# Toward Local Madelung Mechanics in Spacetime

Mordecai Waegell  Institute for Quantum Studies, Chapman University, Orange,
CA 92866, USA ORCID: 0000-0002-1292-6041  Corresponding Author, email:
<EMAIL_ADDRESS>
###### Abstract

It has recently been shown that relativistic quantum theory leads to a local
interpretation of quantum mechanics wherein the universal wavefunction in
configuration space is entirely replaced with an ensemble of local fluid
equations in spacetime. For want of a fully relativistic quantum fluid
treatment, we develop a model using the nonrelativistic Madelung equations,
and obtain conditions for them to be local in spacetime. Every particle in the
Madelung fluid is equally real, and has a definite position, momentum, kinetic
energy, and potential energy. These are obtained by defining quantum momentum
and kinetic energy densities for the fluid and separating the momentum into
average and symmetric parts, and kinetic energy into classical kinetic and
quantum potential parts. The two types of momentum naturally give rise to a
single classical kinetic energy density, which contains the expected kinetic
energy, even for stationary states, and we define the reduced quantum
potential as the remaining part of the quantum kinetic energy density. We
treat the quantum potential as a novel mode of internal energy storage within
the fluid particles, which explains most of the nonclassical behavior of the
Madelung fluid. For example, we show that in tunneling phenomena the quantum
potential negates the barrier so that nothing prevents the fluid from flowing
through. We show how energy flows and transforms in this model, and that
enabling local conservation of energy requires defining a quantum potential
energy current that flows through the fluid rather than only flowing with it.
The nonrelativistic treatment generally contains singularities in the velocity
field, which undermines the goal of local dynamics, but we expect a proper
relativistic treatment will bound the fluid particle velocities at $c$.

##  1 Introduction

Since the early days of quantum theory, researchers like Madelung [1] have
recognized that the Schrödinger dynamics can be recast as equations of fluid
dynamics, where the conservation of probability current is mapped to
conservation of fluid current. This leads naturally to a many-worlds
interpretation of Born rule probabilities as proportions of fluid current [2],
where individual experiences of collapse are mapped to individual particles
within the fluid (e.g., if we observe a quantum particle to reflect off a beam
splitter with twice the probability that it is transmitted, then twice as many
of its constituent fluid particles are reflected as are transmitted).

For a single quantum particle (say, an electron), this dynamics occurs in the
familiar three spatial dimensions and one time dimension, but for $N$
entangled particles, the standard nonrelativistic treatment occurs in a
configuration space with $3N$ spatial dimensions, although there is still only
one time dimension. The fluid analogy still holds in this space, but now a
single ‘particle’ in the fluid describes all $N$ particles in 3-space at once.
For a fluid in 3-space, we can consider the kinetic and potential energies of
the individual particles in the fluid, and corresponding energy densities for
the bulk fluid itself, but these classical quantities are not well-defined for
the ‘particles’ in $3N$-dimensional space, which significantly undermines the
analogy.

However, the local interpretation of relativistic quantum physics from [2]
restores all of the dynamics to 3-space, which allows the full classical
analogy to be recovered. In this treatment a single quantum particle typically
comprises several different fluids in spacetime, with indexes to indicate
which distinct fluid each particle belongs to. The different fluids of a given
quantum system only interact with one another when there is a local coupling
to another quantum system. The indexed fluids of a single quantum system allow
for the treatment of entanglement and internal degrees of freedom like spin,
with all of the dynamics occurring in 3-space where the particle energies and
fluid energy densities are all well-defined. These are essentially classical
fluids from a mathematical standpoint, but they behave quite differently from
any standard classical liquid or gas because of the quantum potential.

The purpose of this research program is to interpret the quantum potential as
a standard classical energy in 3-space and ideally to identify the local
interaction rules for the particles in the fluid. We will begin with the
nonrelativistic Schrödinger equation and the corresponding Madelung equations,
with the ultimate aim of carrying the physical intuition we develop here over
to the relativistic case.

##  2 Model Overview

We begin by defining a classical momentum density $p_{q}(\vec{x},t)$ using the
integrand of the expectation value of the quantum momentum operator of single-
particle Madelung fluid in 3-space, $\psi^{*}\hat{p}\psi$, with the
wavefunction expressed in eikonal form, $\psi(\vec{x},t)\equiv
R(\vec{x},t)e^{iS(\vec{x},t)/\hbar}$. The real part of this density is the
fluid current, with particle velocity $\vec{\nabla}S/m$, and the imaginary
part is associated with additional momentum that averages to zero over all of
the fluid particles at a given event in spacetime, with particle velocity
$\Big{|}\frac{\hbar}{m}\frac{\vec{\nabla}R}{R}\Big{|}\hat{r}_{i}$, and unit
vector $\hat{r}_{i}$. This gives us a distribution of definite particle
velocities at each event, such that $\sum_{i}\hat{r}_{i}=0$, so the imaginary
part contributes nothing to the net current of the fluid.

Next, we define a classical energy density using the integrand of the
expectation value of the quantum kinetic energy operator,
$\psi^{*}\hat{K}\psi$, and separate it into real positive terms that
correspond to the classical kinetic energy density associated with the
particle velocities found above, and other real terms that can become
negative, which we identify as the (reduced) quantum potential energy density.
There are also several imaginary terms we ignore, which integrate to zero.
When the Schrödinger equation density is expanded out, those imaginary terms
belong to the continuity equation, while the real terms relate to the
evolution of the fluid current. We thus find that the ‘quantum kinetic energy
density’ is really a sum of a classical kinetic and quantum potential energy
densities, which suggests that the quantum potential has a kinetic character,
even though it can become negative.

There are two ways to group the terms that appear in the quantum kinetic
energy density $k_{q}$, and each has its own conceptual advantages. The
standard way [1, 3] is to define a kinetic energy density
$k_{a}=\frac{1}{2m}R^{2}|\vec{\nabla}S|^{2}$ corresponding to the net fluid
flow, and the quantum potential energy density as
$q=-\frac{\hbar^{2}}{2m}R\nabla^{2}R$, such that $k_{q}=k_{a}+q$. However, the
quantum potential density can be broken down into a kinetic energy density
$k_{s}=\frac{\hbar^{2}}{2m}|\vec{\nabla}R|^{2}$ corresponding to the motion
that averages to zero over the velocity distribution, and a reduced quantum
potential $q_{r}=-\frac{\hbar^{2}}{4m}\nabla^{2}R^{2}$, such that
$q=k_{s}+q_{r}$, and $k_{q}=k_{a}+k_{s}+q_{r}$. Averaging over the velocity
distribution, all of the cross terms cancel so that the classical kinetic
energy density associated with the total velocity
$\frac{1}{m}\big{(}\vec{\nabla}S+\hbar\frac{\vec{\nabla}R}{R}\hat{r}_{i}\Big{)}$
of each fluid particle can be written as $k_{c}=k_{a}+k_{s}$, so the second
way we can group terms is $k_{q}=k_{c}+q_{r}$. This is probably the more
physically correct grouping, since $k_{a}=0$ for stationary states, while
$k_{s}$ (and thus $k_{c}$) integrates to the expectation value of kinetic
energy. Furthermore, there is experimental evidence that there is motion in
stationary states [4], because the laboratory-frame half-lives of muons in
stationary states of muonic atoms are time-dilated in a way that appears to
relate to the speed $\Big{|}\frac{\hbar}{m}\frac{\vec{\nabla}R}{R}\Big{|}$.

However, as we will show, the reduced quantum potential acts as an
intermediary between the two kinetic energy terms, so relative to the average
kinetic energy $k_{a}$, it is reasonable to think of both the symmetric
kinetic energy $k_{s}$ and reduced quantum potential $q_{r}$ as a single
entity, the usual quantum potential $q$, and we would use the standard
grouping. To see how the energy flows between these types, we construct the
continuity equations for each of the above energy densities, and match their
source/sink terms, which give us
$k_{s}{\color[rgb]{0,.5,.5}\leftrightarrow}q_{r}{\color[rgb]{.75,0,.25}\leftrightarrow}k_{a}{\color[rgb]{0,0,1}\leftrightarrow}u$,
where $u=R^{2}U$ is the external potential energy density.

