# Madelung Mechanics and Superoscillations

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

In single-particle Madelung mechanics, the single-particle quantum state
$\Psi(\vec{x},t)=R(\vec{x},t)e^{iS(\vec{x},t)/\hbar}$ is interpreted as
comprising an entire conserved fluid of classical point particles, with local
density $R(\vec{x},t)^{2}$ and local momentum $\vec{\nabla}S(\vec{x},t)$
(where $R$ and $S$ are real). The Schrödinger equation gives rise to the
continuity equation for the fluid, and the Hamilton-Jacobi equation for
particles of the fluid, which includes a new density-dependent quantum
potential energy term
$Q(\vec{x},t)=-\frac{\hbar^{2}}{2m}\frac{\vec{\nabla}R(\vec{x},t)}{R(\vec{x},t)}$,
which is all that makes the fluid behavior nonclassical. In particular, the
quantum potential can become negative and create a nonclassical boost in the
kinetic energy. This boost is related to superoscillations in the
wavefunction, where the local frequency of $\Psi$ exceeds its global band
limit. Berry showed that for states of definite energy $E$, the regions of
superoscillation are exactly the regions where $Q(\vec{x},t)<0$. For energy
superposition states with band-limit $E_{+}$, the situation is slightly more
complicated, and the bound is no longer $Q(\vec{x},t)<0$. However, the fluid
model provides a definite local energy for each fluid particle which allows us
to define a local band limit for superoscillation, and with this definition,
all regions of superoscillation are again regions where $Q(\vec{x},t)<0$ for
general superpositions. Detailed examples are given which illustrate the role
of the quantum potential and superoscillations in a range of scenarios.

##  1 Introduction

Superoscillation [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11] is a wave phenomenon
which occurs in both classical and quantum contexts wherein the local
oscillation frequency or wave vector of a wavefunction exceeds the global band
limit of that function (e.g. a wave superposition composed entirely of
frequencies of smaller magnitude than $f_{0}$ may, in some small region,
oscillate faster than $f_{0}$). This phenomenon is known to occur around
vortices where successive crests of the wave get squeezed together, and such
vortices are known to occur around nodes in the wavefunction, and also where
the wave passes through or around obstructions.

For this article, we will focus on wavefunctions $\Psi(\vec{x},t)$ satisfying
the single-particle Schrödinger equation, and their Madelung interpretation
[12, 13, 14, 15, 16, 17, 18, 19, 20], which describes a conserved flow of a
fluid in spacetime, starting from
$\Psi(\vec{x},t)=R(\vec{x},t)e^{iS(\vec{x},t)/\hbar}$, where the fluid density
is $\rho\equiv R^{2}$ and the fluid momentum is
$\vec{\nabla}S(\vec{x},t)-q_{0}\vec{A}(\vec{x},t)$, with $q_{0}$ the charge of
the particle, and $\vec{A}$ the vector potential. For simplicity, we begin
with the case $\vec{A}=0$. The fluid picture is nearly classical, if one
considers a new type of internal potential energy possessed by the elemental
particles it comprises, called the (reduced) quantum potential energy. It is
reasonable to say that all of the apparently nonclassical behavior of this
fluid is due to the role of the quantum potential energy, and if this is
interpreted as a new type of classical potential energy, then the behavior of
the fluid is entirely classical. There has been some recent renewed interest
in understanding the role and applications of the quantum potential [21, 22,
23, 24, 25, 26, 27, 20, 28]. Importantly, there is also a relatively new local
quantum formalism [29] which replaces the configuration space wavefunction
with a set of single-particle wavefunctions in spacetime, so even entangled
many-particle systems can be understood using this fluid picture.

Michael Berry recently identified a connection between superoscillations and
the quantum potential [10, 30], which has already attracted some interest
[31], and it is the main purpose of this article to explore and generalize
these results. Berry showed that for energy eigenstates with a single fixed
value of $E$, the quantum potential $Q$ is negative in exactly the regions
where the wavefunction is superoscillatory, in the sense that the local
momentum of the fluid elements exceeds a corresponding ‘expected classical
momentum,’ given the energy $E$ and external potential $U$. This also means
that curves where $Q=0$ bound regions of superoscillation in the wavefunction.

While this result follows for waves of fixed energy $E$ (even superpositions
with the momentum in different directions), it is no longer valid for general
energy superpositions, where the quantum momentum becomes a complicated
function of the different energies — a detail which Berry seems to have
overlooked in [10].

However, we can still consider the boundary $Q=0$ for general superpositions,
and we will see that this fits with a generalization of the concept of the
‘expected classical energy’ which incorporates all of the energies in the
superposition. We will therefore define a new class of superoscillations that
are always bounded by $Q=0$.

