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WKB approximation for abruptly varying potential wells

We present an approach to obtain eigenfunctions and eigenenergies for abruptly varying potentials in the framework of the Wentzel–Kramers–Brillouin (WKB) approximation. To illustrate it, two examples of the potentials are studied. The first one is the combination of a step barrier and a harmonic oscillator potential, and the second one consists of a step barrier and a linear potential. The formulation of a WKB quantization rule is proposed. Our approach shows that WKB energies and those from numerical calculation are in good agreement. According to matching conditions used, WKB wavefunctions in this present work are violated at only one classical turning point, but they behave well at another point where the potentials are discontinuous.