This calculation also reveals that energy can only be locally conserved if the
(reduced) quantum potential energy can flow between particles in the fluid, in
addition to flowing with them. We compute the necessary energy current for
local energy conservation in the case that the reduced quantum potential
$q_{r}$ flows through the fluid. We also compute it for the case that the
entire standard quantum potential $q$ flows through the fluid, which is
consistent with a model in which all of $q$ is treated as an internal
potential energy, but inconsistent with our model, since the symmetric kinetic
energy $k_{s}$ should flow with the fluid, not through it.

We will show that any discontinuity in $u$ is exactly canceled by an opposite
discontinuity in $q_{r}$, so the total potential energy density $q_{r}+u$ is a
continuous function, so the two types of potential energy seem to naturally
lock together. It is also interesting to note that if we also include $k_{s}$
as part of the potential energy relative to the flow energy $k_{a}$, the
potential energy density $q+u$ is (mostly) smooth, and appears to coincide
more naturally with the fluid density, and physical intuition about how it
flows. We will demonstrate these features in our tunneling example, and show
how the quantum potential negates the barrier, providing a simple explanation
of how the otherwise-classical fluid flows through.

Finally, we have made two somewhat arbitrary assumptions in constructing this
model, either of which might be subject to experimental falsification. The
first is that there is motion related to speed
$\Big{|}\frac{\hbar}{m}\frac{\vec{\nabla}R}{R}\Big{|}$ even in stationary
states. The past experiments with muonic atoms were of low fidelity, but a new
set of muon experiments in which the shape of stationary wavefunctions is
carefully controlled could allow us to measure whether the average half-life
of muons in the fluid is really consistent with this velocity distribution. If
this first assumption seems to be borne out, this would also lend support for
the idea that only the reduced quantum potential is a true internal energy,
which is our second assumption. Either way, these experiments would probe an
important and under-explored interface between quantum mechanics and
relativity, which could help us to reconcile stationary quantum states with
time dilation, even if this fluid model is falsified. Beyond this, We believe
a new generation of muon experiments would be invaluable for shedding light on
the interplay between quantum mechanics and relativity, both special and
general [5].

##  3 The Formalism

###  3.1 The quantum momentum density

We define the quantum momentum density $\vec{p}_{q}(\vec{x},t)$ as
$\psi^{*}(\vec{x},t)\hat{p}\psi(\vec{x},t)$, in the sense the integral of this
quantity over space gives the expectation value of the momentum. Taking the
eikonal form of the wavefunction, $\psi=Re^{iS/\hbar}$,

$\vec{p}_{q}(\vec{x},t)=\psi^{*}(\vec{x},t)\hat{p}\psi(\vec{x},t)=-i\hbar
R(\vec{x},t)e^{-iS(\vec{x},t)/\hbar}\vec{\nabla}R(\vec{x},t)e^{iS(\vec{x},t)/\hbar}$
(1) $=-i\hbar
Re^{-iS/\hbar}\big{(}\vec{\nabla}R+\frac{i}{\hbar}R\vec{\nabla}S\big{)}e^{iS/\hbar}$
$=R\big{(}R\vec{\nabla}S-i\hbar\vec{\nabla}R\big{)}.$

We define the average momentum density as the real quantum momentum density,

$\vec{p}_{a}(\vec{x},t)\equiv R^{2}\vec{\nabla}S=m\rho\vec{v}_{a}=m\vec{j},$
(2)

where the density $\rho\equiv R^{2}$ is understood as describing a locally
conserved fluid, and the average local velocity of the fluid-particles is
$\vec{v}_{a}(\vec{x},t)=\vec{j}(\vec{x},t)/\rho(\vec{x},t)=\vec{\nabla}S(\vec{x},t)/m$,
where $\vec{j}$ is the probability (fluid) current.

As a brief aside and cautionary note, the spatial integral over this density
gives the expectation value of the momentum, which is consistent with the
ensemble average after measuring the particle momentum many times, where each
result appears to collapse to a random momentum eigenstate with Born rule
probability. However, unlike a position measurement, where the ensemble
average probability to find the particle near $\vec{x}$ matches the density
$\rho(\vec{x})$, the momentum eigenstates are spread over space. To find the
probability of each outcome one must decompose the entire fluid state into
this set of delocalized states, which has little to do with the density
$\vec{p}_{q}(\vec{x})$ at each location $\vec{x}$, so there is no joint
probability distribution for which the outcomes of both position and momentum
measurements are marginals (Wigner functions are as close as one can get). In
general, it is the measurement apparatus which determines the eigenbasis into
which the fluid must be decomposed, and with the exception of position
measurements, these eigenstates are spread out in space. However, the local
density $\vec{p}_{q}(\vec{x})$ is of physical importance, because it correctly
describes where the momentum can be found, regardless of what basis is used to
measure it. For example, if the fluid is coherently separated into multiple
regions, and local momentum modes are measured in those separate regions, the
expectation value in each region is still the integral of
$\vec{p}_{q}(\vec{x})$ over that region.

Now, the expected classical kinetic energy of a fluid particle with velocity
$\vec{v}_{a}(\vec{x},t)$ at $\vec{x}$ and $t$ is then
$K_{a}(\vec{x},t)=\frac{1}{2}m\vec{v}_{a}(\vec{x},t)\cdot\vec{v}_{a}(\vec{x},t)$,
so the expected classical kinetic energy density associated with the average
local velocity of the fluid is,

$k_{a}(\vec{x},t)\equiv\rho(\vec{x},t)K_{a}(\vec{x},t)=\frac{1}{2m}R^{2}(\vec{\nabla}S\cdot\vec{\nabla}S).$
(3)

We are making progress, but there still seems to be a discrepancy in this
analysis having to do with stationary states. The classical analog of the
energy eigenstates describe particles in motion, and experiments with muonic
atoms show that this motion slows the proper time of the muons relative to the
lab frame, but stationary states have $\vec{v}_{a}=0$, so this motion seems to
be missing.

To resolve this issue, we define the imaginary part of the quantum momentum
density as the symmetric momentum density,

$\vec{p}_{s}(\vec{x},t)\equiv-\hbar
R(\vec{x},t)\vec{\nabla}R(\vec{x},t)=m\rho\vec{v}_{s}(\vec{x},t),$ (4)

so the symmetric velocity is

$\vec{v}_{s}(\vec{x},t)=-\frac{\hbar}{m}\frac{\vec{\nabla}R(\vec{x},t)}{R(\vec{x},t)}.$
(5)

The kinetic energy and kinetic energy density associated with this velocity
are then,

$\begin{array}[]{ccc}K_{s}=\frac{1}{2}m\vec{v}_{s}\cdot\vec{v}_{s}=\frac{\hbar^{2}}{2m}\frac{\vec{\nabla}R\cdot\vec{\nabla}R}{R^{2}},&&k_{s}=\rho
K_{s}=\frac{\hbar^{2}}{2m}\vec{\nabla}R\cdot\vec{\nabla}R.\end{array}$ (6)

This energy is not zero for stationary states, and for simple cases like the
infinite square well, all of the energy is of this type, so it seems we are on
the right track to identify the motion in this state. However, this energy is
not associated with a change in the fluid density $R^{2}$. To explain this we
argue that, unlike a classical continuum fluid, the quantum fluid does not
have a single velocity at each event in spacetime $(\vec{x},t)$, but rather a
distribution of velocities,

$\vec{v}_{i}(\vec{x},t)=\vec{v}_{a}(\vec{x},t)+|\vec{v}_{s}(\vec{x},t)|\hat{r}_{i}(\vec{x},t),$
(7)

with unit vectors $\hat{r}_{i}(\vec{x},t)$ such that,

$\sum_{i}\hat{r}_{i}(\vec{x},t)=0$ (8)

(thus symmetric). We then have $\vec{v}_{a}=(\vec{v}_{i})_{\textrm{ave}}$
because all of $|\vec{v}_{s}(\vec{x},t)|\hat{r}_{i}(\vec{x},t)$ components
cancel and contribute nothing to the net flux of fluid particles at event
$(\vec{x},t)$, and $\vec{v}_{a}(\vec{x},t)$ really is the average of the local
velocity distribution, which gives the (net) local fluid current
$\vec{j}(\vec{x},t)=\rho(\vec{x},t)\vec{v}_{a}(\vec{x},t).$ In this picture,
the fluid particles are essentially of zero volume, and do not exert direct
forces on one another. All of their interactions are collective and occur
through the quantum potential, which we will demonstrate in later sections.
Effectively, all of the fluid particles contribute to creating a collective
quantum potential surface, and in turn, each of the fluid particles moves in
that potential.

