A new method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods have been discussed critically.
In this article, the following results are obtained: the process of a randomly wandering particle having a size and a continuous trajectory of motion is considered; (b) based on the study of this probabilistic process, a derivation of the Schrodinger equation is proposed; the Planck constant h is expressed in terms of the characteristics of the stationary stochastic process under consideration; the probability distribution density of the n-order derivative for a n-times differentiable stationary stochastic process is determined; it is shown that the obtained generalized Schrodinger equation is applicable not only to describe the possible states of microcosm objects, but also for discrete states of a number of macrocosm stochastic systems