In this chapter, we will show that the two Eqs. (15) and (28) being related to BM can be transformed in two quantum hydrodynamic equations: The continuity equation of BM (15) can be transformed into the continuity equation of MPQHD and the Eulerian version of the Bohmian equation of motion (28) can be transformed into the quantum Cauchy equation. Hereby, both the continuity equation of MPQHD and the quantum Cauchy equation only depend on a single position vector \(\mathbf {q}\). As a contrast, the two Eqs. (15) and (28) being related to BM depend on the complete set of particle coordinates \(\mathbf {Q}\).
Please note that we derived the continuity equation of MPQHD and the quantum Cauchy equation already in [9], but in the derivations and discussions there, the connection between BM and MPQHD was not cleared—this gap is filled in this work. Before we start with the derivations in this chapter, we point out these two details:
The first detail is that in [9], the continuity equation of MPQHD was named in another manner as the many-particle continuity equation because we had not to distinguish there between a Bohmian and a quantum hydrodynamic version of the continuity equation.
The second detail is that we presented in [9] two different versions of the continuity equation of MPQHD and the quantum Cauchy equation—namely, for both the continuity equation of MPQHD and the quantum Cauchy equation there is one version for an individual particle sort \(\text {A}\) and another version for the total particle ensemble. We will show that the continuity equation of BM (15) can be transformed into both versions of the continuity equation of MPQHD. Analogously, we will show that the Eulerian version of the Bohmian equation of motion (28) can be transformed into both versions of the quantum Cauchy equation.
Now, we will show the connection between BM and MPQHD. We start to carry out this task by transforming the continuity equation of BM (15) into the continuity equation of MPQHD for the particle sort \(\text {A}\). Therefore, we multiply the continuity equation of BM (15) by the factor \(N(\text {A}) \hspace{0.05 cm} m_\text {A} \delta (\mathbf {q} - \mathbf {q}_1^{ \text {A}})\) and integrate it over the complete set of particle coordinates \(\mathbf {Q}\). So, we find:
$$\begin{aligned} N(\text {A}) \hspace{0.05 cm}m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \frac{\partial D(\mathbf {Q},t)}{\partial t}&= - N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _{\mathbf {Q}} \mathbf {J}(\mathbf {Q},t). \end{aligned}$$
(32)
Then, using the fact that the MPQHD mass density \(\rho _m^\text {A}(\mathbf {q},t)\) for the particles of the sort \(\text {A}\) is given by [9]
$$\begin{aligned} \rho _m^\text {A}(\mathbf {q},t)&= N(\text {A}) \hspace{0.05 cm} m_\text {A} \; \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D(\mathbf {Q},t), \end{aligned}$$
(33)
we find for the left side of Eq. (32):
$$\begin{aligned} N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \frac{\partial D(\mathbf {Q},t)}{\partial t}&= \frac{\partial }{\partial t} \left[ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q}\; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D(\mathbf {Q},t) \right] \nonumber \\&= \frac{\partial \rho _m^\text {A}(\mathbf {q},t)}{\partial t}. \end{aligned}$$
(34)
For the transformation of the right side of Eq. (32), we regard that
$$\begin{aligned} \nabla _{\mathbf {Q}} \mathbf {J}(\mathbf {Q},t)&= \sum _{\text {B},i} \nabla _i^\text {B} \mathbf {J}_i^{\text {B}}(\mathbf {Q},t), \end{aligned}$$
(35)
so, it is given by:
$$\begin{aligned}&- N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _{\mathbf {Q}} \mathbf {J}(\mathbf {Q},t) \nonumber \\&\quad = - N(\text {A}) \hspace{0.05 cm} m_\text {A} \sum _{\text {B},i} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _i^\text {B} \mathbf {J}_i^{\text {B}}(\mathbf {Q},t). \end{aligned}$$
(36)
Using the volume element \(\text{d} \mathbf {Q}_{i}^{\text {B}}\) for all coordinates except for the coordinate \(\mathbf {q}_i^{\text {B}}\), we can write the volume element \(\text {d} \mathbf {Q}\) for the complete set of particles as
$$\begin{aligned} \text{d} \mathbf {Q}&= \text{d} \mathbf {Q}_{i}^{\text {B}} \; \text{d} \mathbf {q}_{i}^{\text {B}}. \end{aligned}$$
(37)
As the next step, we transform Eq. (36) by splitting up the sum \(\sum _{\text {B},i}\) into the summand for the special case \(\{ \text {B} = \text{A}, i = 1 \}\) and a sum over all the remaining summands. In addition, we use Eq. (37) for these remaining summands. Then, by applying the divergence theorem, we find that the integration over the coordinate \(\mathbf {q}_i^{\text {B}}\) for all these remaining summands can be transformed into an integral over the system boundary surface where the wave function vanishes. Thus, these remaining summands vanish, and only the summand for the special case \(\{ \text {B} = \text{A}, i = 1 \}\) remains. By considering these ideas, we find:
$$\begin{aligned}&- N(\text {A}) \hspace{0.05 cm} m_\text {A} \sum _{\text {B},i} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _i^\text {B} \mathbf {J}_i^{\text {B}}(\mathbf {Q},t) \nonumber \\&\quad = - N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _1^\text {A} \mathbf {J}_1^\text {A}(\mathbf {Q},t) \nonumber \\&\quad \quad - \underset{\{ \text {B},j \} \ne \{ \text {A}, 1 \}}{\sum _{\text {B}=1}^{{N_S}} \sum _{i=1}^{N(\text {B})}} \int \text {d} \mathbf {Q}_{i}^{\text {B}} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \underbrace{\int \text {d} \mathbf {q}_{i}^{\text {B}} \; \nabla _i^\text {B} \mathbf {J}_i^{\text {B}}(\mathbf {Q},t)}_{=\; 0} \nonumber \\&\quad = \; - N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _1^\text {A} \mathbf {J}_1^\text {A}(\mathbf {Q},t) \nonumber \\&\quad = \; - \nabla _{\mathbf {q}} \left[ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta( \mathbf {q} - \mathbf {q}_1^{ \text {A}}) \; \mathbf {J}_1^\text {A}(\mathbf {Q},t) \right] . \end{aligned}$$
(38)
Now, the MPQHD mass current density \(\mathbf {j}_m^\text {A}(\mathbf {q},t)\) for the particles of the sort \(\text {A}\) given by [9]Footnote 2
$$\begin{aligned} \mathbf {j}_m^\text {A}(\mathbf {q},t)&= N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \mathbf {J}_1^\text {A}(\mathbf {Q},t) \end{aligned}$$
(39)
can be inserted in Eq. (38), and we find the following result for the right side of Eq. (32):
$$\begin{aligned} - N(\text {A}) \hspace{0.05 cm} m_\text {A} \sum _{\text {B},i} \int \text {d} \mathbf {Q} \; \delta ( \mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _i^\text {B} \mathbf {J}_i^{\text {B}}&= - \nabla _{\mathbf {q}} \mathbf {j}_m^\text {A}(\mathbf {q},t). \end{aligned}$$
(40)
Finally, we can combine the results (34) and (40) for the left side and the right side of Eq. (32) and get as a result the continuity equation of MPQHD for the particle sort \(\text {A}\) [9]:
$$\begin{aligned} \frac{\partial \rho _m^\text {A}(\mathbf {q},t)}{\partial t}&= - \nabla _{\mathbf {q}} \mathbf {j}_m^\text {A}(\mathbf {q},t). \end{aligned}$$
(41)
Thus, we have proven that the continuity equation of MPQHD for the particle sort \(\text {A}\) (41) can be derived from the continuity equation of BM (15).
