Soliton Solutions for a Quantum Particle in One-dimensional Boxes

In this study, we present a new analytical solution for the time-dependent Schrödinger equation for a free particle in one-dimensional case. The solution is derived by doing a non-linear transform to the linear Schrödinger equation and converting it into a Burger-like equation. We obtained an interesting non-stationary wave function where our soliton solution moves in time while maintaining its shape. The new solution is then analysed for three different cases: a periodic box, a box with hard wall boundary conditions and a periodic array of Dirac delta potentials. The resulting analytical solutions exhibit several interesting features including quantized soliton velocity and velocity bands. The analytical soliton solution that has been proposed, in our opinion, makes an important contribution to the study of quantum mechanics and we believe it will contribute significantly to our understanding of how particles behave in one-dimensional box potentials.