  A Laplace transform that maps the topological recursion (TR) wavefunction to
its $x$-$y$ swap dual is defined. This transform is then applied to the
construction of quantum curves. General results are obtained, including a
formula for the quantisation of many spectral curves of the form $e^xP_2(e^y) -
P_1(e^y) = 0$ where $P_1$ and $P_2$ are coprime polynomials; an important class
that contains interesting spectral curves related to mirror symmetry and knot
theory that have, heretofore, evaded the general TR-based methods previously
used to derive quantum curves. As an application, the quantum spectral curve
for the Gromov-Witten theory of the complex projective line is derived directly
from the theory of TR. The results seem to suggest a link between the
topological recursion and the Gromov-Witten invariants of the weighted complex
projective line $\mathbb{P}[r]$.
