  We show that the all-orders WKB periods of one-dimensional quantum mechanical
oscillators are governed by the refined holomorphic anomaly equations of
topological string theory. We analyze in detail the double-well potential and
the cubic and quartic oscillators, and we calculate the WKB expansion of their
quantum free energies by using the direct integration of the anomaly equations.
We reproduce in this way all known results about the quantum periods of these
models, which we express in terms of modular forms on the WKB curve. As an
application of our results, we study the large order behavior of the WKB
expansion in the case of the double well, which displays the double factorial
growth typical of string theory.
