The Schrödinger Equation for a Free Particle: Generalized Bessel Solutions

We demonstrate the time evolution of a free particle in a three-dimensional space given that the initial condition is any arbitrary radial function $f\left(r\right)$. The solution is expressed as a series expansion in terms of generalized Bessel functions derived using a 3D recursion formula for Bessel functions in the radial coordinate r. Additionally, we establish that these generalized Bessel functions can be represented through intricate double series, which ultimately enable the construction of the full solution. This work presents a novel solution to the problem, as previous approaches were limited to expressions involving only spherical Bessel functions.