary transformation of the wavefunction and variable transformations to change the Schrodinger equation to be time-independent in new variables.These variables are then determined by solving a set of simple differential equations.This way we are able to find and to obtain analytical solutions for time-dependent potentials which we mention above.
A simple procedure has been found for the general solution of the time-independent Schrödinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential wells as examples. Their energies and normalized wave functions were found as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One was seen to be very fit.