Vol. 7, Issue 2, Part D (2025)

We obtain a mathematical and algebraic expression of the Green-function method in treating quantum tunnelling in one-dimensional barrier systems. The retarded Green's function of the free-particle resolvent is then obtained consistently to obtain scattering amplitudes using the Lippmann-Schwinger equation. The technique is used to apply to the d-function barrier, the finite square barrier, and the double-barrier quantum-well structure in order to describe the transmission, penetration of the barrier, and resonance. With the Green, the pole structure in the case of the double-barrier directly traces the quasi-bound and the Breit-Wigner-type resonances.