With this velocity, the classical kinetic energy of a particle in the fluid is
$K_{i}(\vec{x},t)=\frac{1}{2}m\vec{v}_{i}(\vec{x},t)\cdot\vec{v}_{i}(\vec{x},t)$,
and the average classical kinetic energy at event $(\vec{x},t)$ is
$K_{c}=(K_{i})_{\textrm{ave}}=K_{a}+K_{s}$ (the cross-terms cancel), and thus
the classical kinetic energy density is,

$k_{c}(\vec{x},t)=\rho(\vec{x},t)K_{c}(\vec{x},t)=k_{a}+k_{s},$ (9)

We have inferred this classical kinetic energy, and the two terms it
comprises, from the quantum momentum density, and some considerations about
motion in stationary states. Next we want to see if this kinetic energy
appears in the quantum kinetic energy density.

###  3.2 The quantum kinetic energy, classical kinetic energy, and quantum
potential energy densities

We define the quantum kinetic energy density $k_{q}(\vec{x},t)$ in standard
quantum theory as the real part of
$\psi^{*}(\vec{x},t)\hat{K}\psi(\vec{x},t)$, in the sense the integral of this
quantity over space gives the expectation value of the kinetic energy.

$\psi^{*}(\vec{x},t)\hat{K}\psi(\vec{x},t)=-\frac{\hbar^{2}}{2m}R(\vec{x},t)e^{-iS(\vec{x},t)/\hbar}\nabla^{2}R(\vec{x},t)e^{iS(\vec{x},t)/\hbar}$
(10)
$=-\frac{\hbar^{2}}{2m}Re^{-iS/\hbar}\vec{\nabla}\cdot\big{(}\vec{\nabla}R+\frac{i}{\hbar}R\vec{\nabla}S\big{)}e^{iS/\hbar}$
$=-\frac{\hbar^{2}}{2m}Re^{-iS/\hbar}\bigg{[}\big{(}\nabla^{2}R+\frac{i}{\hbar}\vec{\nabla}R\cdot\vec{\nabla}S+\frac{i}{\hbar}R\nabla^{2}S\big{)}e^{iS/\hbar}+\frac{i}{\hbar}\big{(}\vec{\nabla}R+\frac{i}{\hbar}R\vec{\nabla}S\big{)}\cdot(\vec{\nabla}S)e^{iS/\hbar}\bigg{]}$
$=-\frac{\hbar^{2}}{2m}R\bigg{(}\nabla^{2}R-\frac{1}{\hbar^{2}}R(\vec{\nabla}S\cdot\vec{\nabla}S)+\frac{2i}{\hbar}\vec{\nabla}R\cdot\vec{\nabla}S+\frac{i}{\hbar}R\nabla^{2}S\bigg{)}.$

The imaginary parts always integrate to zero, so while they are related to the
continuity equation, dropping them here gives a real quantum kinetic energy
density,

$k_{q}(\vec{x},t)\equiv-\frac{\hbar^{2}}{2m}R\nabla^{2}R+\frac{1}{2m}R^{2}(\vec{\nabla}S\cdot\vec{\nabla}S).$
(11)

The apparent trouble with this expression is that it can be negative, which
makes it difficult to interpret as a kinetic energy density. The resolution is
that it is not, in fact, only a kinetic energy, but rather sum of a true
classical kinetic energy density, which is always positive, and the quantum
potential energy density, which may be negative.

We can identify the second term of the quantum kinetic energy density above as
the classical kinetic energy density $k_{a}$ we were expecting, which is
always positive, so we will define the first as the quantum potential energy
density. From Bohm/Madelung’s definition of the quantum potential energy of a
single classical particle at $\vec{x}$,

$Q(\vec{x},t)\equiv-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R}{R}=\frac{\hbar^{2}}{2m}\frac{\vec{\nabla}R\cdot\vec{\nabla}R}{R^{2}}-\frac{\hbar^{2}}{4m}\frac{\nabla^{2}(R^{2})}{R^{2}},$
(12)

we see that this really is the density of quantum potential energy in the
fluid of particles,

$q(\vec{x},r)\equiv\rho(\vec{x},t)Q(\vec{x},t)=-\frac{\hbar^{2}}{2m}R\nabla^{2}R.$
(13)
$=\frac{\hbar^{2}}{2m}\vec{\nabla}R\cdot\vec{\nabla}R-\frac{\hbar^{2}}{4m}\nabla^{2}(R^{2}).$

Note that in the expanded form of the quantum potential, we can now identify
the first term as our other classical kinetic energy density $k_{s}$, so if we
collect $k_{s}$ and $k_{a}$ together into a single classical kinetic energy
density $k_{c}$, we are left with a different term that we call the reduced
quantum potential energy density,

$q_{r}\equiv q-k_{s}=-\frac{\hbar^{2}}{4m}\nabla^{2}(R^{2}),$ (14)

corresponding to reduced quantum potential energy per fluid particle,

$Q_{r}=\frac{q_{r}}{R^{2}}=-\frac{\hbar^{2}}{4m}\frac{\nabla^{2}(R^{2})}{R^{2}}.$
(15)

Thus we have broken the quantum kinetic energy density down into a well-
behaved positive kinetic energy density and a new reduced quantum potential
which represents a new type of internal energy carried by particles in the
fluid,

$k_{q}=k_{a}+q=k_{a}+k_{s}+q_{r}=k_{c}+q_{r}.$ (16)

Defining the external potential energy density as
$u(\vec{x},t)\equiv\rho(\vec{x},t){U}(\vec{x},t)$, where $\hat{U}(\vec{x},t)$
is the external potential in which the particle (fluid) moves (assumed to be
non-differential for this article), the total energy density is then,

$e=k_{a}+q+u=k_{c}+q_{r}+u,$ (17)

and the integral over this density is the constant energy expectation value.

##  4 Local Conservation of Fluid Density and Energy Density

###  4.1 Time Evolution

Now that we have this new type of potential energy in play, it will be helpful
to consider how it moves within the fluid, and how energy is transferred from
one type to another. To begin, we return to the Schrödinger equation, and
left-multiply by $\psi^{*}$ to get the evolution equations for the densities,

$i\hbar\psi^{*}\frac{\partial\psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\psi^{*}\hat{K}\psi+\psi^{*}\hat{U}\psi$ (18)
$=-\frac{\partial S}{\partial t}R^{2}+i\hbar R\frac{\partial R}{\partial t}$
$=-\frac{\hbar^{2}}{2m}R\nabla^{2}R+\frac{1}{2m}R(\vec{\nabla}S\cdot\vec{\nabla}S)+R^{2}U-\frac{i\hbar}{2m}\big{(}2\vec{\nabla}R\cdot\vec{\nabla}S+R\nabla^{2}S\big{)}$

As discussed, the imaginary part gives the continuity equation for the fluid
density, which can be seen from,

$\frac{\partial\rho}{\partial
t}=-\vec{\nabla}\cdot\vec{j}=-\frac{1}{m}\vec{\nabla}\cdot(\rho\vec{\nabla}S)=-\frac{1}{m}\big{(}\vec{\nabla}\rho\cdot\vec{\nabla}S+\rho\nabla^{2}S\big{)}$
(19)
$=-\frac{1}{m}\big{(}2R\vec{\nabla}R\cdot\vec{\nabla}S+R^{2}\nabla^{2}S\big{)}=2R\frac{\partial
R}{\partial t},$

from which it follows that,

$\frac{\partial R}{\partial
t}=-\frac{1}{2m}\big{(}2\vec{\nabla}R\cdot\vec{\nabla}S+R\nabla^{2}S\big{)}=-\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR},$
(20)

so we have the evolution equation for $R$, and the amount of fluid is a
locally conserved quantity.