For the analysis in this article, we work directly in terms of the energy of
the fluid, rather than focusing on the momenta of elemental particles in the
fluid, since the boundaries for (semiclassical [30]) superoscillation can be
expressed just as easily in terms of energies as in terms of momenta. In
general, a lowercase variable represents a density and the corresponding
capital variable the quantity belonging to an individual elemental particle in
the fluid, and these always differ by a factor of $R^{2}$ (e.g.,
$k_{a}=R^{2}K_{a}$ for average kinetic energy density). The densities have the
virtue of being well-behaved finite functions.

This article is organized as follows. In Sec. 2 we review Berry’s results and
examine the case of real-valued energy eigenstates which solve the time-
independent Schrödinger equation, and the case of complex degenerate
superpositions of such states. In Sec. 3 we examine the case of general energy
superpositions, and introduce our generalized concepts of the local particle
energy, the local classical kinetic energy bound, and the local bound for
superoscillations. In Sec. 4 we revisit and expand upon the notion of a
reduced quantum potential and a symmetric kinetic energy [20], which is a
different way to interpret the terms in the Hamilton-Jacobi equation, which
leads to different concepts of the local classical kinetic energy bound, and
the local bound for superoscillations. In Sec. 5 we go an extensive set of
examples designed to build physical intuition about the fluid model and role
of superoscillations, with diagrams showing various energy and energy density
curves for real eigenstates and degenerate superpositions, and showing
momentum streamlines for moving energy superposition for cases with both
tunneling and interference. The best visualizations for building intuition
about this picture are the animations in the Supplementary Materials, with one
of tunneling in a simple superpositions state of a double-well potential, and
two of tunneling and reflection as a Gaussian packet is incident on a finite
barrier tuned as a 50/50 beam splitter. We conclude in Sec. 6 with some
summarizing comments and outlook for future research.

##  2 Superoscillations in Energy Eigenstates

Recasting Berry’s original analysis for a fixed energy $E$, the time-
independent Schrödinger equation and a fixed potential $U(\vec{x})$, leads to
the relation,

$K_{a}+Q=E-U,$ (1)

where $K_{a}=K_{a}=\frac{1}{2m}|\vec{\nabla}S(\vec{x})|^{2}$ is the (average)
kinetic energy,
$Q\equiv-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}R(\vec{x})}{R(\vec{x})}$ is the
quantum potential energy, and $K_{\textrm{cl}}=E-U(\vec{x})$ is the ‘expected
classical kinetic energy.’ The expected classical kinetic energy is negative
in classically forbidden regions ($E<U(\vec{x})$), so while it follows
directly from energy conservation, it is somewhat dubious to call it a
classical kinetic energy.

It is also worth noting that for stationary energy eigenstates of energy $E$,
with $S=-Et$, it follows from $K_{a}=0$ that the total potential
$Q(\vec{x})+U(\vec{x})=E$ is flat. That is, the energy of every elemental
particle in the fluid is $E$, and the quantum and external potential energies
add up to this value everywhere in the fluid. It then follows that the force
on each elemental particle is $-\vec{\nabla}(Q+U)=0$, which gives a classical
explanation for why the fluid remains in equilibrium. Multiplying this
flatness/equilibrium condition for the fluid by $R(\vec{x})$ gives back the
time-independent Schrödinger equation for $R(\vec{x})$,

$Q(\vec{x})R(\vec{x})+U(\vec{x})R(\vec{x})=-\frac{\hbar^{2}}{2m}\nabla^{2}R(\vec{x})+U(\vec{x})R(\vec{x})=ER(\vec{x}),$
(2)

so we can think of the quantization condition as identifying the only $R_{n}$
functions for which $Q+U=E_{n}$ is flat.

Now, it is easy to see that if $K_{a}=E-U$, then $Q=0$, so this is the
boundary of the superoscillatory region ($K_{a}>K_{\textrm{cl}}=E-U$
$\longrightarrow$ $Q<0$) for superpositions of degenerate energy eigenstates,
which necessarily includes all classically forbidden regions
($K_{\textrm{cl}}=E-U<0$), since $K_{a}\geq 0$. Note that $K_{a}$ can be
nonzero for degenerate superpositions of energy $E$, but the momentum
$\vec{\nabla}S$ flows in closed loops such that $R^{2}$, $k_{a}$, and $q$ stay
constant. In such cases, $K_{a}+Q+U=E$, and thus every elemental particle in
the fluid has energy $E$.

