Continuum limit for volume

Given the probability density

$$rho = frac{ihbar}{2mc^2}(psi^*partial_tpsi – (partial_tpsi^*)psi) = |mathcal{N}|^2|Xi|^2$$

where $mathcal{N}$ is a (real) normalization constant and $Xi$ does not depend on $vec{x}$.

If we want to normalize it such that we find one particle in a volume $V$, we demand

$$int_V mathrm{d}x^3|mathcal{N}|^2|Xi|^2 = V |mathcal{N}|^2|Xi|^2$$

and therefore $mathcal{N} = frac{1}{sqrt{V|Xi|^2}}$

How can I get the continuum limit of this? I’ve seen $Vrightarrow (2pihbar)^3$ but this does not work out with the units. Also, what would be the idea behind this?