  We study the WKB periods for the $(r+1)$-th order ordinary differential
equation (ODE) which is obtained by the conformal limit of the linear problem
associated with the $A_r^{(1)}$ 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 $A_r$. 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.