In general, degenerate superpositions can contain nodes which produce negative
singularities in the quantum potential, and the elemental fluid particles
orbit in the potential wells surrounding these singularities. Near the
singularities the orbits are approximately circular, and the total force is
approximately central, and thus it follows that
$-\vec{\nabla}(Q+U)=\vec{\nabla}K_{a}=-\frac{Z}{r^{3}}\hat{r}$ and thus
$-Q-U=K_{a}=\frac{Z}{2r^{2}}=\frac{1}{2m}|\vec{\nabla}S|^{2}$, where $Z$ is a
constant and $r$ is the radius of the circular orbit around the singularity in
$Q$ [32]. So we have an inverse square potential well near the singularity,
with orbital speed $|\vec{v}_{a}|=|\vec{\nabla}S|/m=\frac{\sqrt{Z/m}}{r}$. See
the example in Sec. 5.10. The properties of the quantized vortices that form
at nodes in the fluid density, and their relation to the Aharonov-Bohm effect,
have been the studied somewhat extensively [33, 34, 35, 36, 37, 38, 39, 19,
25].

##  3 Superoscillations in Superpositions of Energy Eigenstates

Next we consider the corresponding relation from the time-dependent
Schrödinger equation,

$K_{a}+Q=-\frac{\partial S}{\partial t}-U,$ (3)

which has the form of a Hamilton-Jacobi equation, $-\frac{\partial S}{\partial
t}=H=K_{a}+Q+U$, where the quantum potential energy is explicitly included. It
is easy to see that if $K_{a}=-\frac{\partial S}{\partial t}-U$, then $Q=0$.
For an energy eigenstate, or a superposition of terms with the same energy
$E$, this reduces to Eq. 1, since $S(\vec{x},t)=-Et+f(\vec{x})$ for such
cases. For a general superposition, the classical energy of a particle in the
fluid is,

$E_{p}(\vec{x},t)\equiv-\frac{\partial}{\partial
t}S(\vec{x},t)=K_{a}(\vec{x},t)+Q(\vec{x},t)+U(\vec{x},t),$ (4)

which is no longer uniform over the entire fluid. This suggests that a natural
generalization of the quantity ‘expected classical kinetic energy’ is,

$K_{\textrm{cl}}(\vec{x},t)=E_{p}(\vec{x},t)-U(\vec{x},t)=-\frac{\partial
S(\vec{x},t)}{\partial t}-U(\vec{x},t),$ (5)

which again makes the boundary of superoscillation
$K_{a}>K_{\textrm{cl}}=-\frac{\partial S}{\partial t}-U=K_{a}+Q$, or simply
$Q<0$. We can also identify the expected classical kinetic energy as equal to
the quantum kinetic energy, $K_{\textrm{cl}}=K_{q}=K_{a}+Q$ derived in [20].

It is natural to ask how this relates to the the maximum energy $E_{+}$ in a
band-limited superposition of energy eigenstates. This generalized definition
will identify some regions where the local energy $K_{a}$ does not exceed
$E_{+}-U$ as superoscillatory, which may seem like a problem given the usual
notion of a band limit. However, the fluid particles cannot have energy
$E_{+}$ everywhere, since this would make the integral over the whole fluid
too large, so $E_{+}-U$ is arguably too high a global bound. The local
classical energy of a fluid particle is $E_{p}$, and this integrates to the
expectation value $\langle E\rangle$, so this generalized definition of
superoscillation using $K_{\textrm{cl}}=E_{p}-U$ as the local bound seems
quite reasonable, and it also reduces back to $K_{\textrm{cl}}=E-U$ for energy
eigenstates.

That said, both bounds are still of general interest, so we define regions
which exceed the local band limit but not the global band limit,
$K_{\textrm{cl}}<K_{a}\leq E_{+}-U$, as regions of soft superoscillation (or
simply $-\frac{\partial S}{\partial t}-E_{+}\leq Q<0$), and regions that
exceed the global band limit, $K_{a}>E_{+}-U$, as regions of hard
superoscillation (or $Q<-\frac{\partial S}{\partial t}-E_{+}$).

Note that because $K_{a}\geq 0$, there is always hard superoscillation in
classically forbidden regions, i.e., $K_{a}>E_{+}-U<0$.

Finally, since the local energy of a particle in the fluid is
$E_{p}(\vec{x},t)$ we can reasonably define $K_{\textrm{cl}}(\vec{x},t)<0$ as
the local classically forbidden regions, and there is always soft
superoscillation in these regions, since $K_{a}>K_{\textrm{cl}}<0$.

To summarize, by using $E_{p}(\vec{x},t)$ as the local energy of fluid
particles, and $K_{\textrm{cl}}(\vec{x},t)$ as their expected kinetic energy,
we obtain refined concepts of where the particles are in classically forbidden
regions ($K_{\textrm{cl}}(\vec{x},t)<0$) and where they are superoscillatory
($K_{\textrm{cl}}(\vec{x},t)<K_{a}(\vec{x},t)$ or equivalently,
$Q(\vec{x},t)<0$).

