Laplace transform analysis of laser's second threshold

M. Zarezadeh, Mohammad Kazem Tavassoly

In this paper, we have solved the Schrödinger equation for a particular kind of Morse potential and find its normalized eigenfunctions and eigenvalues, exactly. Our work is based on the Laplace transform technique which reduces the second-order differential equation to a first-order.

А. Н. Ораевский

Zidong Chen

AbstrcatThe second-order one-dimensional Schrdinger differential equation for the harmonic oscillator is reduced to a first-order differential equation by means of the Laplace transform,and then,its solutions are obtained by means of directly making use of an integral.The energy spectrum is derived from the condition that the wave functions must be single-valued.The required eigenfunctions are obtained by applying the expansions of series and inverse Laplace transform.

F. T. Arecchi, G.S. Rodari, A. Sona

Luis M Báez, Andrés Santos

Abstract The Laplace transform is a valuable tool in physics, particularly in solving differential equations with initial or boundary conditions. A 2014 study by Tsaur and Wang (2014 Eur. J. Phys. 35 015006) introduced a Laplace-transform-based method to solve the stationary Schrödinger equation for various potentials. However, their approach contains critical methodological flaws: the authors disregard essential boundary conditions and apply the residue theorem incorrectly in the inverse transformation process. These errors ultimately cancel out, leading to correct results despite a flawed derivation. In this paper, we revisit the use of the Laplace transform for the one-dimensional Schrödinger equation, clarifying correct practices in handling boundary conditions and singularities. This analysis offers a sound and consistent framework for the application of Laplace transforms in stationary quantum mechanics, underscoring their educational utility in quantum mechanics coursework.

A. A. Bakasov, N. B. Abraham