[@Thi]. To recall that construction briefly, if $\{\phi_m\}_{m=0}^\infty $ is an orthonormal basis of an abstract, complex separable Hilbert space $\mathfrak{H}$ and $\{\chi^1, \chi^2\}$ is an orthonormal basis of $\mathbb{C}^2$, then the VCS take the form, $$\mid {\mathbf{q}},j \rangle = \mathcal{N}_1(\vert {\mathbf{q}}\vert)^{-1/2}
\sum_{m=0}^\infty \frac{{\mathbf{q}}^m}{\sqrt{x_m!}} \chi^j \otimes \phi_m
\in \mathbb C^2 \otimes \mathfrak H, \quad j=1,2,
 \label{qv2}$$ where $\mathcal{N}_1(\vert {\mathbf{q}}\vert)$ and $x_m! = \rho(m)$ can be chosen appropriately. Using (\[san1\]) we can determine the normalization constant $\mathcal N_1(\vert {\mathbf{q}}\vert)$, and the resolution of the identity (for details see [@Thi]). First note that, in order for the norm of the vector $\mid {\mathbf{q}},j \rangle$ to be finite, we must have, $$\langle {\mathbf{q}},j \mid {\mathbf{q}},j \rangle = \mathcal N_1(\vert {\mathbf{q}}\vert)^{-1} \sum_{m=0}^\infty \frac {r^{2m}}{x_m!} < \infty.$$ Thus, if $\lim_{m \rightarrow \infty} x_m = x$, we need to restrict $r$ to $0 \leq r < L = \sqrt{x}$ for the convergence of the above series. In this case, define $$\label{domain}
\mathcal D = \{(r, \theta, \phi, \psi) \mid 0 \leq r < L, \; 0
\leq \phi \leq \pi, \; 0 \leq \theta, \psi < 2\pi\},$$ The resolution of the identity is then given with respect to a measure of the type $$\label{measure}
d\varsigma(r, \theta,
