METHOD OF FINDING THE WAVE FUNCTION OF A SYSTEM OF PARTICLES
DOI:
https://doi.org/10.31110/2413-1571-2023-038-2-001Keywords:
system of quantum particles, stationarity, state, wave function, iterationAbstract
This paper analyzes the integral equations corresponding to the wave function of a system of particles in a bound state. The equivalence of previously obtained integral equations of the Fredholm and Volterra type is shown. It is proved that homogeneous integral equations for the wave function of a system of interacting particles in a bound state have only trivial solutions. For the iteration of integral equations and finding the energy spectrum, a spherically symmetric form of free terms is proposed, which takes into account the symmetry of the wave function.
Formulation of the problem. Clarifying the possibility and creating methods of applying integral equations corresponding to the Schrödinger equation for a system of particles to finding the wave functions of a system of quantum particles.
Materials and methods. Application of the Fourier transform in the study of multidimensional integral equations and the use of Fredholm's theorems in the general theory of integral equations.
Results. The analysis of the integral equations corresponding to the wave function of the bound state of the system of particles was carried out, and the correctness of the way of their obtaining was shown. According to Fredholm's alternative, it is proved that only wave functions corresponding to inhomogeneous equations have a physical meaning. To find the wave function from integral equations by iteration, a spherically symmetric form of free terms is proposed, which implicitly takes into account the spin of the system particles.
Conclusions. The proposed method of finding the wave function of the particle system is promising since the iteration series for many types of interaction potential energy will converge, due to the fact that the proposed integral equation refers to Volterra-type equations. It should be noted that the proposed form of free terms is not the only possible form. When modeling particle systems of various types, free terms must reflect the characteristic features of the system.
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