<jats:title>A<jats:sc>bstract</jats:sc>
</jats:title><jats:p>We study the WKB periods for the (<jats:italic>r</jats:italic> + 1)-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {A}_r^{(1)} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msubsup>
<mml:mi>A</mml:mi>
<mml:mi>r</mml:mi>
<mml:mfenced>
<mml:mn>1</mml:mn>
</mml:mfenced>
</mml:msubsup>
</mml:math></jats:alternatives></jats:inline-formula> affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra <jats:italic>A</jats:italic><jats:sub><jats:italic>r</jats:italic></jats:sub>. For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and the WKB period, which is confirmed by solving the TBA equation numerically.</jats:p>
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