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We propose an extension of the Wentzel-Kramers-Brillouin (WKB) approximationfor solving the Schr\"odinger equation. A set of coupled differential equationshas been obtained by considering an ansatz of wave function with two auxiliaryconditions on gauging the first and the second derivatives of the wavefunction, respectivey. It is shown that an alternating perturbation method canbe used to decouple this set of equations, yielding the well known Bremmerseries. We derive an analytical scheme to approximate the wave function whichconsists of the usual WKB approximation as the zeroth order solution with extracorrection terms up to the first order perturbation. Therefore, we call thisalternative methodology the alternating WKB (a-WKB) approximation. We apply thea-WKB method to solve the eigenvalue problem of a harmonic oscillator and thescattering problem of a repulsive inverse-square (centrifugal) potential anddemonstrate its supremacy over the usual WKB approximation.

Alternating Wentzel-Krammers-Brillouin Approximation for the Schr\"odinger Equation: A Rediscovering of the Bremmers Series