Path integral solution for a Klein–Gordon particle in vector and scalar deformed radial Rosen–Morse-type potentials

Abstract

The problem of a Klein–Gordon particle moving in equal vector and scalar Rosen–Morse-type potentials is solved in the framework of Feynman’s path integral approach. Explicit path integration leads to a closed form for the radial Green’s function associated with different shapes of the potentials. For $$q\le -1$$ , and $$\frac{1}{2\alpha }\ln \left| q\right|<r<+\infty$$ , the energy equation and the corresponding wave functions are deduced for the l states using an appropriate approximation to the centrifugal potential term. When $$-1<q<0$$ or $$q>0$$ , it is shown that the quantization conditions for the bound state energy levels $$E_{n_{r}}$$ are transcendental equations which can be solved numerically. Three special cases such as the standard radial Manning–Rosen potential $$(\left| q\right| =1)$$ , the standard radial Rosen–Morse potential $$(V_{2}\rightarrow -V_{2},q=1)$$ and the radial Eckart potential $$(V_{1}\rightarrow -V_{1},q=1)$$ are also briefly discussed.