##  4 The Reduced Quantum Potential Energy and the Symmetric Kinetic Energy

Now, there is a different interpretation of the quantities we have been
discussing, introduced in [20], which breaks the quantum potential energy $Q$
apart as $Q=K_{s}+Q_{r}$, with symmetric kinetic energy
$K_{s}=\frac{\hbar^{2}}{2m}\frac{\vec{\nabla}R\cdot\vec{\nabla}R}{R^{2}}$ and,
reduced quantum potential energy
$Q_{r}=-\frac{\hbar^{2}}{4m}\frac{\nabla^{2}(R^{2})}{R^{2}}$. In that model,
the symmetric kinetic energy $K_{s}$ is due to the presence of a local
velocity distribution in the fluid whose symmetric part sums to zero, leaving
only the average flow associated with the average kinetic energy $K_{a}$. As a
result of this symmetry, the total classical kinetic energy density $k_{c}$ of
the fluid breaks apart as $k_{c}=k_{a}+k_{s}$. The primary virtue of this
alternative picture is that real energy eigenstates with $k_{a}=0$ still
contain kinetic energy $k_{s}\neq 0$, as we would expect for the corresponding
classical system, and this motion is more consistent with time-dilation in
muonic atoms [40]. Note that in this picture we work in terms of the
densities, because even at a given event $(\vec{x},t)$, there is a
distribution of individual particle energies $K_{c_{i}}$, which are averaged
over in the local density $k_{c}\equiv R^{2}\langle K_{c_{i}}\rangle_{i}$.
This means that the total quantum kinetic energy density can be expressed in
the original Madelung/Bohm form, or in a new form using the classical kinetic
energy density and the reduced quantum potential energy density,
$k_{q}=k_{a}+q=k_{c}+q_{r}$. The total energy density of elemental particles
in the fluid can then be expressed as
$e_{p}(\vec{x},t)=k_{c}(\vec{x},t)+q_{r}(\vec{x},t)+u(\vec{x})=-R(\vec{x},t)^{2}\frac{\partial}{\partial
t}S(\vec{x},t)$.

The symmetric kinetic energy is suggested by the imaginary part of the quantum
momentum density, and treats the energy content of each eigenstate as due to
motion of the elemental particles, rather than being primarily stored in the
(standard) quantum potential, with the fluid at rest. The symmetric kinetic
energy is also more directly related to the energy eigenvalue, and likewise
for the momentum. For example, for an eigenstate of the 1D infinite square
well
$\psi_{n}(x,t)=R_{n}(x,t)e^{iS_{n}(x,t)/\hbar}=A\sin(k_{n}x)e^{-iE_{n}t/\hbar}$,
the symmetric momentum density is,

$\vec{p}_{s}(x,t)=-\hbar R\vec{\nabla}R(x,t)=-\hbar
k_{n}A^{2}\cos(k_{n}x)\sin(k_{n}x)\hat{x},$ (6)

and the symmetric and average kinetic energies are,

$\begin{array}[]{ccc}k_{s}=\frac{\hbar^{2}}{2m}{\vec{\nabla}R\cdot\vec{\nabla}R}=A^{2}\frac{\hbar^{2}k_{n}^{2}}{2m}\cos^{2}(k_{n}x),&&k_{a}=\frac{1}{2m}R^{2}(\vec{\nabla}S\cdot\vec{\nabla}S)=0,\end{array}$
(7)

with energy eigenvalue $E_{n}=\frac{\hbar^{2}k_{n}^{2}}{2m}$, and $\hbar
k_{n}$ the corresponding classical momentum of the mode. Thus, all of the
energy in this eigenstate resides in the symmetric kinetic energy. The reduced
quantum potential energy density is not zero for energy eigenstates, but it
always integrates to zero, so all net energy comes from the symmetric kinetic
and external potential energy densities.

To see this, note that in 1D, $R(\pm\infty)=0$ because $R^{2}$ is a normalized
density, and thus,

$\langle
Q_{r}\rangle\equiv\int_{-\infty}^{\infty}Q_{r}R^{2}dx=\int_{-\infty}^{\infty}q_{r}dx=-\frac{\hbar^{2}}{4m}\int_{-\infty}^{\infty}\frac{\partial^{2}(R^{2})}{\partial
x^{2}}dx$ (8) $=-\frac{\hbar^{2}}{4m}\bigg{[}\frac{\partial(R^{2})}{\partial
x}\bigg{]}^{\infty}_{-\infty}=-\frac{\hbar^{2}}{4m}\bigg{[}2R\frac{\partial
R}{\partial x}\bigg{]}^{\infty}_{-\infty}=0,$

and this result generalizes trivially to 3D. For real energy eigenstates, we
also have $K_{a}=0$, so

$E=\langle\hat{K}\rangle+\langle\hat{U}\rangle=\langle K_{a}\rangle+\langle
K_{s}\rangle+\langle Q_{r}\rangle+\langle U\rangle=\langle
K_{s}\rangle+\langle U\rangle,$

and thus $\langle\hat{K}\rangle=\langle
K_{s}\rangle\equiv\int_{\textrm{space}}K_{s}R^{2}dV$, so all of the kinetic
energy in the eigenstate is apparently of the symmetric form, as promised.

