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Analytical solutions to the Schrodinger equation for some bounded systems

Richa Rani

Published Jan 1, 2018• 0 citations
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The most physically interesting systems are not exactly solvable in quantum mechanics. For one-dimensional bound systems without exact solutions, we analytically and numerically find that the Rayleigh-Schrödinger perturbed series sensitively depends on an unsolvable integration, which leads to numerical instability in quantum mechanics. By using an exact formal solution of the non-homogeneous Schrödinger equation, we demonstrate the existence of analytically bound states and propose a simple scheme to truncate infinity so that the instability difficulty is avoided.
We propose a new method to obtain approximate solutions for the Schrödinger equation with an arbitrary potential that possesses bound states. This method, relying on the auxiliary field technique, allows to find in many cases, analytical solutions. It offers a convenient way to study the qualitative features of the energy spectrum of bound states in any potential. In particular, we illustrate our method by solving the case of central potentials with power-law form and with logarithmic form. For these types of potentials, we propose very accurate analytical energy formulae which greatly improves the corresponding formulae that can be found in the literature.