In addition, having completed this proof, it is trivial that the continuity equation of MPQHD for the total particle ensemble can be derived from the continuity equation of BM (15), too, since in [9], it was already shown that the continuity equation of MPQHD for the total particle ensemble can be derived just by summing up the continuity equation of MPQHD for particles of a certain sort \(\text {A}\) over all sorts of particle. This continuity equation of MPQHD for the total particle ensemble has the form
$$\begin{aligned} \frac{\partial \rho _m^\text {tot}(\mathbf {q},t)}{\partial t}&= - \nabla _{\mathbf {q}} \mathbf {j}_m^\text {tot}(\mathbf {q},t), \end{aligned}$$
(42)
where the quantity \(\rho _m^\text {tot}(\mathbf {q},t)\) is the MPQHD total mass density given by:
$$\begin{aligned} \rho _m^\text {tot}(\mathbf {q},t)&= \sum _{\text {A}=1}^{N_S} \rho _m^\text {A}(\mathbf {q},t), \end{aligned}$$
(43)
and the quantity \(\mathbf {j}_m^\text {tot}(\mathbf {q},t)\) is the MPQHD total mass current density given by:
$$\begin{aligned} \mathbf {j}_m^\text {tot}(\mathbf {q},t)&= \sum _{\text {A}=1}^{N_S} \mathbf {j}_m^\text {A}(\mathbf {q},t). \end{aligned}$$
(44)
So, we have shown that one can take the continuity equation of BM (15) as a starting point to derive both the continuity equation of MPQHD for the particle sort \(\text {A}\) (41) and the continuity equation of MPQHD for the total particle ensemble (42).
In the following calculation, we will take the Eulerian version of the Bohmian equation of motion (28) as a starting point to derive both the quantum Cauchy equation for the particle sort \(\text {A}\) and the quantum Cauchy equation for the total particle ensemble.
We start this calculation by taking into account the three vector components of the Eulerian version of the Bohmian equation of motion (28), which are related to the \((\text {A},1)\) particle. For these three vector components, the following differential equation holds:
$$\begin{aligned} m_{\text{ A }} \left[ \frac{\partial }{\partial t} + \sum _{\text {B},i} \left( \mathbf {w}_i^\text {B}(\mathbf {Q},t) \nabla _i^\text {B} \right) \right] \mathbf {w}_i^\text {A}(\mathbf {Q},t)&= - \nabla _1^\text {A} \left[ V_{\text {qu}}(\mathbf {Q},t) + V(\mathbf {Q},t) \right] . \end{aligned}$$
(45)
Now, we multiply Eq. (45) by \(N(\text {A}) \hspace{0.05 cm} D(\mathbf {Q},t) \hspace{0.05 cm} \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}})\) and integrate it over the complete set of coordinates \(\mathbf {Q}\). So, we find:
$$\begin{aligned}&N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \left[ \frac{\partial }{\partial t} + \sum _{\text {B},i} \left( \mathbf {w}_i^\text {B}(\mathbf {Q},t) \nabla _i^\text {B} \right) \right] \mathbf {w}_1^\text {A}(\mathbf {Q},t) \nonumber \\&\quad = \; - N(\text {A}) \int \text {d} \mathbf {Q} \; \delta \hspace{0.05 cm} (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D \; \nabla _1^\text {A} \left[ V_{\text {qu}} + V \right] . \end{aligned}$$
(46)
As the next step, we introduce the force density \(\mathbf {f}^\text {A}(\mathbf {q},t)\) [8, 9]Footnote 3 and the quantum force density \(\mathbf {f}^\text {A}_{\text {qu}}(\mathbf {q},t)\) for the particles of the sort \(\text {A}\) ([8]Footnote 4 and cf. [7], p. 326):
$$\begin{aligned} \mathbf {f}^\text {A}(\mathbf {q},t)&= N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D(\mathbf {Q},t) \left[ - \nabla _1^\text {A} V(\mathbf {Q},t) \right] , \end{aligned}$$
(47)
$$\begin{aligned} \mathbf {f}^\text {A}_{\text {qu}}(\mathbf {q},t)&= N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D(\mathbf {Q},t) \hspace{0.05 cm} \left[ - \nabla _1^\text {A} V_{\text {qu}}(\mathbf {Q},t) \right] . \end{aligned}$$
(48)
Using Eqs. (47) and (48), we can rewrite the right side of Eq. (46) in the following form:
$$\begin{aligned}&- N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D(\mathbf {Q},t) \; \nabla _1^\text {A} \left[ V_{\text {qu}} (\mathbf {Q},t) + V (\mathbf {Q},t) \right] \; \nonumber \\&\quad = \; \mathbf {f}^\text {A}_{\text {qu}}(\mathbf {q},t) + \mathbf {f}^\text {A}(\mathbf {q},t). \end{aligned}$$
(49)
As an intermediate step, we introduce the momentum flow density tensor \({\varvec{\Pi }}^\text {A}(\mathbf {q},t)\) for the sort of particle \(\text {A}\).