The symmetric kinetic energy density arises because the elemental fluid
particles have a symmetric velocity distribution
$|\vec{v}_{s}(\vec{x},t)|\hat{r}_{i}(\vec{x},t)$, with
$\sum_{i}\hat{r}_{i}(\vec{x},t)=0$ at each point in spacetime, and this is why
it is unrelated to the average flow $\vec{v}_{a}(\vec{x},t)$. Thus, relative
to the average flow, the symmetric kinetic energy effectively behaves as
another local potential energy, which is why the $(q,k_{a})$ picture is often
more intuitive than the $(q_{r},k_{c})$ picture.

One speculative model that would explain this is that the symmetric velocity
distributions are such that the elemental fluid particles at each location
$\vec{x}$ fall into a circular potential well with infinitesimal radius, where
they orbit at speed $|\vec{v}_{s}(\vec{x},t)|$ with nearly infinite frequency.
This model is inspired by the macroscopic vortex in the 2D square well
discussed in the example of Sec. 5.10, where the average flow velocity
$\vec{v}_{a}(\vec{x},t)$ is associated with the circuitous motion around the
singularity in $Q$. Here we would say that the portion of $Q=Q_{r}+K_{s}$
associated with $Q_{r}$ is a local internal energy storage mechanism, while
the portion associated with $K_{s}$ is comprised of a sea of infinitesimal
vortices where some of the kinetic energy of the particle can be trapped in
the infinitesimal orbits ($K_{s}$), and some can be involved in net flow
($K_{a}$). Finally, we have only used the symmetric speed
$|\vec{v}_{s}(\vec{x},t)|$ in the model so far, but the bare symmetric
velocity $i\vec{v}_{s}(\vec{x},t)$ emerges from the quantum momentum density
with a direction, so we interpret it as a rotational vector, which points
along the axis of rotation, and interpret the $i$ as indicating that the
particle orbits the axis $\hat{v}_{s}(\vec{x},t)$ counterclockwise with speed
$|\vec{v}_{s}(\vec{x},t)|$ at an infinitestimal radius.

We now consider the circumstances of superoscillation in this alternative
interpretation, where $q_{r}$ and $k_{c}$ are the physically significant
quantities. The arguments for the expected classical kinetic energy are
unchanged, so we have the lower bound
$k_{\textrm{cl}}=e_{p}-u=k_{q}=q_{r}+k_{c}$ for soft superoscillation, and the
lower bound $R^{2}E_{+}-u$ for hard superoscillation. However, in this
picture, we must check the total classical kinetic energy density $k_{c}$ for
superoscillation, rather than only the average kinetic energy $k_{a}$ (which
is zero for stationary states anyway), and thus we define regions of soft
superoscillation to be where $k_{c}>k_{\textrm{cl}}=q_{r}+k_{c}$, or
$q_{r}<0$, so $q_{r}=0$ defines the boundary for soft superoscillation in the
$(q_{r},k_{c})$ picture, just as $q$ does in the original $(q,k_{a})$ picture.
The regions of soft superoscillation give way to regions of hard
superoscillation where $k_{c}>R^{2}E_{+}-u$. Hard superoscillation occurs
where $q_{r}<-R^{2}\big{(}\frac{\partial S}{\partial t}+E_{+}\big{)}$.

Note that because $k_{c}\geq 0$, there is always hard superoscillation in
classically forbidden regions, i.e., $k_{c}>R^{2}E_{+}-u<0$, and soft
superoscillation in the local classically forbidden regions regions, since
$k_{c}(\vec{x},t)>k_{\textrm{cl}}<0$, analogous to the $(q,k_{a})$ picture.

Finally, because $k_{c}\geq k_{a}$, the region of superoscillation is
generally larger in the $(q_{r},k_{c})$ picture than in the $(q,k_{a})$
picture, which can be seen throughout Sec. 5.

##  5 Examples

It is very helpful to go through several key examples to build up a physical
intuition for how things work in the fluid picture, and under what
circumstances we will find superoscillations.