The real part of Eq. 18 then gives us the evolution equation for $S$,

$\frac{\partial S}{\partial
t}=-\Big{(}\frac{1}{2m}(\vec{\nabla}S\cdot\vec{\nabla}S)-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R}{R}+U\Big{)},$
(21)

from which we can get the evolution for $\vec{\nabla}S$, which will be useful
when we consider the time evolution of the energy densities,

$\frac{\partial\vec{\nabla}S}{\partial
t}=-\vec{\nabla}\bigg{(}\frac{1}{2m}(\vec{\nabla}S\cdot\vec{\nabla}S)-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R}{R}+U\bigg{)}.$
(22)

###  4.2 Energy Density Continuity Equations

Now, the continuity equations for the kinetic energy density, quantum
potential energy density, and external potential energy density are,
respectively,

$\begin{array}[]{ccccc}\frac{\partial k_{a}}{\partial
t}+\vec{\nabla}\cdot\vec{j}_{k}=\kappa,&&\frac{\partial q}{\partial
t}+\vec{\nabla}\cdot\vec{j}_{q}=\alpha,&&\frac{\partial u}{\partial
t}+\vec{\nabla}\cdot\vec{j}_{u}=\beta.\end{array}$ (23)

where $\kappa$, $\alpha$, and $\beta$ represents the source/sink terms, and
$\vec{j}_{k}\equiv k_{a}\frac{\vec{\nabla}S}{m}$ , $\vec{j}_{u}\equiv
u\frac{\vec{\nabla}S}{m}$, and $\vec{j}_{q}\equiv
q\Big{(}\frac{\vec{\nabla}S}{m}+\vec{v}\Big{)}$ are the energy current
densities, where the kinetic and potential energy are external properties of
the particles in the fluid, and thus their densities flow with the average
fluid velocity $\frac{\vec{\nabla}S}{m}$. The (internal) quantum potential
energy density flows with the fluid, but can also flow through the fluid (from
particle to particle) with an additional current $q\vec{v}$, which we have not
yet defined.

Note that we have begun with the case that all of the quantum potential energy
$q$ is internal, and can flow between particles in the fluid, which is
inconsistent with our model, but this still lays the mathematical groundwork
for the case of our model, where only the reduced quantum potential energy
$q_{r}$ is internal and flows between fluid particles with relative current
$q_{r}\vec{v}_{r}$.

We will show that in either case, the requirement that energy is locally
conserved fixes this additional current up to an additional divergence-free
term.

So starting with the $q\vec{v}$ case, the three source/sink terms are,

$\kappa=\frac{1}{m}R^{2}\vec{\nabla}S\cdot\frac{\partial\vec{\nabla}S}{\partial
t}+\frac{1}{m^{2}}R^{2}|\vec{\nabla}S|\vec{\nabla}S\cdot\vec{\nabla}|\vec{\nabla}S|$
(24)
$={\color[rgb]{.75,0,.25}\frac{\hbar^{2}}{2m}R^{2}\vec{\nabla}S\cdot\vec{\nabla}\Big{(}\frac{\nabla^{2}R}{mR}\Big{)}}-{\color[rgb]{0,0,1}\frac{1}{m}R^{2}\vec{\nabla}S\cdot\vec{\nabla}U},$
$\alpha=-\frac{\hbar^{2}}{2m}\bigg{(}R\nabla^{2}\frac{\partial R}{\partial
t}-\frac{\partial R}{\partial
t}\nabla^{2}R+{\color[rgb]{.75,0,.25}R^{2}\vec{\nabla}S\cdot\vec{\nabla}\Big{(}\frac{\nabla^{2}R}{mR}\Big{)}}+\vec{\nabla}\cdot\big{(}\vec{v}R\nabla^{2}R\big{)}\bigg{)},$
(25)

and

$\beta=R^{2}\frac{\partial U}{\partial
t}+{\color[rgb]{0,0,1}\frac{1}{m}R^{2}\vec{\nabla}S\cdot\vec{\nabla}U},$ (26)

where we have used the continuity equation $\frac{\partial R^{2}}{\partial
t}+\vec{\nabla}\cdot\Big{(}\frac{R^{2}\vec{\nabla}S}{m}\Big{)}=0$ to eliminate
some terms in these expressions, and substituted in Eq. 22. The colors in the
three equations above show the matched terms that represent local energy
exchanges between the kinetic energy and the potential energies.

###  4.3 Local Energy Conservation

If we add all of the source/sink terms up, we see that the matched terms
cancel as expected, but a few terms remain,

$\kappa+\alpha+\beta=-\frac{\hbar^{2}}{2m}\bigg{(}R\nabla^{2}\frac{\partial
R}{\partial t}-\frac{\partial R}{\partial
t}\nabla^{2}R+\vec{\nabla}\cdot\big{(}\vec{v}R\nabla^{2}R\big{)}\bigg{)}+R^{2}\frac{\partial
U}{\partial t}.$ (27)

If energy is to be locally conserved when $\frac{\partial U}{\partial t}=0$,
then the term in parentheses must be zero. Substituting in Eq. 20 we have,

$0=-R\nabla^{2}\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR}\bigg{)}+\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR}\bigg{)}\nabla^{2}R+\vec{\nabla}\cdot\big{(}\vec{v}R\nabla^{2}R\big{)}$
(28)
$=\vec{\nabla}\cdot\bigg{(}-R\vec{\nabla}\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR}\bigg{)}+\vec{\nabla}R\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR}\bigg{)}\bigg{)}+\vec{\nabla}\cdot\big{(}\vec{v}R\nabla^{2}R\big{)},$
$0=\vec{\nabla}\cdot\bigg{(}-R^{2}\vec{\nabla}\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR^{2}}\bigg{)}+\vec{v}R\nabla^{2}R\bigg{)}.$

Thus we have the additional current,

$q\vec{v}=\frac{\hbar^{2}}{2m}R^{2}\vec{\nabla}\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR^{2}}\bigg{)}+\vec{F},$
(29)

as the condition for local energy conservation in the quantum fluid, where
$\vec{F}$ is any field satisfying $\vec{\nabla}\cdot\vec{F}=0$. If we assume a
current that satisfies this condition then source/sink term for the quantum
potential energy density simplifies to,

${\color[rgb]{.75,0,.25}\alpha_{\textrm{lec}}}={\color[rgb]{.75,0,.25}-\frac{\hbar^{2}}{2m}R^{2}\vec{\nabla}S\cdot\vec{\nabla}\Big{(}\frac{\nabla^{2}R}{mR}\Big{)}}$
(30)

Now we move on to the case where only $q_{r}$ in an internal energy that can
flow between particles in the fluid, while $k_{s}$ is now treated as external
energy that only flows with the fluid. We can use the calculation we’ve
already done as a shortcut for finding the additional current
$q_{r}\vec{v}_{r}$ need for local energy conservation. The continuity
equations for $k_{s}$ and $q_{r}$ are,

$\begin{array}[]{ccc}\frac{\partial k_{s}}{\partial
t}+\vec{\nabla}\cdot\big{(}k_{s}\frac{\vec{\nabla}S}{m}\big{)}=\alpha_{s},&&\frac{\partial
q_{r}}{\partial
t}+\vec{\nabla}\cdot\Big{(}q_{r}\big{(}\frac{\vec{\nabla}S}{m}+\vec{v}\big{)}\Big{)}=\alpha_{r},\end{array}$
(31)

where $\alpha_{s}$ and $\alpha_{r}$ are the source/sink terms, and $\vec{v}$
is again treated as unknown. Now, noting that
$\alpha_{0}\equiv\alpha_{s}+\alpha_{r}=\alpha-\vec{\nabla}\cdot\big{(}k_{s}\vec{v}\big{)}$,
where this is the old $\alpha$ from Eq. 27, the new sum of sinks and sources
is,