As we discussed already in [9], there are several versions of this tensor \({\varvec{\Pi }}^\text {A}(\mathbf {q},t)\) because it is not this tensor itself which is physically relevant but only its tensor divergence \(\nabla {\varvec{\Pi }}^\text {A}(\mathbf {q},t)\). Moreover, we pointed out in the mentioned reference that we recommend to use the Wyatt version of this tensor, and therefore, in this work, we will only use this tensor version. The naming of this tensor version is motivated by the form of the formula (1.57) in R. E. Wyatts book [7], p. 31, and it is given by (hereby, \(\mathbf{1}\) is the unit matrix, and the operation symbol \(\otimes\) interrelates two vectors to a dyadic product) [9]:
$$\begin{aligned} {\varvec{\Pi }}^{\text {A}}(\mathbf {q},t)&= \mathbf{1} P_\text {A} + N(\text {A}) \hspace{0.05 cm} m_{\text {A}} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A} } ) \; D \; \left[ \left( \mathbf {w}_{1}^\text {A} \otimes \mathbf {w}_{1}^\text {A} \right) + \left( \mathbf {d}_{1}^{\text {A}} \otimes \mathbf {d}_{1}^{\text {A}} \right) \right] . \end{aligned}$$
(50)
Therefore, its Cartesian tensor elements are given by:
$$\begin{aligned} \Pi _{\alpha \beta }^{\text {A}}(\mathbf {q},t)&= P_\text {A} \delta _{\alpha \beta } + N(\text {A}) \; m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A} } ) \; D \; \left( w_{1 \alpha }^\text {A} w_{1 \beta }^\text {A} + d_{1 \alpha }^\text {A} d_{1 \beta }^\text {A} \right) . \end{aligned}$$
(51)
The quantity \(P_\text {A}(\mathbf {q},t)\), which appears in Eqs. (50) and (51), is the scalar quantum pressure for the particles of the sort \(\text {A}\), and this quantity is given by ([9] and cf. [7], p. 326):
$$\begin{aligned} P_\text {A}(\mathbf {q},t)&= - N(\text {A}) \frac{\hbar ^2}{4 m_\text {A}} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; \Delta _1^\text {A} D. \end{aligned}$$
(52)
In addition, in Eq. (50), the dyadic product of a vector \(\mathbf {d}_i^{\text {A}}(\mathbf {Q},t)\) for the case \(i=1\) appears; this vector is defined by
$$\begin{aligned} \mathbf {d}_i^{\text {A}}(\mathbf {Q},t)&= - \frac{\hbar }{2 m_\text {A}} \frac{\nabla _i^\text {A} {D}}{D}. \end{aligned}$$
(53)
The quantity \(\mathbf {d}_i^{\text {A}}(\mathbf {Q},t)\) is named Bohmian osmotic velocity of the \((\text {A},i)\) particle corresponding to the nomenclature in [7], p. 327 and [9]. The Bohmian osmotic velocity \(\mathbf {d}_i^{\text {A}}(\mathbf {Q},t)\) is the quantum analog to the Bohmian particle velocity \(\mathbf {w}_i^\text {A}(\mathbf {Q},t)\), and in addition, this quantity is related to the shape of \(D(\mathbf {Q},t)\).
Moreover, in [9], we discussed that one can split the momentum flow density tensor \({\varvec{\Pi }}^\text {A}(\mathbf {q},t)\) for the particles of the sort \(\text {A}\) in a classical part and a quantum part:
$$\begin{aligned} {\varvec{\Pi }}^\text {A}(\mathbf {q},t)&= {\varvec{\Pi }}^\text {A,cl}(\mathbf {q},t) + {\varvec{\Pi }}^\text {A,qu}(\mathbf {q},t). \end{aligned}$$
(54)
Hereby, the classical part is given by
$$\begin{aligned} {\varvec{\Pi }}^\text {A,cl}(\mathbf {q},t)&= N(\text {A}) \hspace{0.05cm} m_{\text {A}} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A} }) \; D \; \left( \mathbf {w}_{1}^\text {A} \otimes \mathbf {w}_{1}^\text {A} \right) , \end{aligned}$$
(55)
and its matrix elements are
$$\begin{aligned} \Pi ^\text {A,cl}_{\alpha \beta }(\mathbf {q},t)&= N(\text {A}) \hspace{0.05 cm} m_{\text {A}} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A} }) \; D \; w_{1\alpha }^\text {A} w_{1\beta }^\text {A}. \end{aligned}$$
(56)
And the quantum part is given by
$$\begin{aligned} {\varvec{\Pi }}^\text {A,qu}(\mathbf {q},t)&= \mathbf{1} P_\text {A} + N(\text {A}) \hspace{0.05 cm} m_{\text {A}} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A} }) \; D \; \left( \mathbf {d}_{1}^{\text {A}} \otimes \mathbf {d}_{1}^{\text {A}} \right) \end{aligned}$$
(57)
with according matrix elements
$$\begin{aligned} \Pi ^\text {A,qu}_{\alpha \beta }(\mathbf {q},t)&= P_\text {A} \delta _{\alpha \beta } + N(\text {A}) \hspace{0.05 cm} m_{\text {A}} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A} }) \; D \; d_{1\alpha }^\text {A} d_{1\beta }^\text {A}. \end{aligned}$$
(58)
Now, we will prove that the quantum force \(\mathbf {f}_{\text {qu}}^\text {A}(\mathbf {q},t)\) is equal to the negative tensor divergence of the quantum part \({\varvec{\Pi }}^{\text {A,qu}}(\mathbf {q},t)\) of the momentum flow density tensor. So, we have to show that the following equation holds, where \(\mathbf {e}_\beta\), \(\beta \in \{x,y,z\}\) are the Cartesian basis vectors:
$$\begin{aligned} \mathbf {f}_{\text {qu}}^{\text {A}}(\mathbf {q},t)&= - \nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,qu}(\mathbf {q},t) \equiv - \sum _{\alpha ,\beta } \frac{\partial \Pi _{\alpha \beta }^\text {A,qu}(\mathbf {q},t)}{\partial q_{\alpha }} \mathbf {e}_\beta . \end{aligned}$$
(59)
Therefore, we transform Eq. (48) for the quantum force density \(\mathbf {f}^\text {A}_{\text {qu}}(\mathbf {q},t)\) using Eq. (17) for the quantum potential \(V_{\text {qu}}(\mathbf {q},t)\):
$$\begin{aligned} \mathbf {f}^\text {A}_{\text {qu}}(\mathbf {q},t)&= \; N(\text {A}) \int \text {d} \mathbf {Q}\; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \; \left[ - \nabla _1^\text {A} V_{\text {qu}} \right] \nonumber
\\&= \; -N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _1^\text {A} \left( D \hspace{0.05 cm}V_{\text {qu}} \right) + N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; V_{\text {qu}} \nabla _1^\text {A} D \nonumber
