Exact solutions of the Schrödinger equation for zero energy

Hasan Hüseyin Erbil, Ahmet Saim Selvi

In some molecules such as ammonia, double vibrations are observed at small energies. These small vibrations are thought to arise from double well potentials. To explain these small energy gaps, double potential wells are investigated, and complex calculations are made. Until recently, since the exact solution of the Schrödinger equation was not known, approximate solutions were always obtained. The exact solution of the Schrödinger equation is now known. In previous studies, a simple procedure for the general solution of the radial Schrödinger Equation has been found for spherical symmetric potentials without making any approximation. In this article, the Schrodinger equation was solved with this new solution method. It has been applied to the oscillations through a barrier tunneling potential for a particle in a central quadratic deformed potential well (double potential well). It is seen that there is no need to look for complex potentials to explain the double wells observed in small vibrations in ammonia and similar molecules, they were compared with experimentally measured energies. The results were found to be fully compatible. It has been seen that there can always be double well potentials at small energies for all molecules, atomic nuclei, and similar particles with angular momentum other than zero.

Anatoly Georgievich Meshkov, N.V. Sukhomlin, V. N. Shapovalov

Rajan N. Iyer

This paper presents a novel and exact solution of the Schrödinger equation with zero potential energy, based on the analogy with Fick’s equation of diffusion. The author derived the wavefunction of a particle in a vacuum as an error function of the rotational phase angle, which corresponds to the geometry of a flat space. This solution reduces to the sinusoidal form of the wavefunction commonly used in quantum mechanics. The author also explored the implications of this solution for the connection between quantum physics and gas mechanics, and the possibility of using error functions and general transforms to model past, present, and future events in physics. The author used computer simulations to compare the error function solution and the sinusoidal solution in terms of response time, accuracy, and fit to the geometry of a flat space. The results showed that the error function solution was superior to the sinusoidal solution in all aspects, and that it had a stronger link to the gas mechanics, useful to Quantum ASTROPHYSICS. The results also suggested that the error function solution could be used to model past, present, and future events in physics, using error functions and general transforms. This paper is a preliminary analysis of the deeper physics underlying the error function solution of the Schrödinger equation, based on the author’s previous publications on point physics generalizing to Hod-PDP mechanism, field tensor modeling, string-metrics, and information-time event matrix formulations. The paper recommends that future research should extend the error function solution to space time geometry higher dimensions, non-zero potentials, and more diverse experiments. The paper also recommends that future research should explore the applications of error functions and general transforms within physics and other mathematical physical general related fields of sciences.

Millard F. Manning

H. R. Krishnamurthy, Hamdi Mani, Harish Chandra Verma

The authors obtain the exact solution of the Schrodinger equation for a particle confined to (i) an equilateral triangle, (ii) a tetrahedral box with corners (- pi / square root 2,- pi / square root 2,- pi / square root 2), ( pi / square root 2,- pi / square root 2, pi /2); (- pi / square root 2, pi / square root 2, pi /2) and ( pi square root 2, pi / square root 2,- pi /2). The energies are: for (i) E nl =(8h(cross) 2 pi 2 /9mL 2 )(n 2 +l 2 -ln) where L is the side of the triangle and l, n are distinct non-zero integers and for (ii), E lmn =(h(cross) 2 /8m)*(3(l 2 +m 2 +n 2 )-2lm-2mn-2nl) where l, m and n are distinct non-zero integers. The wavefunctions have been classified according to the irreducible representation of the corresponding symmetry groups.