$\kappa+\alpha_{0}+\beta=\kappa+\alpha+\beta-\vec{\nabla}\cdot\big{(}k_{s}\vec{v}\big{)}$
(32) $=-\frac{\hbar^{2}}{2m}\bigg{(}R\nabla^{2}\frac{\partial R}{\partial
t}-\frac{\partial R}{\partial
t}\nabla^{2}R+\vec{\nabla}\cdot\big{(}\vec{v}(R\nabla^{2}R+\vec{\nabla}R\cdot\vec{\nabla}R)\big{)}\bigg{)}+R^{2}\frac{\partial
U}{\partial t}.$

Following the same reasoning as before, to locally conserve energy we must now
have,

$0=\vec{\nabla}\cdot\bigg{(}-R^{2}\vec{\nabla}\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{2mR^{2}}\bigg{)}+\vec{v}\big{(}\vec{\nabla}\cdot(R\vec{\nabla}R)\big{)}\bigg{)},$
(33)

and noting
$\nabla^{2}(R^{2})=2\big{(}\vec{\nabla}\cdot(R\vec{\nabla}R)\big{)}$, the new
solutions are,

$q_{r}\vec{v}_{r}=\frac{\hbar^{2}}{2m}R^{2}\vec{\nabla}\bigg{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{mR^{2}}\bigg{)}+\vec{F},$
(34)

for any field $\vec{F}$ such that $\vec{\nabla}\cdot\vec{F}=0$.

###  4.4 Energy Exchanges

From $\kappa$, $\alpha$, and $\beta$ we can see the power density for local
transfers between the kinetic, quantum potential, and external potential
energies.

The term,

$p_{u}=R^{2}\frac{\partial U}{\partial t},$ (35)

represents the power delivered to the fluid by a time-dependent external
potential. Next, we can identify the term,

${\color[rgb]{0,0,1}p_{k_{a}u}}=-{\color[rgb]{0,0,1}R^{2}\vec{\nabla}S\cdot\vec{\nabla}U},$
(36)

which is the power density being converted from external potential to kinetic
energy. Lastly we have,

${\color[rgb]{.75,0,.25}p_{k_{a}q}}={\color[rgb]{.75,0,.25}\frac{\hbar^{2}}{2m^{2}}R^{2}\vec{\nabla}S\cdot\vec{\nabla}\Big{(}\frac{\nabla^{2}R}{R}\Big{)}},$
(37)

which is the power density being converted from quantum potential to kinetic
energy.

Now, to separate the quantum potential into the symmetric kinetic energy and
reduced quantum potential, and identify how energy flows between them, and how
it flows out to the average kinetic energy. Assuming the local energy
conservation condition is satisfied, we have to
$\alpha_{r}=\alpha|_{\vec{v}=\vec{v}_{r}}-{\color[rgb]{0,.5,.5}\alpha_{s}}-\vec{\nabla}\cdot(k_{s}\vec{v}_{r})$,
so we can see that $\alpha_{r}$ has the terms from $\alpha$ plus a direct
energy exchange ${\color[rgb]{0,.5,.5}\alpha_{s}}$ with the symmetric kinetic
energy, where the power density delivered from the reduced quantum potential
density to the symmetric kinetic energy density is,

${\color[rgb]{0,.5,.5}p_{q_{r}k_{s}}}={\color[rgb]{0,.5,.5}\alpha_{s}}={\color[rgb]{0,.5,.5}\frac{\hbar^{2}}{2m^{2}}\bigg{(}\vec{\nabla}\cdot\Big{(}\vec{\nabla}S(\vec{\nabla}R\cdot\vec{\nabla}R)\Big{)}-\vec{\nabla}R\cdot\vec{\nabla}\Big{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{R}\Big{)}\bigg{)}}.$
(38)

This gives us

$\alpha_{r,\textrm{lec}}=-{\color[rgb]{.75,0,.25}\frac{\hbar^{2}}{2m}R^{2}\vec{\nabla}S\cdot\vec{\nabla}\Big{(}\frac{\nabla^{2}R}{mR}\Big{)}}$
(39)
$-{\color[rgb]{0,.5,.5}\frac{\hbar^{2}}{2m^{2}}\bigg{(}\vec{\nabla}\cdot\Big{(}\vec{\nabla}S(\vec{\nabla}R\cdot\vec{\nabla}R)\Big{)}-\vec{\nabla}R\cdot\vec{\nabla}\Big{(}\frac{\vec{\nabla}\cdot(R^{2}\vec{\nabla}S)}{R}\Big{)}\bigg{)}},$

and thus the average kinetic energy is coupled to the reduced quantum
potential, but not the symmetric kinetic energy, and the power density
delivered from the reduced quantum potential to the average kinetic energy is,

${\color[rgb]{.75,0,.25}p_{k_{a}q_{r}}}={\color[rgb]{.75,0,.25}p_{k_{a}q}}={\color[rgb]{.75,0,.25}\frac{\hbar^{2}}{2m^{2}}R^{2}\vec{\nabla}S\cdot\vec{\nabla}\Big{(}\frac{\nabla^{2}R}{R}\Big{)}},$
(40)

The full pattern of local energy exchanges is then,

$k_{s}{\color[rgb]{0,.5,.5}\longleftrightarrow}q_{r}{\color[rgb]{.75,0,.25}\longleftrightarrow}k_{a}{\color[rgb]{0,0,1}\longleftrightarrow}u,$
(41)

and for our two groupings it reduces to,

$\begin{array}[]{ccc}q{\color[rgb]{.75,0,.25}\longleftrightarrow}k_{a}{\color[rgb]{0,0,1}\longleftrightarrow}u,&\textrm{
or,
}&q_{r}{\color[rgb]{0,0,0}\longleftrightarrow}k_{c}{\color[rgb]{0,0,1}\longleftrightarrow}u,\end{array}$
(42)

where $\longleftrightarrow$ includes both
${\color[rgb]{0,.5,.5}\longleftrightarrow}$ and
${\color[rgb]{.75,0,.25}\longleftrightarrow}$.

##  5 What does the quantum potential do?

Once we understand the quantum potential as a new type of local interaction
within the fluid, we can start to look at how this interaction causes the
fluid to behave in ways that a classical liquid or gas classical fluid never
would. Two important examples are how it explains phenomena like tunneling,
where fluid is found in a classically forbidden region, and its role in
quantum interference within the fluid.

The quantum potential has a tendency to cancel out jumps in the external
potential, so for the fluid, it is as though the jumps aren’t even there, or
not entirely there. To see this, consider the 1D Schrödinger Equation with a
constant external potential, $U(x,t)=U_{0}\Theta(x)$, where the $\Theta(x)$ is
the Heaviside step function. The boundary conditions at the finite
discontinuity are $\psi_{l}(0,t)=\psi_{r}(0,t)$, from which it follows that
$R_{l}(0,t)=R_{r}(0,t)$ and $e^{-iS_{l}(0,t)/\hbar}=e^{-iS_{r}(0,t)/\hbar}$,
and $\vec{\nabla}\psi_{l}(0,t)=\vec{\nabla}\psi_{r}(0,t)$. We begin from the
Schrödinger Equations for the left ($l$) and right ($r$) sides of the
discontinuity,

$\begin{array}[]{c}\psi_{l}^{*}\sum_{n}E_{n}\psi_{n,l}e^{-iE_{n}t/\hbar}=\psi_{l}^{*}\hat{K}\psi_{l}+U_{l}R_{l}^{2},\\\
\\\
\psi_{r}^{*}\sum_{n}E_{n}\psi_{n,r}e^{-iE_{n}t/\hbar}=\psi_{r}^{*}\hat{K}\psi_{r}+U_{r}R_{r}^{2},\end{array}$
(43)

where $\psi(x,t)=\sum_{n}\psi_{n}(x)e^{-iE_{n}t/\hbar}$, and $\psi_{n}(x)$ are
the energy eigenstates. Next we consider the difference of these two
equations, making use of the fact that $U_{l}=0$ and $U_{r}=U_{0}$, and
substituting in the quantum kinetic energy density,