\\&= -N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _1^\text {A} \left\{ -\sum _{\text {B},i} \frac{\hbar ^2}{4 m_\text {B}} \left[ \mathop {}\!\mathbin \Delta _i^\text {B} D - \frac{\left( \nabla _i^\text {B} D \right) ^2}{2 D} \right] \right\} \nonumber
\\& \quad - N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \sum _{\text {B},i} \frac{\hbar ^2}{4 m_\text {B}} \left[ \frac{\mathop {}\!\mathbin \Delta _i^\text {B} D}{D} - \frac{\left( \nabla _i^\text {B} D \right) ^2}{2 D^2} \right] \nabla _1^\text {A} D \nonumber
\\& = -N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _1^\text {A} \left[ - \sum _{\text {B},i} \frac{\hbar ^2}{4 m_\text {B}} \mathop {}\!\mathbin \Delta _i^\text {B} D \right] \nonumber \\&\quad-N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \sum _{\text {B},i} \frac{\hbar ^2}{4 m_\text {B}} \; \nonumber
\\&\quad \times \left\{ \frac{ \left( \Delta _i^\text {B} D \right) \left( \nabla _1^\text {A} D \right) }{D} - \frac{ \left( \nabla _i^\text {B} D \right) ^2 \left( \nabla _1^\text {A} D \right) }{2 D^2} + \nabla _1^\text {A} \left[ \frac{ \left( \nabla _i^\text {B} D \right) ^2 }{2 D} \right] \right\} . \end{aligned}$$
(60)
Now, we transform the term appearing in the first line of Eq. (60) by regarding in a argumentation similar to the transformation of Eq. (36) to (38) that only the summand for \(\{\text {B}=\text {A}, i=1\}\) does not vanish for this term. In addition, we use the definition (52) for the scalar pressure \(P_\text {A}(\mathbf {q},t)\) and find:
$$\begin{aligned}&- N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _1^\text {A} \left[ - \sum _{\text {B},i} \frac{\hbar ^2}{4 m_\text {B}} \Delta _i^\text {B} D \right] \nonumber \\&\quad = - N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \nabla _1^\text {A} \left[ - \frac{\hbar ^2}{4 m_\text {A}} \Delta _1^\text {A} D \right] \nonumber \\&\quad = - \nabla _{\mathbf {q}} \left[ - N(\text {A}) \frac{\hbar ^2}{4 m_\text {A}} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \Delta _1^\text {A} D \right] \nonumber \\&\quad = - \nabla _{\mathbf {q}} P_\text {A} \; = \; - \sum _\alpha \frac{\partial }{\partial q_\alpha } \left( \delta _{\alpha \beta } P_\text {A} \right) \; = \; - \nabla _{\mathbf {q}} \left( \mathbf{1} P_\text {A} \right) . \end{aligned}$$
(61)
Note that in the last line of the calculation above, we rewrote the gradient of the scalar quantity \(P_\text {A}(\mathbf {q},t)\) into the divergence of the tensor \(\left[ \mathbf{1} P_\text {A}(\mathbf {q},t) \right]\).
Our next step is to transform the term in curly brackets in Eq. (60). Using the abbreviation \(\frac{\partial }{\partial q_{i \alpha }^\text {A}} = \partial _{i \alpha }^\text {A}\) and the definition (53) of the Bohmian osmotic velocity \(\mathbf {d}_i^{\text {A}}(\mathbf {Q},t)\), we find in a straightforward calculation the following result for the x component of the term in curly brackets in Eq. (60):
$$\begin{aligned}&\left\{ \frac{ \left( \Delta _i^\text {B} D \right) \left( \nabla _1^\text {A} D \right) }{D} - \frac{ \left( \nabla _i^\text {B} D \right) ^2 \left( \nabla _1^\text {A} D \right) }{2 D^2} + \nabla _1^\text {A} \left[ \frac{ \left( \nabla _i^\text {B} D \right) ^2 }{2 D} \right] \right\} _x \nonumber \\&\quad =\ \sum _\alpha \left\{ \frac{ \left( \partial _{i \alpha }^\text {B} \partial _{i \alpha }^\text {B} D \right) \left( \partial _{1 x}^\text {A} D \right) }{D} - \frac{\left( \partial _{i \alpha }^\text {B} D \right) ^2 \left( \partial _{1 x}^\text {A} D \right) }{2 D^2} + \partial _{1 x}^\text {A} \left[ \frac{ \left( \partial _{i \alpha }^\text {B} D \right) ^2}{2 D} \right] \right\} \nonumber \\&\quad =\; \sum _\alpha \left\{ \partial _{i \alpha }^\text {B} \left[ \frac{ \left( \partial _{i \alpha }^\text {B} D \right) \left( \partial _{1 x}^\text {A} D \right) }{D} \right] - \frac{\left( \partial _{i \alpha }^\text {B} D \right) \left( \partial _{i \alpha }^\text {B} \partial _{1 x}^\text {A} D \right) }{D} + \frac{\left( \partial _{i \alpha }^\text {B} D \right) ^2 \left( \partial _{1 x}^\text {A} D \right) }{D^2} \right. \nonumber \\&\quad \quad \left. - \frac{\left( \partial _{i \alpha }^\text {B} D \right) ^2 \left( \partial _{1 x}^\text {A} D \right) }{2 D^2} + \frac{\left( \partial _{i \alpha }^\text {B} D \right) \left( \partial _{1 x}^\text {A} \partial _{i \alpha }^\text {B} D \right) }{D} - \frac{\left( \partial _{i \alpha }^\text {B} D \right) ^2 \left( \partial _{1 x}^\text {A} D \right) }{2 D^2} \nonumber \right\} \\&\quad = \sum _\alpha \partial _{i \alpha }^\text {B} \left[ \frac{ \left( \partial _{i \alpha }^\text {B} D \right) \left( \partial _{1 x}^\text {A} D \right) }{D} \right] \nonumber \\&\quad = \frac{4 m_\text {A} m_\text {B}}{\hbar ^2} \sum _\alpha \partial _{i \alpha }^\text {B} \left( D \; d_{i \alpha }^\text {B} \; d_{1 x}^\text {A} \right) . \end{aligned}$$
(62)
So, using a dyadic product \(\mathbf {d}_{i}^{\text {B}} \otimes \mathbf {d}_{1}^{\text {A}}\), we can write the term in curly brackets in Eq. (60) as:
$$\begin{aligned}&\frac{ \left( \Delta _i^\text {B} D \right) \left( \nabla _1^\text {A} D \right) }{D} - \frac{ \left( \nabla _i^\text {B} D \right) ^2 \left( \nabla _1^\text {A} D \right) }{2 D^2} + \nabla _1^\text {A} \left[ \frac{ \left( \nabla _i^\text {B} D \right) ^2 }{2 D} \right] \nonumber \\&\quad = \frac{4 m_\text {A} m_\text {B}}{\hbar ^2} \nabla _{i}^\text {B} \left[ D \left( \mathbf {d}_{i}^{\text {B}} \otimes \mathbf {d}_{1}^{\text {A}} \right) \right] . \end{aligned}$$
