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The Picard-Fuchs equation is a powerful mathematical tool which has numerousapplications in physics, for it allows to evaluate integrals without resortingto direct integration techniques. We use this equation to calculate both theclassical action and the higher-order WKB corrections to it, for the sexticdouble-well potential and the Lam\'e potential. Our development rests on thefact that the Picard-Fuchs method links an integral to solutions of adifferential equation with the energy as a parameter. Employing the sameargument we show that each higher-order correction in the WKB series for thequantum action is a combination of the classical action and its derivatives.From this, we obtain a computationally simple method of calculatinghigher-order quantum-mechanical corrections to the classical action, anddemonstrate this by calculating the second-order correction for the sextic andthe Lam\'e potential. This paper also serves as a self-consistent guide to theuse of the Picard-Fuchs equation.

The Picard-Fuchs equation in classical and quantum physics: Application to higher-order WKB method