###  5.1 Energy Superposition in the Infinite Square Well

One of the simplest cases of a bound state which varies in time is an equal
superposition of the ground state and the first excited state in the 1D
infinite square well. For a well with ends at $x=-L$ and $x=L$, this state is,

$\psi(x,t)=\frac{1}{\sqrt{2L}}\bigg{(}\cos\Big{(}\frac{\pi
x}{2L}\Big{)}e^{-iE_{1}t/\hbar}+\sin\Big{(}\frac{2\pi
x}{2L}\Big{)}e^{-4iE_{1}t/\hbar}\bigg{)},$ (9)

where $E_{1}=\frac{\pi^{2}\hbar^{2}}{8mL^{2}}$ is the ground state energy, and
the first excited energy is $E_{2}=4E_{1}$. These parameters were chosen for
easy comparison with the next example where a finite barrier is added around
$x=0$. This state evolves periodically with angular frequency $3E_{1}/\hbar$,
with nodes appearing on either side once every cycle. The situation is
illustrated in Fig. 1, with streamlines indicating the motion of the fluid and
shading indicating areas of superoscillation.

---
Figure 1: Streamline diagrams for one period of evolution for an equal
superposition of the ground state and first excited state of a 1D infinite
square well of width $2L$. Neighboring streamlines are separated by equal
proportions of fluid, and so they depict the fluid density. The nodes that
appear once per cycle are shown with asterisks. upper: The lighter gray
shading indicates regions of soft superoscillation, $Q<0$ and the darker gray
shading indicates regions of hard superoscillation $K_{a}>4E_{1}$ ($U=0$
inside the well). lower: The reduced quantum potential picture. The darker
gray shading indicates regions of soft superoscillation, $Q_{r}<0$ and the
darker gray shading indicates regions of hard superoscillation $K_{c}>4E_{1}$.

Now, considering the upper diagram in Fig. 1, superoscillation appears to play
a direct role in this motion, and the cyclic motion appears to be driven by a
dip, where the quantum potential becomes negative, that appears to ‘bounce’
back and forth between the two events where the nodes form. The dip ($Q<0$, so
soft superoscillation) creates a local boost in the flow kinetic energy
$K_{a}$ as the fluid is pulled across, which is what really drives the motion
of the fluid, and in some regions this is enough kinetic energy to constitute
hard superoscillation.

Considering the lower diagram in Fig. 1, in the $(Q_{r},K_{c})$ picture, we
see the same role for the reduced quantum potential in driving the flow, but
the wells are significantly deeper in some places, and some of the energy is
stored locally in the form of symmetric kinetic energy $K_{s}$, so the regions
of soft superoscillation where $Q_{r}<0$, or hard superoscillation where
$K_{c}=K_{a}+K_{s}>4E_{1}$ are generally larger than in the standard
$(Q,K_{a})$ picture.

###  5.2 Tunneling Through a Finite Barrier in the Infinite Square Well

---
Figure 2: Streamline diagrams for one period of evolution for an equal
superposition of the ground state and first excited state of a 1D infinite
square well of width $2L$ with a finite barrier of height $U_{0}$ and width
$L/5$ in the center. Neighboring streamlines are separated by equal
proportions of fluid, and so they depict the fluid density. The nodes that
appear once per cycle are shown with asterisks. upper: The lighter gray
shading indicates regions of soft superoscillation, $Q<0$ and the darker gray
shading indicates regions of hard superoscillation $K_{a}>E_{2}^{\prime}-U$.
lower: The reduced quantum potential picture. The darker gray shading
indicates regions of soft superoscillation, $Q_{r}<0$ and the darker gray
shading indicates regions of hard superoscillation $K_{c}>E_{2}^{\prime}-U$.
The fluid tunnels through the barrier within continuous central region of
superoscillation.

We continue with the 1D infinite square well, but we have now added a finite
rectangular barrier of height $U_{0}=15\hbar^{2}/mL^{2}$ and width $L/5$ in
the center. The new energy eigenstates and energy eigenvalues were determined
numerically for this case. We again consider an equal superposition of the
ground state and first excited state, which is again periodic with angular
frequency $(E_{2}^{\prime}-E_{1}^{\prime})/\hbar$. Both energies are
significantly less than $U_{0}$, so any flow across the barrier must be due to
tunneling. The situation is illustrated in Fig. 2, with streamlines indicating
the motion of the fluid and shading indicating areas of superoscillation.

Considering the upper diagram in Fig. 2, we can see that the situation
resembles the previous case without the barrier in that the cyclic motion is
driven by a dip where the quantum potential becomes negative, which bounces
back and forth between the two events where the nodes form. The dip ($Q<0$, so
soft superoscillation) creates a local boost in the flow kinetic energy
$K_{a}$ as the fluid is pulled across, resulting in some regions of hard
superoscillation, including the classically forbidden region within the
barrier. The quantum potential cancels out the external potential barrier, so
the fluid flows through in almost the same way as if the barrier were absent,
except that it is slower.