(63)
As the next step, we transform Eq. (60) using the intermediate results (61) and (63), obtaining:
$$\begin{aligned} \mathbf {f}^\text {A}_{\text {qu}}(\mathbf {q},t) =&- \nabla _{\mathbf {q}} \left( \mathbf{1} P_\text {A} \right) - N(\text {A}) \hspace{0.05 cm} m_\text {A} \sum _{\text {B},i} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; \nabla _i^\text {B} \left[ D \left( \mathbf {d}_{i}^{\text {B}} \otimes \mathbf {d}_{1}^{\text {A}} \right) \right] . \end{aligned}$$
(64)
Now, we again use a similar argumentation to the transformation of Eq. (36) to (38) that in Eq. (64) only the summand for \(\{\text {B}=\text {A}, i=1\}\) does not vanish. Finally, using the definition (57) for the quantum part \({\varvec{\Pi }}^\text {A,qu}(\mathbf {q},t)\) of the momentum flow density tensor for the particle of the sort A, Eq. (59) can be proven as:
$$\begin{aligned} \mathbf {f}^\text {A}_{\text {qu}}(\mathbf {q},t)&= - \nabla _{\mathbf {q}} \left( \mathbf{1} P_\text {A} \right) - \nabla _{\mathbf {q}} \left[ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q}\; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \left( \mathbf {d}_{1}^{\text {A}} \otimes \mathbf {d}_{1}^{\text {A}} \right) \right] \nonumber \\&= - \nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,qu}(\mathbf {q},t). \end{aligned}$$
(65)
Combining Eqs. (49) and (59), we now find for the right side of Eq. (46) the following result:
$$\begin{aligned}&- N(\text {A}) \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D(\mathbf {Q},t) \; \nabla _1^\text {A} \left[ V_{\text {qu}}(\mathbf {Q},t) + V(\mathbf {Q},t) \right] \; = \nonumber \\&\quad = \; \mathbf {f}^\text {A}(\mathbf {q},t) - \nabla _{\mathbf {q}} {\varvec{\Pi }}^{\text {A,qu}}(\mathbf {q},t). \end{aligned}$$
(66)
As the next task, we transform the left side of Eq. (46) by analyzing the x component of this equation in detail:
$$\begin{aligned}&\left\{ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \left[ \frac{\partial }{\partial t} + \sum _{{\text{B}},{i}} \left( \mathbf {w}_i^\text {B} \nabla _i^\text {B} \right) \right] \mathbf {w}_1^\text {A} \right\} _x \nonumber
\\&\quad = N(\text {A}) \hspace{0.05 cm}m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \left[ \frac{\partial }{\partial t} + \sum _{\text {B},i} \left( \mathbf {w}_i^\text {B} \nabla _i^\text {B} \right) \right] w_{1x}^\text {A} \; \nonumber
\\&\quad = N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; \left[ \frac{\partial }{\partial t} \left( D \hspace{0.05 cm} w_{1x}^\text {A} \right) - w_{1x}^\text {A} \frac{\partial D}{\partial t}\right] \nonumber
\\&\quad \quad + \; N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \hspace{0.05cm} \left\{ \sum _{\text {B},i} \left[ \sum _\alpha \frac{\partial }{\partial q_{i \alpha }^{\text {B}}} \left( w_{1 x}^{\text {A}} w_{i \alpha }^{\text {B}} \right) \right] - w_{1 x}^{\text {A}} \nabla _i^{\text {B}} \mathbf {w}_i^{\text {B}} \right\} \nonumber \\&\quad = \underbrace{\frac{\partial }{\partial t} \left[ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \hspace{0.05 cm} w_{1 x}^\text {A} \right] }_{ \overset{{(39)}}{\underset{\text {}}{=}} \; \frac{\partial }{\partial t} j_{m,x}^\text {A}} \nonumber
\\& \quad \quad - N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; w_{1 x}^\text {A} \hspace{0.05 cm} \underbrace{\frac{\partial D}{\partial t}}_{\overset{{(12)}}{\underset{\text {}}{=}} \; - \sum _{\text {B},i} \nabla _i^\text {B} \left( D \hspace{0.05 cm} \mathbf {w}_i^\text {B} \right) } \nonumber
\\&\quad \quad + \; \underbrace{N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; \sum _{\text {B},i} \left[ \sum _{\alpha } \frac{\partial }{\partial q_{i \alpha }^\text {B}} \left( D \hspace{0.05 cm} w_{1 x}^\text {A} w_{i \alpha }^\text {B} \right) \right] }_{= \; \sum _{\alpha } \frac{\partial }{\partial q_\alpha } \left[ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \hspace{0.05 cm} \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \hspace{0.05 cm} D \hspace{0.05 cm} w_{1 x}^\text {A} w_{1 \alpha }^\text {A} \right] } \nonumber
\\&\quad \quad - \; N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; w_{1 x}^\text {A} \sum _{\text {B},i} \underbrace{\left( \sum _{\alpha } w_{i \alpha }^\text {B} \frac{\partial D}{\partial q_{i \alpha }^\text {B} } \right) }_{= \; \mathbf {w}_{i}^\text {B} \nabla _i^\text {B} D} \nonumber
\\&\quad \quad - \; N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D \hspace{0.05 cm} w_{1 x}^\text {A} \sum _{\text {B},i} \nabla _i^\text {B} \mathbf {w}_i^\text {B}. \end{aligned}$$
(67)
So, we got an intermediate result (67) for this vector component that ranges over five equation lines. As indicated by lowered braces, the first of these five equation lines can be transformed using the x component of the vector equation (39) for the MPQHD mass current density \(\mathbf {j}^\text {A}_m(\mathbf {q},t)\), the second equation line can be transformed using the continuity equation of BM (12), the third equation line can be simplified by regarding the fact that due to the divergence theorem only the summand for \(\{\text {B}=\text {A},i=1\}\) in the double sum over \(\text {B}\) and i does not vanish, and the fourth equation line is rewritten by using a vector notation instead of the sum over \(\alpha\). In addition, we permute the second and the third of these five equation lines and find:
$$\begin{aligned}&\left\{ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \left[ \frac{\partial }{\partial t} + \sum _{\text {B},i} \left( \mathbf {w}_i^\text {B} \nabla _i^\text {B} \right) \right] \mathbf {w}_1^\text {A} \right\} _x \nonumber
\\&\quad = \frac{\partial j_{m,x}^\text {A}}{\partial t} \nonumber
\\&\quad \quad + \underbrace{\sum _{\alpha } \frac{\partial }{\partial q_\alpha } \left[ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q}\; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D \hspace{0.05cm} w_{1 x}^\text {A} w_{1 \alpha }^\text {A} \right] }_{\overset{{(56)}}{\underset{\text {}}{=}} \; \left( \nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,cl} \right) _x} \nonumber
\\&\quad \quad + \; N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; w_{1 x}^\text {A} \sum _{\text {B},i} \nabla _i^\text {B} \left( D \hspace{0.05 cm} \mathbf {w}_i^\text {B} \right) \nonumber
\\&\quad \quad - \; N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; w_{1 x}^\text {A} \sum _{\text {B},i} \mathbf {w}_{i}^\text {B} \nabla _i^\text {B} D \nonumber \\&\quad \quad - \; N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}}) \; D \hspace{0.05cm} w_{1 x}^\text {A} \sum _{\text {B},i} \nabla _i^\text {B} \mathbf {w}_i^\text {B}. \end{aligned}$$
(68)
Above we indicated, using within a lowered brace Eq. (56), that one finds the term appearing in the second line of the right side of Eq. (68) to be equal to the x component of the divergence of classical part \({\varvec{\Pi }}^{\text {A,cl}}(\mathbf {q},t)\) of the momentum flow density tensor for the particles of the sort \(\text {A}\). Moreover, we merge the terms appearing in the third, fourth, and fifth line of the right side of Eq. (68) and recognize that the sum of these terms vanishes. Thus, we get the following result for the x component of the expression on the left side of Eq. (46):
$$\begin{aligned}&\left\{ N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \left[ \frac{\partial }{\partial t} + \sum _{\text {B},i} \left( \mathbf {w}_i^\text {B} \nabla _i^\text {B} \right) \right] \mathbf {w}_1^\text {A} \right\} _x \nonumber
\\& \quad = \frac{\partial j_{m,x}^\text {A}}{\partial t} + \left( \nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,cl} \right) _x \nonumber
\\& \quad \quad + \; N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q} \; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; w_{1 x}^\text {A} \sum _{\text {B},i} \underbrace{\left[ \nabla _i^\text {B} \left( D \hspace{0.05 cm} \mathbf {w}_i^\text {B} \right) - \mathbf {w}_i^\text {B} \nabla _i^\text {B} D - D \hspace{0.05 cm} \nabla _i^\text {B} \mathbf {w}_i^\text {B} \right] }_{= \; 0} \nonumber \\&\quad = \frac{\partial j_{m,x}^\text {A}}{\partial t} + \left( \nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,cl} \right) _x. \end{aligned}$$
(69)
Therefore, the expression on the left side of Eq. (46) is given by:
$$\begin{aligned}&N(\text {A}) \hspace{0.05 cm} m_\text {A} \int \text {d} \mathbf {Q}\; \delta (\mathbf {q} - \mathbf {q}_1^{\text {A}} ) \; D \left[ \frac{\partial }{\partial t} + \sum _{\text {B},i} \left( \mathbf {w}_i^\text {B} \nabla _i^\text {B} \right) \right] \mathbf {w}_1^\text {A} \nonumber \\&\quad = \frac{\partial \mathbf {j}_{m}^\text {A} (\mathbf {q},t)}{\partial t} + \nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,cl}(\mathbf {q},t). \end{aligned}$$
(70)
Now, we combine the result (70) for the left side of Eq. (46) and the result (66) for the right side of Eq. (46) and get:
$$\begin{aligned} \frac{\partial \mathbf {j}_{m}^\text {A} (\mathbf {q},t)}{\partial t} + \nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,cl}(\mathbf {q},t) =&\; \mathbf {f}^\text {A}(\mathbf {q},t) - \nabla _{\mathbf {q}} {\varvec{\Pi }}^{\text {A,qu}}(\mathbf {q},t). \end{aligned}$$
(71)
By shifting the term \(\nabla _{\mathbf {q}} {\varvec{\Pi }}^\text {A,cl}(\mathbf {q},t)\) to the right side of Eq. (71) and by applying \({\varvec{\Pi }}^{\text {A}}(\mathbf {q},t) = {\varvec{\Pi }}^{\text {A,cl}}(\mathbf {q},t) + {\varvec{\Pi }}^{\text {A,qu}}(\mathbf {q},t)\), one gets the Ehrenfest equation of motion for the particles of the sort \(\text {A}\) that we presented already in [9]:
$$\begin{aligned} \frac{\partial \mathbf {j}_{m}^\text {A} (\mathbf {q},t)}{\partial t} =&\; \mathbf {f}^\text {A}(\mathbf {q},t) - \nabla _{\mathbf {q}} {\varvec{\Pi }}^{\text {A}}(\mathbf {q},t). \end{aligned}$$
(72)
In addition, in [9], it was shown that performing some mathematical operations using the continuity equation of MPQHD for the particles of the sort \(\text {A}\) (41), one can transform the Ehrenfest equation of motion for the particles of the sort \(\text {A}\) (72) into the quantum Cauchy equation for the particles of this sort, which is given by:
$$\begin{aligned} \rho ^\text {A}_m(\mathbf {q},t) \left[ \frac{\partial }{\partial t} + \left( \mathbf {v}^\text {A}(\mathbf {q},t) \nabla _{\mathbf {q}} \right) \right] \mathbf {v}^\text {A}(\mathbf {q},t)&= \mathbf {f}^{\text {A}}(\mathbf {q},t) - \nabla _{\mathbf {q}} \mathbf{p}^\text {A}(\mathbf {q},t). \end{aligned}$$
(73)
In the quantum Cauchy equation for the particles of the sort \(\text {A}\) the MPQHD particle velocity \(\mathbf {v}^\text {A}(\mathbf {q},t)\) for the particles of the sort \(\text {A}\) and the pressure tensor \(\mathbf{p}^\text {A}(\mathbf {q},t)\) for the particles of this sort appear. The definition of these quantities was discussed already in our prework [9]. For the following discussions, there is no need to recapitulate these definitions again, thus, we skip them in this work.
