We present a formulation wave-particle complementarity and the de Broglie relation \lambda=h/p within the framework of categorical quantum mechanics on rigged Hilbert spaces. Position and momentum are represented by continuous dagger-Frobenius struc
We present a formulation wave-particle complementarity and the de Broglie relation \lambda=h/p within the framework of categorical quantum mechanics on rigged Hilbert spaces. Position and momentum are represented by continuous dagger-Frobenius structures associated with the locally compact abelian group R and its Pontryagin dual. The Fourier transform is the unitary implementation of Pontryagin duality and relates the two observable structures. We show that a general character pairing \chi_p^{(\alpha)}(x)=\exp(ipx/\alpha) yields a one-parameter family of unitarily equivalent Fourier transforms. Pontryagin duality therefore determines the structural form of the complementarity, but not the numerical value of Planck's constant. The identification \alpha=\hbar is a physical input fixed by the Weyl commutation relations. With this input, the spatial period of the character yields \lambda=h/p.