$\sum_{n}E_{n}e^{-iE_{n}t/\hbar}\big{(}\psi_{r}^{*}\psi_{n,r}-\psi_{l}^{*}\psi_{n,l}\big{)}=q_{r,r}-q_{r,l}+k_{a,r}-k_{a,l}+k_{s,r}-k_{s,l}+U_{0}R_{r}^{2}.$
(44)

The boundary conditions apply to both the state and the eigenstates, so we can
see that the term in the parentheses in Eq. 44 is zero at the boundary.
Furthermore, recalling that $k_{a}=\frac{1}{2m}R^{2}|\vec{\nabla}S|^{2}$ and
$k_{s}=\frac{\hbar^{2}}{2m}|\vec{\nabla}R|^{2}$, and using the identities,

$\vec{\nabla}R=\frac{1}{2}\Big{(}e^{-iS/\hbar}\vec{\nabla}\psi+e^{iS/\hbar}\vec{\nabla}\psi^{*}\Big{)}=\textrm{Re}\Big{(}e^{-iS/\hbar}\vec{\nabla}\psi\Big{)},$
(45)

and,

$\vec{\nabla}S=\frac{1}{2iR}\Big{(}e^{-iS/\hbar}\vec{\nabla}\psi-e^{iS/\hbar}\vec{\nabla}\psi^{*}\Big{)}=\frac{1}{R}\textrm{Im}\Big{(}e^{-iS/\hbar}\vec{\nabla}\psi\Big{)},$
(46)

the boundary conditions give us $k_{a,r}=k_{a,l}$ and $k_{s,r}=k_{s,l}$, so
Eq. 44 reduces to,

$\begin{array}[]{ccc}q_{r,r}-q_{r,l}=-U_{0}R^{2},&\textrm{ or,
}&Q_{r,r}-Q_{r,l}=-U_{0}.\end{array}$ (47)

Thus, as claimed, the discontinuity in the reduced quantum potential is equal
and opposite to the discontinuity in the external potential, and so the total
potential energy density $q_{r}+u$ is a continuous function, and by extension,
so is $q+u$.

This effect is particularly relevant for tunneling through a rectangular
barrier, and a detailed example showing the roles of the different types of
energy during such tunneling is given in the Appendix. While truly
discontinuous potentials are probably nonphysical, they are still an
interesting limiting case. Perhaps most importantly, we have also proved that
if the external potential is continuous, then so is the reduced quantum
potential.

The Appendix also contains a detailed example of reflection off an infinite
barrier.

##  6 Discussion

So far we have ignored a conceptual issue that may be problematic for this
model, which is that the velocities and energies of the particles generally
contain singularities, as does the additional current $q_{r}\vec{v}_{r}$ of
the reduced quantum potential energy also. Despite this, all of the energy
densities we have considered, and all of the currents associated with velocity
$\vec{\nabla}S/m$, are well-behaved finite functions, which has allowed for
our analysis. Furthermore, while the fluid current, average kinetic energy
density, and quantum potential energy density are all zero where the fluid
density is zero, the symmetric velocity is singular, so the symmetric current,
momentum, and kinetic energy densities, and the reduced quantum potential
energy density can all be finite where the fluid density has an isolated zero
(although not where it is smoothly zero). The fact that there can be current,
momentum, and energy where there are no fluid particles seems like a
conceptual problem, but since this can only happen at isolated zeroes in the
density, the overall amount of current, momentum, and energy at these
locations is of measure zero, maybe it is nothing to worry about.

In general, the infinite velocities undermine the premise that this model is
local, but we think this is likely to be an artifact of the nonrelativistic
treatment we have used. Our hope is that in a proper relativistic treatment,
these singularities will vanish and the particle velocity in the fluid will be
bounded by $c$ (this seems to work out in [6]). Restricting $\vec{v}_{s}$ in
this way could also make it so that all current, momentum, and energy
densities are zero where the fluid density is zero.

In conclusion, while the present model has many satisfying features, it also
has some apparent problems that need to be addressed. However, there is good
reason to think that a relativistic picture will overcome these issues, since
we are looking for a fluid model consistent with the local Heisenberg picture
used in relativistic quantum field theory. We also suspect that in the
generalized treatment, the fluid particles will be of definite energy rather
than definite mass, which will mix up several of the fluid properties we have
considered here, and also allow for the treatment of massless particles. In
general, energy is a locally conserved quantity in (special) relativistic
theories, so it is quite natural to think of it as a fluid anyway.

In the local many worlds picture, the conserved fluid of each quantum system
is separated into multiple branches during an interaction between systems,
each corresponding to a different outcome, and the relative amounts of fluid
in each branch give rise to the Born rule probability of observing that
outcome at macroscopic scale. If each fluid particle is taken to be of
definite energy (in a given frame), then this is just another manifestation of
local conservation of energy, whereas in our present treatment it is
effectively conservation of mass, which we should not expect to survive a
generalization to special relativity. The fluid particles should also possess
definite values for any other locally conserved property, whether frame
invariant like charge, or frame-dependent like momentum.

Finally, it is worth noting that many other works have explored different ways
to interpret or modify Madelung’s original fluid equations (or their Bohm
equivalent) [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23,
24, 25, 26, 27, 28, 29, 30, 6, 31, 32, 33, 34, 35], so even if the present
model fails, there are plenty of alternative ideas available.