Considering the lower diagram in Fig. 2, in the $(Q_{r},K_{c})$ picture, we
see the same role for the reduced quantum potential in driving the flow, but
the wells are significantly deeper in some places, and some of the energy is
stored locally in the form of symmetric kinetic energy $K_{s}$, so the regions
of soft superoscillation where $Q_{r}<0$, or hard superoscillation where
$K_{c}=K_{a}+K_{s}>E_{2}^{\prime}$ are larger than in the standard $(Q,K_{a})$
picture.

###  5.3 Energy Eigenstates of the Harmonic Oscillator

---
Figure 3: Energy densities $q$, $k_{a}$, $q_{r}$, $k_{c}$, $k_{s}$, and $v$
for the ground state (upper) and first excited state (lower) of the 1D
harmonic oscillator. The external potential $U$ is superimposed on the plot in
arbitrary units, and the zero of the energy densities is set relative to $U$
at the energy eigenvalue. The darker red shading shows regions where $q<0$
(hard/soft superoscillation in the ($q,k_{a}$) picture), which coincides with
the classically forbidden region, and the lighter red shows regions where
$q_{r}<0$ (in the ($q_{r},k_{c}$) picture). For the ground state
$k_{c}=k_{s}=u$, so these are all plotted as one curve, whereas for the first
excited state $k_{c}=k_{s}\neq u$. Also, because $k_{a}=0$ for energy
eigenstates, superoscillation in the $(q,k_{a})$ picture occurs where
$0=k_{a}>e-u=q$, or simply $q<0$, where the central inequality shows kinetic
energy exceeding the expected classical bound. Likewise, for the
($q_{r},k_{c}$) picture, superoscillation occurs where
$k_{s}=k_{c}>e-u=q_{r}+k_{s}$, or simply $q_{r}<0$.

The energy eigenstates of the 1D harmonic oscillator have fluid in the
classically forbidden regions at the outside edges, which are associated with
penetration depth rather than tunneling. For eigenstates there is no
difference between hard and soft superoscillations, so we depict both the
($Q,K_{a}$) and ($Q_{r},K_{c}$) pictures in the same plots in Fig. 3, using
different shades of red. For these example plots, we used natural frequency
$\omega=10\hbar/mL^{2}$. This example gives a clear idea of how ubiquitous
superoscillations are in the physical world, since eigenstates of typical
physical systems have tails in the classically forbidden region, which always
corresponds to superoscillation. In the $(Q_{r},K_{c})$ picture, there is
superoscillation near all minima of the fluid density $R^{2}$ for all energy
eigenstates, which also includes the classically forbidden regions. Regions of
superoscillation can be seen in the plots where the kinetic energy exceeds the
expected classical kinetic energy, $k_{a}>e-u$ (also $q<0$), or $k_{c}>e-u$
(also $q_{r}<0$), in the respective pictures.

###  5.4 Energy Superposition in the Harmonic Oscillator

We now consider an equal superposition of the ground state and first excited
state of the 1D harmonic oscillator, which is periodic with angular frequency
$(E_{1}-E_{0})/\hbar$, with $E_{0}=\hbar\omega/2$ and $E_{1}=3\hbar\omega/2$.
The situation is illustrated in Fig. 4, with streamlines indicating the motion
of the fluid and shading indicating areas of superoscillation.

---
Figure 4: Streamline diagrams for one period of evolution for an equal
superposition of the ground state and first excited state of a 1D harmonic
oscillator with natural frequency $\omega=10\hbar/mL^{2}$. Neighboring
streamlines are separated by equal proportions of fluid, and so they depict
the fluid density. The nodes that appear once per cycle are shown with
asterisks. upper: The lighter gray shading indicates regions of soft
superoscillation, $Q<0$ and the darker gray shading indicates regions of hard
superoscillation $K_{a}>E_{1}-U$. lower: The reduced quantum potential
picture. The darker gray shading indicates regions of soft superoscillation,
$Q_{r}<0$ and the darker gray shading indicates regions of hard
superoscillation $K_{c}>E_{1}-U$.

Considering the upper diagram in Fig. 4, we can see that the situation
resembles the cases we saw for an energy superposition in the 1D infinite
square well, with or without a finite barrier in the center, in that the
cyclic motion is driven by a dip, where the quantum potential becomes
negative, which bounces back and forth between the two events where the nodes
form. The dip ($Q<0$, so soft superoscillation) creates a local boost in the
flow kinetic energy $K_{a}$ as the fluid is pulled across, resulting in some
regions of hard superoscillation. There is also the ubiquitous
superoscillation in (or near) the classically forbidden region.

Considering the lower diagram in Fig. 4, in the $(Q_{r},K_{c})$ picture, we
see the same role for the reduced quantum potential in driving the flow, but
the wells are significantly deeper in some places, and some of the energy is
stored locally in the form of symmetric kinetic energy $K_{s}$, so the regions
of soft superoscillation where $Q_{r}<0$, or hard superoscillation where
$K_{c}=K_{a}+K_{s}>E_{1}$ are larger than in the standard $(Q,K_{a})$ picture.