Taking into account that the derivation mentioned above, i. e. the derivation of the quantum Cauchy equation for the particles of the sort \(\text {A}\) (73), was already worked out in detail in our previous work, we have proven that this quantum hydrodynamic equation can be derived from the Eulerian version of the Bohmian equation of motion (28) as a starting point.
Now, the only remaining open task of the outlined tasks at the end of Sect. 2 is the proof that the quantum Cauchy equation for the total particle ensemble can be derived from the Eulerian version of the Bohmian equation of motion (28) as a starting point. Therefore, we regrad as shown above that Eq. (28) can be used as a starting point to derive the Ehrenfest equation of motion for the particles of the sort \(\text {A}\) (72).
As the next step, we here recapitulate shortly that we have proven already in our prework [9] that one can derive the quantum Cauchy equation for the total particle ensemble using the Ehrenfest equation of motion for the particles of the sort \(\text {A}\) (72) and the continuity equation of MPQHD for the total particle ensemble (42):
Therefore, we take into account that, analogously to the definition (44) of the MPQHD total mass current density \(\mathbf {j}^{\text {tot}}_m(\mathbf {q},t)\), the momentum flow density tensor \({\varvec{\Pi }}^\text {tot}(\mathbf {q},t)\), and the force density \(\mathbf {f}^{\text {tot}}(\mathbf {q},t)\) for the total particle ensemble are given by the sum over the correspondent quantities for the particular particle sorts:
$$\begin{aligned} {\varvec{\Pi }}^\text {tot}(\mathbf {q},t)&= \sum _{\text {A}=1}^{N_S} {\varvec{\Pi }}^\text {A}(\mathbf {q},t), \end{aligned}$$
(74)
$$\begin{aligned} \mathbf {f}^{\text {tot}}(\mathbf {q},t)&= \sum _{\text {A}=1}^{N_S} \mathbf {f}^\text {A}(\mathbf {q},t). \end{aligned}$$
(75)
Now, we can sum up the Ehrenfest equation of motion for a certain sort of particle \(\text {A}\) (72) over all sorts of particle and using the Eqs. (44), (74), and (75), one gets the Ehrenfest equation of motion for the total particle ensemble as a result. This equation is given by:
$$\begin{aligned} \frac{\partial \mathbf {j}_{m}^{\mathrm{tot}} (\mathbf {q},t)}{\partial t} =&\; \mathbf {f}^\text {{tot}}(\mathbf {q},t) - \nabla _{\mathbf {q}} {\varvec{\Pi }}^{\mathrm{tot}}(\mathbf {q},t). \end{aligned}$$
(76)
In addition, in a calculation using the continuity equation of MPQHD for the total particle ensemble (42), one can now transform the Ehrenfest equation of motion for the total particle ensemble (76) into the quantum Cauchy equation for the total particle ensemble. Here, we skip the details of this calculation and only state its result, the quantum Cauchy equation for the total particle ensemble [9]:
$$\begin{aligned} \rho ^\text {tot}_m(\mathbf {q},t) \left[ \frac{\partial }{\partial t} + \left( \mathbf {v}^\text {tot}(\mathbf {q},t) \nabla _{\mathbf {q}} \right) \right] \mathbf {v}^\text {tot}(\mathbf {q},t)&= \mathbf {f}^{\text {tot}}(\mathbf {q},t) - \nabla _{\mathbf {q}} \mathbf{p}^\text {tot}(\mathbf {q},t). \end{aligned}$$
(77)
In Eq. (77), the quantity \(\mathbf {v}^{\text {tot}}(\mathbf {q},t)\) is the MPQHD particle velocity for the total particle ensemble, and \(\mathbf{p}^\text {tot}(\mathbf {q},t)\) is the pressure tensor for the total particle ensemble. Like for the corresponding quantities for a single sort of particle \(\text {A}\), we refer for the definitions of the quantities \(\mathbf {v}^{\text {tot}}(\mathbf {q},t)\) and \(\mathbf{p}^\text {tot}(\mathbf {q},t)\) to our previous work [9].
So, we now summarized how the quantum Cauchy equation for the total particle ensemble (77) can be derived using the Eulerian version of the Bohmian equation of motion (28) as a starting point.
As an additional remark to this proof, we mention that the reader might wonder why this proof was not performed by trying to find the quantum Cauchy equation for the total particle ensemble (77) by summing up the quantum Cauchy equation for a certain sort of particle \(\text {A}\) (73) over all sorts of particle instead of summing up the Ehrenfest equation of motion for a certain sort of particle A (72) over all sorts of particle.
The reason why we did not choose this proceeding in our analysis above is that one does not get the quantum Cauchy equation for the total particle ensemble (77) just by summing up the quantum Cauchy equation for a certain sort of particle \(\text {A}\) (73) over all sorts of particle because the Eq. (73) is a non-linear differential equation where the superposition principle is not valid.
For a good overview of the equations of BM, which we used as a starting point in this work, and the equations of MPQHD, which we here found as final results, we present them again in Table 2.
Table 2 On the left side of this table, we present the Bohmian equations, where all quantities depend on the complete set of particle coordinates \(\mathbf {Q}\) and the time t, and on the right side the corresponding quantum hydrodynamic equations, where all quantities depend on a single position vector \(\mathbf {q}\) and the time t As an additional point, we mention this analogy between classical mechanics and quantum mechanics:
In a classic system with several interacting particles, solving the coupled Newtonian equations of motion to receive the trajectories of these particles is a promising strategy for small systems with only few particles because already for relative large systems with tens or hundreds of particles the system of the Newtonian equations of motion is very hard to solve due to its big size. Thus, for such large many-particle systems, using classical hydrodynamics instead is a better approach because for these systems, the size of the hydrodynamical differential equations is much smaller than the size of the system of the Newtonain differential equations.