## References

  * [1] Erwin Madelung “Quantum theory in hydrodynamical form” In _z. Phys_ 40, 1927, pp. 322
  * [2] Mordecai Waegell “Local Quantum Theory with Fluids in Space-Time” In _Quantum Reports_ 5.1 MDPI, 2023, pp. 156–185
  * [3] David Bohm “A suggested interpretation of the quantum theory in terms of “hidden” variables. I” In _Physical review_ 85.2 APS, 1952, pp. 166
  * [4] Mark P Silverman “Relativistic time dilatation of bound muons and the Lorentz invariance of charge” In _American Journal of Physics_ 50.3 American Association of Physics Teachers, 1982, pp. 251–254
  * [5] Iarley P Lobo and Christian Pfeifer “Muon accelerators–Muon lifetime measurements as window to Planck scale physics” In _arXiv preprint arXiv:2306.07210_ , 2023
  * [6] Arquimedes Ruiz-Columbié, Hira Farooq and Luis Grave Peralta “Direct relativistic extension of the Madelung-de-Broglie-Bohm reformulations of quantum mechanics and quantum hydrodynamics” In _Journal of Modern Physics_ 12.10 Scientific Research Publishing, 2021, pp. 1418–1434
  * [7] David Bohm and Jean-Pierre Vigier “Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations” In _Physical Review_ 96.1 APS, 1954, pp. 208
  * [8] M Schönberg “On the hydrodynamical model of the quantum mechanics” In _Il Nuovo Cimento (1943-1954)_ 12 Springer, 1954, pp. 103–133
  * [9] EA Spiegel “Fluid dynamical form of the linear and nonlinear Schrödinger equations” In _Physica D: Nonlinear Phenomena_ 1.2 Elsevier, 1980, pp. 236–240
  * [10] Giovanni Salesi “Spin and Madelung fluid” In _Modern Physics Letters A_ 11.22 World Scientific, 1996, pp. 1815–1823
  * [11] DH Delphenich “The geometric origin of the Madelung potential” In _arXiv preprint gr-qc/0211065_ , 2002
  * [12] Oktay K Pashaev and Jyh-Hao Lee “Resonance solitons as black holes in Madelung fluid” In _Modern Physics Letters A_ 17.24 World Scientific, 2002, pp. 1601–1619
  * [13] M Caliari, G Inverso and LM Morato “Dissipation caused by a vorticity field and generation of singularities in Madelung fluid” In _New Journal of Physics_ 6.1 IOP Publishing, 2004, pp. 69
  * [14] Bill Poirier “Bohmian mechanics without pilot waves” In _Chemical Physics_ 370.1-3 Elsevier, 2010, pp. 4–14
  * [15] Bill Poirier “Reconciling semiclassical and Bohmian mechanics. I. Stationary states” In _The Journal of chemical physics_ 121.10 American Institute of Physics, 2004, pp. 4501–4515
  * [16] TF Nonnenmacher “Hamiltonian models for the Madelung fluid and generalized Langevin equations” In _Stochastic Processes in Classical and Quantum Systems: Proceedings of the 1st Ascona-Como International Conference, Held in Ascona, Ticino (Switzerland), June 24–29, 1985_ , 2005, pp. 470–480 Springer
  * [17] Robert E Wyatt “Quantum dynamics with trajectories: introduction to quantum hydrodynamics” Springer Science & Business Media, 2005
  * [18] Corey Trahan and Bill Poirier “Reconciling semiclassical and Bohmian mechanics. III. Scattering states for continuous potentials” In _The Journal of Chemical Physics_ 124.3 AIP Publishing, 2006, pp. 034116
  * [19] Bill Poirier and Gérard Parlant “Reconciling semiclassical and Bohmian mechanics: IV. Multisurface dynamics” In _The Journal of Physical Chemistry A_ 111.41 ACS Publications, 2007, pp. 10400–10408
  * [20] Laura M Morato “Formation of singularities in Madelung fluid: a nonconventional application of Itô calculus to foundations of quantum mechanics” In _Stochastic Analysis and Applications: The Abel Symposium 2005_ , 2007, pp. 527–540 Springer
  * [21] Bill Poirier “Reconciling semiclassical and Bohmian mechanics. V. Wavepacket dynamics” In _The Journal of chemical physics_ 128.16 AIP Publishing, 2008
  * [22] Bill Poirier “Reconciling semiclassical and Bohmian mechanics. VI. Multidimensional dynamics” In _The Journal of chemical physics_ 129.8 AIP Publishing, 2008
  * [23] Agung Budiyono and Ken Umeno “Madelung fluid model for the most likely wave function of a single free particle in a two-dimensional space with a given average energy” In _Physical Review A_ 79.4 APS, 2009, pp. 042104
  * [24] Bill Poirier and David Tannor “An action principle for complex quantum trajectories” In _Molecular Physics_ 110.9-10 Taylor & Francis, 2012, pp. 897–908
  * [25] Bill Poirier “Trajectory-based theory of relativistic quantum particles” In _arXiv preprint arXiv:1208.6260_ , 2012
  * [26] Eyal Heifetz and Eliahu Cohen “Toward a thermo-hydrodynamic like description of Schrödinger equation via the Madelung formulation and Fisher information” In _Foundations of Physics_ 45 Springer, 2015, pp. 1514–1525
  * [27] Peter Vadasz “Rendering the Navier–Stokes Equations for a Compressible Fluid into the Schrödinger Equation for Quantum Mechanics” In _Fluids_ 1.2 MDPI, 2016, pp. 18
  * [28] Eyal Heifetz, Roumen Tsekov, Eliahu Cohen and Zohar Nussinov “On entropy production in the Madelung fluid and the role of Bohm’s potential in classical diffusion” In _Foundations of Physics_ 46 Springer, 2016, pp. 815–824
  * [29] Maik Reddiger “The Madelung picture as a foundation of geometric quantum theory” In _Foundations of Physics_ 47.10 Springer, 2017, pp. 1317–1367
  * [30] Dennis M Heim “Recursive formulation of Madelung continuity equation leads to propagation equation” In _Journal of Mathematical Physics_ 59.12 AIP Publishing, 2018
  * [31] James P Finley “Refined Madelung Equations” In _arXiv preprint arXiv:2109.12766_ , 2021
  * [32] Luis Grave De Peralta, Luis A Poveda and Bill Poirier “Making relativistic quantum mechanics simple” In _European Journal of Physics_ 42.5 IOP Publishing, 2021, pp. 055404
  * [33] James P Finley “Fields and Equations of Classical Mechanics for Quantum Mechanics” In _arXiv preprint arXiv:2207.04349_ , 2022
  * [34] James P Finley “A fluid description based on the Bernoulli equation of the one-body stationary states of quantum mechanics with real valued wavefunctions” In _Journal of Physics Communications_ 6.4 IOP Publishing, 2022, pp. 045002
  * [35] Maik Reddiger and Bill Poirier “Towards a mathematical theory of the Madelung equations: Takabayasi’s quantization condition, quantum quasi-irrotationality, weak formulations, and the Wallstrom phenomenon” In _Journal of Physics A: Mathematical and Theoretical_ 56.19 IOP Publishing, 2023, pp. 193001
  * [36] Chris Dewdney and Basil J Hiley “A quantum potential description of one-dimensional time-dependent scattering from square barriers and square wells” In _Foundations of Physics_ 12 Springer, 1982, pp. 27–48
  * [37] Eric R Bittner “Quantum tunneling dynamics using hydrodynamic trajectories” In _The Journal of Chemical Physics_ 112.22 American Institute of Physics, 2000, pp. 9703–9710
  * [38] Corey Trahan and Bill Poirier “Reconciling semiclassical and Bohmian mechanics. II. Scattering states for discontinuous potentials” In _The Journal of chemical physics_ 124.3 AIP Publishing, 2006

##  7 Appendix: Tunneling and Interference Examples

The simplest tunneling example is an infinite square well of width $2L$, and
with a finite rectangular barrier of width $2a$ at its center (for other
examples and simulations, see [36, 37, 38]). If the two lowest energy
eigenvalues, $E_{1}$ and $E_{2}$, are below the barrier height $U_{0}$, then
the eigenstates are of the form,

$\psi_{1}(x)=\left\\{\begin{array}[]{lll}A\sin(k_{1}(x+L))&&-L\leq x\leq-a\\\
B\cosh(\kappa_{1}x)&&-a\leq x\leq a\\\ A\sin(k_{1}(L-x))&&a\leq x\leq
L\end{array}\right.,$

and

$\psi_{2}(x)=\left\\{\begin{array}[]{lll}C\sin(k_{2}(x+L))&&-L\leq x\leq-a\\\
D\sinh(\kappa_{2}x)&&-a\leq x\leq a\\\ -C\sin(k_{2}(L-x))&&a\leq x\leq
L\end{array}\right.,$

where $A,B,C,$ and $D$ are fixed by the boundary conditions and normalization,
and $k_{n}=\frac{\sqrt{2mE_{n}}}{\hbar}$, and
$\kappa_{n}=\frac{\sqrt{2m(U_{0}-E_{n})}}{\hbar}$.

If we prepare an initial state
$\psi(x,0)=\frac{1}{\sqrt{2}}\big{(}\psi_{1}(x)+\psi_{2}(x)\big{)}$, most of
the fluid is on one side of the barrier, and as time evolves it tunnels until
most of it is on the other side, and then back.

The example in Fig. 1 shows the energy eigenvalues,
$E_{1}=3.22\hbar^{2}/mL^{2}$ and $E_{2}=5.02\hbar^{2}/mL^{2}$ (found
numerically), for the shown double-well potential with barrier height
$U_{0}=15\hbar^{2}/mL^{2}$, the fluid density, the four energy density terms
$q_{r}$, $k_{a}$, $k_{s}$, and $u$, and the two alternative groupings, all in
the middle of this tunneling process. In both groupings, not only has the
(reduced) quantum potential canceled out the barrier, but it has also created
a depression beneath it, which is compensated by a bump in the kinetic energy
of the fluid within the barrier - so we have a nearly classical explanation of
tunneling. The animation in the Supplementary Material shows the energy
densities for the entire cycle, with streamline markers that move at velocity
$\vec{v}_{a}$ along the fluid streamlines. A $t$ vs. $x$ plot of the
streamlines is given in Fig. 2.

