Traditional first order JWKB method (
) is a conventional semiclassical approximation method mainly used in quantum mechanical systems for accurate solutions.
general solution of the Time Independent Schrodinger’s Equation (TISE) involves application of the conventional asymptotic matching rules to give the accurate wavefunction in the Classically Inaccessible Region (CIR) of the related quantum mechanical system. In this work, Bessel Differential Equation of the first order (
) is chosen as a mathematical model and its
solution is obtained by first transforming into the normal form via the change of independent variable. The
general solution for appropriately chosen initial values in both normal and standard form representations is analyzed via the generalized
asymptotic matching rules regarding the
matrix elements given in the literature. Instead of applying the common
asymptotic matching rules relying on the physical nature of the quantum mechanical system, i.e., a physically acceptable (normalizable) wavefunction, a pure semiclassical analysis is studied via the
model mathematically. Finally, an application to a specific case of the exponential potential decorated quantum mechanical bound state problem is presented.