In an analogous manner, solving the Bohmian equation of motion (30) for a quantum system with several interacting particles to calculate the set \(\mathbf {p}(t)\) of all the individual Bohmian particle momenta is also a promising strategy for small systems only. The reason why solving the Bohmian equation of motion (30) is a good ansatz for small systems only is that already for relative large many-particle systems with tens or hundreds of particles the Bohmian equation of motion (30) is difficult to solve since for these systems, the Bohmian equation of motion (30) is a high-dimensional differential equation with \([\sum _{\text {A}=1}^{N_S} 3 N(\text {A})]\) dimensions. As a contrast, for many-particle systems, solving the Ehrenfest equations of motion (72) and (76) for a calculation of the MPQHD mass current densities \(\mathbf {j}_m^\text {A}(\mathbf {q},t)\) and \(\mathbf {j}_m^\text {tot}(\mathbf {q},t)\), or solving the quantum Cauchy equations (73) and (77) for a calculation of the MPQHD particle velocities \(\mathbf {v}^\text {A}(\mathbf {q},t)\) and \(\mathbf {v}^{\text {tot}}(\mathbf {q},t)\), are more promising approaches than solving the Bohmian equation of motion (73) because the Ehrenfest equations of motion and quantum Cauchy equations (72), (73), (76), and (77) are always three-dimensional equations independent of the size of the system.
Now, the reader might object that applying MPQHD for the calculation of the current densities \(\mathbf {j}_m^\text {A}(\mathbf {q},t)\), \(\mathbf {j}_m^\text {tot}(\mathbf {q},t)\) using Eqs. (72) and (76) is not purposive because in order to apply MPQHD in the manner described in the previous paragraph, the wave function of the system \(\varPsi (\mathbf {Q},t)\) has to be available in order to know the quantities on the right side of the Eqs. (72) and (76)—and knowing the wave function \(\varPsi (\mathbf {Q},t)\) we could calculate the mass current densities \(\mathbf {j}_m^\text {A}(\mathbf {q},t)\) and \(\mathbf {j}_m^\text {tot}(\mathbf {q},t)\) directly, anyway: Hereby, this direct calculation would be done using the Eqs. (5), (9), (10), (11), (39), and (44). However, there are still systems where the application of MPQHD is interesting:
In vibrating molecules within a single electronic state, electronic mass current densities \(\mathbf {j}^e_m(\mathbf {q},t)\) vanish if one calculates them directly within the Born-Oppenheimer approximation (BOA) using the BOA wave function
$$\begin{aligned} \varPsi _{\text {BOA}}(\mathbf {Q}^e,\mathbf {Q}^n,t)&= a_{\text {BOA}}(\mathbf {Q}^e,\mathbf {Q}^n,t) \exp \left[ \frac{\mathrm {i} S_{\text {BOA}}(\mathbf {Q}^n,t)}{\hbar } \right] \end{aligned}$$
(78)
as an input [26], where \(\mathbf {Q}^e\) is the complete set of all electronic coordinates and \(\mathbf {Q}^n\) is the complete set of all nuclear coordinates. The reason for this is that the action \(S_{\text {BOA}}(\mathbf {Q}^n,t)\) for the occupation of a single electronic state is falsely independent of the set of electronic coordinates \(\mathbf {Q}^e\) because of the BOA. So, due to Eq. (10), all the Bohmian electronic velocities \(\mathbf {w}_i^e(\mathbf {Q},t)\), \(i \in \{1,2,\ldots ,N_e\}\) for all electrons vanish.
As a contrast, the total particle density \(D(\mathbf {Q}^e,\mathbf {Q}^n,t)\) can be calculated within the BOA in good accuracy using
$$\begin{aligned} D(\mathbf {Q}^e, \mathbf {Q}^n,t) \approx a_{\text {BOA}}^2(\mathbf {Q}^e, \mathbf {Q}^n,t). \end{aligned}$$
(79)
In addition, if one analyzes molecular vibrations where electrons and nuclei are at rest at the start time \(t=0\), one can assume as a start condition the electronic mass current density \(\mathbf {j}_m^e(\mathbf {q},t)\) to be zero at this start time.
Now, we discuss the application of the Ehrenfest equation of motion (72) for the particle sort \(\text {A}\) for this system: Therefore, regarding the electrons as the particle sort \(\text {A}\) in Eq. (72), we take into account that on the right side of this equation the force density \(\mathbf {f}^e(\mathbf {q},t)\) and the electronic momentum flow density tensor \({\varvec{\Pi }}^e(\mathbf {q},t)\) appear. Hereby, due to Eq. (54), the tensor \({\varvec{\Pi }}^e(\mathbf {q},t)\) can be split into a classical part \({\varvec{\Pi }}^{e,\text {cl}}(\mathbf {q},t)\) and a quantum part \({\varvec{\Pi }}^{e,\text {qu}}(\mathbf {q},t)\). As the next step, we realize that the BOA wave function \(\varPsi _{\text {BOA}}(\mathbf {Q}^e,\mathbf {Q}^n,t)\) has on the quantities \(\mathbf {f}^e(\mathbf {q},t)\) and \({\varvec{\Pi }}^{e,\text {qu}}(\mathbf {q},t)\) only an impact via the total particle density \(D(\mathbf {Q}^e,\mathbf {Q}^n,t)\); this can be checked using the Eqs. (47), (52), (53), and (57). So, we can calculate these quantities \(\mathbf {f}^e(\mathbf {q},t)\) and \({\varvec{\Pi }}^{e,\text {qu}}(\mathbf {q},t)\) within the BOA in good accuracy. However, the BOA wave function \(\varPsi _{\text {BOA}}(\mathbf {Q}^e,\mathbf {Q}^n,t)\) has on the classical tensor \({\varvec{\Pi }}^{e,\text {cl}}(\mathbf {q},t)\) both an impact via the total particle density \(D(\mathbf {Q}^e,\mathbf {Q}^n,t)\) and via the electronic Bohmian velocity \(\mathbf {w}_1^e(\mathbf {Q}^e,\mathbf {Q}^n,t)\), which vanishes untruely in the BOA. As a consequence of Eq. (55), this tensor \({\varvec{\Pi }}^{e,\text {cl}}(\mathbf {q},t)\) vanishes untruely in the BOA, too. However, for situations where the term \(-\nabla {\varvec{\Pi }}^{e,\text {cl}}(\mathbf {q},t)\) is negligible, anyway, compared to the term \([\mathbf {f}^{e}(\mathbf {q},t) - \nabla {\varvec{\Pi }}^{e,\text {qu}}(\mathbf {q},t)]\), the right side of the Ehrenfest equation of motion (72) can be calculated within the BOA in good approximation. Thus, for molecular vibrations where electrons and nuclei are at rest at the start time \(t=0\), this condition is fulfilled at least at small times t after the start time. So, at these times, we can calculate the right side of Eq. (72) in the BOA in good accuracy and are able to solve this differential equation numerically using the start condition \(\mathbf {j}_m^e(\mathbf {q},t=0) = \mathbf {0}\) mentioned above. Further research will now be necessary to assess the extent to which this approach can be used to investigate vibrations in molecules.