|
---|---
|
Figure 1: (top,left) A double-well potential $U$ with infinite walls and a finite rectangular barrier in the center, the two lowest energy eigenvalues, $E_{1}$ and $E_{2}$, and eigenstates, $\psi_{1}(x)$ and $\psi_{2}(x)$, and the fluid density $R^{2}$ for the state $\psi(x,T/4)=(\psi_{1}(x)e^{-iE_{1}T/4\hbar}+\psi_{2}(x)e^{-iE_{2}T/4\hbar})/\sqrt{2}$, with period $T=2\pi\hbar/(E_{2}-E_{1})$. (bottom,left) The reduced quantum potential, $q_{r}=-(\hbar^{2}/4m)\nabla^{2}R^{2}$, average kinetic, $k_{a}=(1/2m)|\vec{\nabla}S|^{2}$, symmetric kinetic, $k_{s}=(\hbar^{2}/2m)|\vec{\nabla}R|^{2}$, and external potential, $u=R^{2}U$, energy densities, for the state $\psi(x,T/4)$. Note the equal and opposite discontinuities in $q_{r}$ and $u$. (top,right) The standard grouping, with quantum potential energy density $q=q_{r}+k_{s}$ added to $u$ to form a single potential energy density, and the average kinetic energy density $k_{a}$ corresponding to the flow. This picture seems to track $R^{2}$ and the current $R^{2}\vec{v}_{a}$ most intuitively, and shows increased kinetic energy where the fluid falls into a dip in the potential energy as it tunnels through the (cancelled) barrier. (bottom,right) The new grouping where the reduced quantum potential $q_{r}$ is added to $u$ to form a single potential energy density, and the average and symmetric kinetic energy densities are combined to form a single classical kinetic energy density, $k_{c}=k_{a}+k_{s}$. This may be the more physically correct picture, where there is motion and kinetic energy within stationary states.  Figure 2: Fluid streamlines for one full period $T=2\pi\hbar/(E_{2}-E_{1})$ of the state $\psi(x,t)=(\psi_{1}(x)e^{-iE_{1}t/\hbar}+\psi_{2}(x)e^{-iE_{2}t/\hbar})/\sqrt{2}$. The fluid density is proportional to the horizontal density of the streamlines. The momentary zeros that form due to the interference of the two eigenstates are also shown. An animation of this motion is given in the Supplementary Material, which shows particles that track the streamlines, the average kinetic energy density, $k_{a}$, and the potential energy density, $q+u$. In the animation, it is apparent that the kinetic energy bump where the fluid falls into a dip in the potential energy at one zero in the fluid density, grows and moves through the barrier, and then shrinks and disappears at the zero that forms on the opposite side. The kinetic bump and potential dip feature seems to appear for dark interference fringes involving multiple energies, which makes their apparent connection to tunneling interesting. |
---|---
|
Figure 3: (top,left) The fluid density $R^{2}$ at time
$t_{1}=0.1907[mL^{2}/\hbar]$ for the Gaussian pulse prepared at $t=0$ as
$\psi(x,0)=Ae^{-x^{2}/(4\Delta x^{2})+ip_{0}x/\hbar}$, with $\Delta
x=L/(4\sqrt{10})$ and $p_{0}=25\hbar/L$, and normalization constant $A$, as it
reflects off an infinite barrier at $x=L/2$. This moment has been chosen just
before the dark interference fringes form, when the densities in the troughs
are still slightly above zero. (bottom,left) The reduced quantum potential,
$q_{r}=-(\hbar^{2}/4m)\nabla^{2}R^{2}$, average kinetic,
$k_{a}=(1/2m)|\vec{\nabla}S|^{2}$, symmetric kinetic,
$k_{s}=(\hbar^{2}/2m)|\vec{\nabla}R|^{2}$, and external potential, $u=R^{2}U$,
energy densities, for the state $\psi(x,t_{1})$. Note the equal and opposite
spikes in $k_{a}$ and $k_{s}$. (top,right) The standard grouping, with quantum
potential energy density $q=q_{r}+k_{s}$ added to $u$ to form a single
potential energy density, and the average kinetic energy density $k_{a}$
corresponding to the flow. This picture seems to track $R^{2}$ and the current
$R^{2}\vec{v}_{a}$ most intuitively, and shows increased kinetic energy where
the fluid falls into a dip in the potential energy as it approaches a dark
interference fringe ($R^{2}=0$). Interestingly, that extra average kinetic
energy appears to truly come out of the symmetric kinetic energy in this case.
(bottom,right) The new grouping where the reduced quantum potential $q_{r}$ is
added to $u$ to form a single potential energy density, and the average and
symmetric kinetic energy densities are combined to form a single classical
kinetic energy density, $k_{c}=k_{a}+k_{s}$. This may be the more physically
correct picture, where there is motion and kinetic energy within stationary
states. It is interesting that in this case, the spikes cancel and we get a
smoothly oscillating $k_{c}$ and $q_{r}$.  Figure 4: Fluid streamlines for a
Gaussian pulse prepared at $t=0$ as $\psi(x,0)=Ae^{-x^{2}/(4\Delta
x^{2})+ip_{0}x/\hbar}$, with $\Delta x=L/(4\sqrt{10})$ and $p_{0}=25\hbar/L$,
and normalization constant $A$, as it reflects off an infinite barrier at
$x=L/2$. The fluid density is proportional to the horizontal density of the
streamlines. Interference between the incoming and reflected portions of the
pulse causes zeros in the fluid density to momentarily form half-way through
the reflection process. An animation of this motion is given in the
Supplementary Material, which shows particles that track the streamlines, the
average kinetic energy density, $k_{a}$, and the potential energy density,
$q+u$. It is interesting to note that as the zero in the fluid density forms,
relatively few fluid particles approach it, and those that do, fall into the
potential dip and are sped across. However, particles that have not made it
past by the time the zero forms are reflected back.

Another important example is the role of the quantum potential in quantum
interference phenomena. Here we consider a 1D Gaussian pulse that is
reflecting off an infinite barrier, creating interference between the incoming
and reflected parts of the pulse. As the destructive fringes form and $R^{2}$
approaches zero, the average kinetic energy density $k_{a}$ forms a spike
above each local minimum of $R^{2}$, and the symmetric kinetic energy density
$k_{s}$ forms a dip at these same locations. These spikes and dips appear as
the destructive interference fringes (zeros) in the fluid density begin to
form, vanish only at the moment the fringes appear, and then reappear as the
fringes vanish.

The example in Fig. 3 shows the fluid density, the four energy density terms
$q_{r}$, $k_{a}$, $k_{s}$, and $u$, and the two alternative groupings, just
before the dark fringes form, as the Gaussian pulse,

$\psi(x,0)=A\exp{\bigg{(}-\frac{x^{2}}{4\Delta
x^{2}}+i\frac{p_{0}x}{\hbar}\bigg{)}},$ (48)

reflects off an infinite barrier at $x=0.5L$ (with $A$ a normalization
constant). At $t=0$ the pulse is centered at the origin, with mean momentum
$p_{0}=25\hbar/L$, and spread $\Delta x=L/(4\sqrt{10})$, and these plots are
at $t=0.01907(mL^{2}/\hbar)$, when the spikes and dips can be seen. The middle
of the reflection occurs at $t=0.0200(mL^{2}/\hbar)$, and the dark fringes
form zeros in the fluid density, which can be seen in Fig. 4 along with fluid
streamlines during the reflection.

However, if we consider the classical kinetic energy density
$k_{c}=k_{a}+k_{s}$, the spike and dip exactly cancel, leaving a smoothly
varying oscillation through the entire pulse, so the two types of kinetic
energy also seem to lock naturally together, similarly to $q$ and $v$ in the
tunneling case. In either grouping, the minima of the fluid density correspond
to depressions in the (reduced) quantum potential which are compensated by
bumps in the kinetic energy.

The animation in the Supplementary Material shows the energy densities for the
reflection process, with streamline markers that move at velocity
$\vec{v}_{a}$.

Looking at both animations, it is interesting to note that the dip beneath the
barrier in the tunneling case actually begins at a zero in the fluid density
outside the barrier and then grows and moves to the center, so there appears
to be some connection between dark interference fringes (zeros) and the
presence of fluid in classically forbidden regions. In both cases, regions
where fluid velocities exceed our classical expectations have corresponding
dips in the quantum potential.