I don't know enough about Langlands reciprocity to answer your question as asked, but the following might be helpful in thinking about the connection you are asking about.

There is by now a well understood connection between the geometric Langlands correspondence and S-duality of a topologically twisted version of $N=4$ Super Yang Mills theory (SYM) formulated on four manifolds of the form $ \Sigma \times X$ with $X$ a classed Riemann surface. Taking $X$ to be "small" leads to a two-dimensional topological field theory on $\Sigma$ and the S-duality of $N=4$ SYM becomes a kind of mirror symmetry of the topological field theory that relates the theory with gauge group $G$ to one with its Langlands dual ${}^LG$. This point of view was developed by Kapustin and Witten and is explained by E. Frenkel in arXiv:0906.2747.

The connection of this to particle-wave duality in quantum mechanics is as follows. S-duality of $N=4$ Super Yang-Mills theory has its origin in the electric-magnetic duality of pure Maxwell theory in $R^4$. Maxwell's equations with vanishing charge density and current sources are invariant under the transformation $\vec E \rightarrow \vec B$, $\vec B \rightarrow - \vec E$ of the electric field $\vec E $ and the magnetic field $\vec B$. This transformation is very analogous to the following transformation on the coordinate $x$ and momentum $p$ of a one-dimensional simple harmonic oscillator (SHO) $ x \rightarrow p$, $p \rightarrow -x$ and this duality transformation is a symmetry of the simple harmonic oscillator Hamiltonian $H= p^2/2+x^2/2$. In the quantum theories this analogy can be made precise by decomposing the electromagnetic field into modes and applying canonical quantization.

Now this symmetry of the SHO should be regarded as the statement that the SHO is self-dual under particle wave duality. The particle aspects are most clearly thought of in coordinate space while the wave aspects are most obvious in momentum space obtained by Fourier transform. Put another way, a particle is localized in $x$, a wave is localized in the canonical dual variable $p$. The self-duality of the SHO under particle-wave duality is manifested in various ways. For example, the ground state wave function of the SHO is a Gaussian in coordinate space. The dual under the above transformation gives the ground state wave function in momentum space since the Fourier transform of a Gaussian is again a Gaussian.

So I claim these is a connection between (geometric) Langlands and wave-particle duality that runs as particle-wave duality of SHO-> electric-magnetic duality of Maxwell theory -> S-duality of N=4 SYM -> Langlands. I leave it someone more knowledgeable to say whether this connection means anything in the number theoretical context.

I don't know enough about Langlands reciprocity to answer your question as asked, but the following might be helpful in thinking about the connection you are asking about.

There is by now a well understood connection between the geometric Langlands correspondence and S-duality of a topologically twisted version of $N=4$ Super Yang Mills theory (SYM) formulated on four manifolds of the form $ \Sigma \times X$ with $X$ a closed Riemann surface. Taking $X$ to be "small" leads to a two-dimensional topological field theory on $\Sigma$ and the S-duality of $N=4$ SYM becomes a kind of mirror symmetry of the topological field theory that relates the theory with gauge group $G$ to one with its Langlands dual ${}^LG$. This point of view was developed by Kapustin and Witten and is explained by E. Frenkel in arXiv:0906.2747.

The connection of this to particle-wave duality in quantum mechanics is as follows. S-duality of $N=4$ Super Yang-Mills theory has its origin in the electric-magnetic duality of pure Maxwell theory in $R^4$. Maxwell's equations with vanishing charge density and current sources are invariant under the transformation $\vec E \rightarrow \vec B$, $\vec B \rightarrow - \vec E$ of the electric field $\vec E $ and the magnetic field $\vec B$. This transformation is very analogous to the following transformation on the coordinate $x$ and momentum $p$ of a one-dimensional simple harmonic oscillator (SHO) $ x \rightarrow p$, $p \rightarrow -x$ and this duality transformation is a symmetry of the simple harmonic oscillator Hamiltonian $H= p^2/2+x^2/2$. In the quantum theories this analogy can be made precise by decomposing the electromagnetic field into modes and applying canonical quantization.

Now this symmetry of the SHO should be regarded as the statement that the SHO is self-dual under particle wave duality. The particle aspects are most clearly thought of in coordinate space while the wave aspects are most obvious in momentum space obtained by Fourier transform. Put another way, a particle is localized in $x$, a wave is localized in the canonical dual variable $p$. The self-duality of the SHO under particle-wave duality is manifested in various ways. For example, the ground state wave function of the SHO is a Gaussian in coordinate space. The dual under the above transformation gives the ground state wave function in momentum space since the Fourier transform of a Gaussian is again a Gaussian.

So I claim these is a connection between (geometric) Langlands and wave-particle duality that runs as particle-wave duality of SHO-> electric-magnetic duality of Maxwell theory -> S-duality of N=4 SYM -> Langlands. I leave it someone more knowledgeable to say whether this connection means anything in the number theoretical context.