## 4 Phase-Current Relation  
In this section, we briefly explore the mutual relation between the phase and current descriptors [see Equation (19)] of the quantum state of an electron,  
\[\psi(\mathbf{q})=\varphi(\mathbf{r})\xi(\sigma),\]  
where  
\[\varphi(\mathbf{r})=m(\mathbf{r})\ \text{exp}[\text{i}\phi(\mathbf{r})]. \tag{36}\]  
Its probability distribution reflects the square of the MO modulus factor, \(p(\mathbf{r})=m(\mathbf{r})^{2}\), while its phase component \(\phi(\mathbf{r})\) determines the state probability current of Equation (19):  
\[\begin{array}{c}j(\mathbf{r})=[\hbar/(2 mi)]\ [\varphi(\mathbf{r})^{*}\ \forall\varphi(\mathbf{r})- \varphi(\mathbf{r})\ \forall\varphi(\mathbf{r})^{*}]\\ =(\hbar/m)\ p(\mathbf{r})\ \
abla\phi(\mathbf{r})\equiv p(\mathbf{r})\ \mathbf{V}(\mathbf{r}).\end{array} \tag{37}\]  
Here, the effective velocity of the probability \"fluid\",  
\[\mathbf{V}(\mathbf{r})=\{V_{u}(\mathbf{r})\equiv V_{u}(u,\,\{v\!\
eq\!u\}),\quad u=x,\,y, \,z\},\]measures the local current-per-particle and reflects the state phase gradient:  
\[\mathbf{V}(\mathbf{r})=j(\mathbf{r})/p(\mathbf{r})=(\hbar/m)\;\
abla\phi(\mathbf{r}). \tag{38}\]  
Therefore, velocity components of the probability flux are determined by the corresponding partials of the state phase function:  
\[V_{u}(\mathbf{r})=(\hbar/m)\;[\partial\phi(\mathbf{r})/\partial u],\quad u=x,y,z. \tag{39}\]  
It follows from this equation that the phase of electronic state can be formally reconstructed by an indefinite integration of the corresponding components of the velocity field of the probability flux:  
\[\phi(u,\{v\
eq u\})=(m/\hbar)\int\limits_{-\infty}^{u}V_{u}(\delta,\{v\
eq u \})d\delta,u=x,y,z. \tag{40}\]  
As an illustrative example, consider the fixed-momentum (free-particle) state  
\[\varphi_{\mathbf{K}}(\mathbf{r})=A\;\exp(\text{i}\mathbf{K}\cdot\mathbf{r})=A\;\exp[\text{i}( K_{x}x+K_{y}y+K_{z}z),\]  
\[K=(\text{m}/\hbar)\mathbf{V},\mathbf{V}=i\mathbf{V}_{\text{x}}+j\mathbf{V}_{\text{y}}+k\mathbf{V}_{ z}=const. \tag{41}\]  
One indeed recovers its phase components,  
\[\phi(\mathbf{r})=K\cdot\mathbf{r}=(m/\hbar)\;(V_{x}x+V_{y}y+V_{z}z), \tag{42}\]  
by straightforward unidimensional integrations of Equation (40).  
Consider now the phase/current dependent term of the overall content of MO resultant gradient-information [53; 54; 55; 56; 57; 58; 59; 60; 61; 28] in Quantum Information Theory (QIT) [62]:  
\[\begin{array}{c}I[\phi]=4\int p(\mathbf{r})\;[\
abla\phi(\mathbf{r})]^{2}d\mathbf{r} \equiv\int p(\mathbf{r})\;I_{\phi}(\mathbf{r})\;d\mathbf{r}\\ =(2m/\hbar)^{2}\int p(\mathbf{r})\;V(\mathbf{r})^{2}\;d\mathbf{r}=(2m/\hbar)^{2}\int p(\mathbf{ r})^{-1}\;j(\mathbf{r})^{2}\;d\mathbf{r}.\end{array} \tag{43}\]  
By analogy to the free-electron wavefunction of Equation (41), one introduces the wave-vector distribution in molecular systems, defined by the local phase gradient  
\[K(\mathbf{r})=(m/\hbar)\;\mathbf{V}(\mathbf{r})=\
abla\phi(\mathbf{r}). \tag{44}\]  
In terms of this local momentum measure, the nonclassical information functional simplifies:  
\[I[\phi]=4\int p(\mathbf{r})\;\mathbf{K}(\mathbf{r})^{2}d\mathbf{r}\equiv 4\langle K^{2}\rangle. \tag{45}\]  
Therefore, in such an inhomogeneous (molecular) electron density, the nonclassical MO information reflects the state average square \(\langle K^{2}\rangle\) of the wave-vector \(\mathbf{K}(\mathbf{r})\). It thus follows from the preceding equation that this nonclassical information measure, related to the current contribution to the state resultant kinetic energy, depends only on the magnitude of the local probability flow, being independent of its direction.