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International Journal of Quantum Chemistry

Two deltons: An inseparable Schrödinger equation

Paul Caylor McKinney

Published Jan 1, 2004• 1 citation
Publisher

Abstract

Abstract Inseparable Schrödinger equations usually cannot be solved analytically. Most often their solutions are obtained numerically. It would be helpful to have an inseparable Schrödinger equation whose solution can be completely calculated. Perhaps more can be learned about appropriate wave function solutions of complex problems if the solutions of an inseparable Schrödinger equation, solved in terms of simple functions, are available. Such a study is possible using the Dirac delta function as the potential energy for two particles moving on a line. The Dirac delta function problem leads naturally to a consideration of the wave functions' symmetry properties. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004

DOI
10.1002/qua.20022

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Abstract Inseparable Schrödinger equations usually cannot be solved analytically. Most often their solutions are obtained numerically. It would be helpful to have an inseparable Schrödinger equation whose solution can be completely calculated. Perhaps more can be learned about appropriate wave function solutions of complex problems if the solutions of an inseparable Schrödinger equation, solved in terms of simple functions, are available. Such a study is possible using the Dirac delta function as the potential energy for two particles moving on a line. The Dirac delta function problem leads naturally to a consideration of the wave functions' symmetry properties. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
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).
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.
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