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Journal of the Arkansas Academy of Science

Schroedinger Equation with Spherically Averaged Potentials

Charles Hughes

Published Jan 1, 1992• 0 citations
OpenAlex

Abstract

Using a method adapted from few—body hyperspherical techniques, an approach to the solution of the Schroedinger equation with nonspherical potentials is discussed. The method is to spherically average the potential over spherical angles and then solve the resulting set of coupled differential equations. A discussion of how this method is applied to the Stark effect is presented.

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Using a method adapted from few—body hyperspherical techniques, an approach to the solution of the Schroedinger equation with nonspherical potentials is discussed. The method is to spherically average the potential over spherical angles and then solve the resulting set of coupled differential equations. A discussion of how this method is applied to the Stark effect is presented.
By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
We use the following approximation to the potential between two /sup 4/He atoms: V(r) = 4420 exp(-r/sup 2//1.69)-1302 exp(-r/sup 2//2.44) (V in K, r in A). We use the hyperspherical harmonics expansion, with Fabre's optimal subset, to solve for the ground state of the /sup 4/He tetramer. The Schroedinger equation is transformed into an infinite set of second order coupled differential equations (CDE), which we truncate to M equations. We test our program for solving CDE in two appendices. We find that the energy for our model is less than -4.3 K.