# Intuition for the spatial density in second quantisation

Suppose that $$\hat{\Psi}^\dagger(x)$$, $$\hat{\Psi}(x)$$ are the usual field operators in second quantisation for some identical particle, and that $$\hat{c}^\dagger_n$$, $$\hat{c}_n$$ are the creation and annihilation operators in some discrete single-particle basis, with spatial wavefunctions $$\phi_n(x)$$ for each mode.

Suppose I know the one body density matrix in that discrete basis - that is, I know $$\langle \hat{c}_n^\dagger \hat{c}_m\rangle$$ for all $$n$$ and $$m$$. I want to use this to calculate the spatial density $$\rho(x)=\langle\hat{\Psi}^\dagger(x)\hat{\Psi}(x)\rangle$$. If I were to guess this by intuition, I would have rather confidently said that it would just be the sum of the spatial densities in each mode - that is: $$\rho(x)=\sum_n |\phi_n(x)|^2\langle\hat{c}^\dagger_n\hat{c}_n\rangle$$ However, this seems to be incorrect. If I take the definition of $$\rho(x)$$ and expand out the field operators as $$\hat{\psi}(x)=\sum_n \phi_n(x)\hat{c}_n$$, $$\hat{\psi}^\dagger(x)=\sum_n \phi_n^*(x)\hat{c}^\dagger_n$$, then I find: $$\rho(x)=\langle\hat{\Psi}^\dagger(x)\hat{\Psi}(x)\rangle=\sum_{nm}\phi_n^*(x)\phi_m(x)\langle \hat{c}_n^\dagger \hat{c}_m\rangle$$ where we have off-diagonal terms for which $$n\neq m$$ contributing! This surprises me. Is there a good physical intuition for why these inter-mode terms contribute to the spatial density (or have I made a mistake, or do they cancel out somehow)?

The $$n=m$$ contribution would give you the "diagonal" density distribution, whereas the $$n\neq m$$ gives you the "off-diagonal" one.
Intuitively, if you only have a diagonal component, then the spatial profile at $$x$$ does not care in the slightest of its surroundings. If you were in a cubic lattice, it would correspond to a state localised onto one lattice site, with no tunnelling into neighbouring minima.
The $$n\neq m$$ contribution would in fact give you a state which is delocalised over the potential landscape, thanks to non-suppressed tunnelling among its minima locations.