###  5.5 Energy Eigenstates of a Quartic Double Well

The energy eigenstates of the 1D quartic double well potential typically have
fluid in the classically forbidden regions, both at the outside edges and
within the central barrier. For eigenstates there is no difference between
hard and soft superoscillations, so we depict both the ($Q,K_{a}$) and
($Q_{r},K_{c}$) pictures in the same plots in Fig. 5, using different shades
of red. For these example plots, we used external potential energy,

$U(x)=\frac{\hbar^{2}}{mL^{2}}\bigg{(}240\frac{x^{4}}{L^{4}}-120\frac{x^{2}}{L^{2}}+15\bigg{)},$
(10)

which has minima $U(\pm 1/2)=0$, and the barrier has peak height
$U_{0}=15\hbar^{2}/mL^{2}$. The energy eigenstates and eigenvalues were found
numerically.

This example is akin to the case we saw for the infinite square well with a
finite barrier, but in a more physically realistic potential. Again, we have
superoscillation $Q<0$ in the classically forbidden regions. In the
$(Q_{r},K_{c})$ picture, there is superoscillation near all minima of the
fluid density $R^{2}$ for all energy eigenstates, which also includes the
classically forbidden regions. Regions of superoscillation can be seen in the
plots where the kinetic energy exceeds the expected classical kinetic energy,
$k_{a}>e-u$ (also $q<0$), or $k_{c}>e-u$ (also $q_{r}<0$), in the respective
pictures.

---
Figure 5: Energy densities $q$, $k_{a}$, $q_{r}$, $k_{c}$, $k_{s}$, and $v$
for the ground state (upper) and first excited state (lower) of a symmetric 1D
quartic double well potential. The external potential $U$ is superimposed on
the plot in arbitrary units, and the zero of the energy densities is set
relative to $U$ at the energy eigenvalue. The darker red shading shows regions
where $q<0$ (hard/soft superoscillation in the ($q,k_{a}$) picture), which
coincides with the classically forbidden regions, and the lighter red shows
regions where $q_{r}<0$ (in the ($q_{r},k_{c}$) picture). For the ground state
$k_{c}=k_{s}=u$, so these are all plotted as one curve, whereas for the first
excited state $k_{c}=k_{s}\neq u$. Also, because $k_{a}=0$ for energy
eigenstates, superoscillation in the $(q,k_{a})$ picture occurs where
$0=k_{a}>e-u=q$, or simply $q<0$, where the central inequality shows kinetic
energy exceeding the expected classical bound. Likewise, for the
($q_{r},k_{c}$) picture, superoscillation occurs where
$k_{s}=k_{c}>e-u=q_{r}+k_{s}$, or simply $q_{r}<0$.

###  5.6 Energy Superpositions in the Quartic Double Well

We now consider an equal superposition of the ground state and first excited
state of the 1D quartic double well potential $U(x)$ from the last section,
which is periodic with angular frequency $(E_{1}-E_{0})/\hbar$, with $E_{0}$
and $E_{1}$ are the numerically-determined energy eigenvalues of the ground
state and first excited state, respectively. Both energies are significantly
less than $U_{0}$, so any flow across the barrier must be due to tunneling.
The situation is illustrated in Fig. 6, with streamlines indicating the motion
of the fluid and shading indicating areas of superoscillation. A full
animation of this time evolution can be found in the supplementary Materials.

---
Figure 6: Streamline diagrams for one period of evolution for an equal
superposition of the ground state and first excited state of a 1D quartic
double well of Eq. 10. Neighboring streamlines are separated by equal
proportions of fluid, and so they depict the fluid density. The nodes that
appear once per cycle are shown with asterisks. upper: The lighter gray
shading indicates regions of soft superoscillation, $Q<0$ and the darker gray
shading indicates regions of hard superoscillation $K_{a}>E_{1}-U$. lower: The
reduced quantum potential picture. The darker gray shading indicates regions
of soft superoscillation, $Q_{r}<0$ and the darker gray shading indicates
regions of hard superoscillation $K_{c}>E_{1}-U$. The fluid tunnels through
the barrier within continuous central region of superoscillation.

Considering the upper diagram in Fig. 6, we see again that the cyclic motion
is driven by a dip where the quantum potential becomes negative, which bounces
back and forth between the two events where the nodes form. The dip ($Q<0$, so
soft superoscillation) creates a local boost in the flow kinetic energy
$K_{a}$ as the fluid is pulled across, resulting in some regions of hard
superoscillation. As in the case of the infinite square well with a
rectangular barrier, the quantum potential cancels out the barrier, and the
flow looks quite similar to the case of the harmonic oscillator with no
barrier. There is also the ubiquitous superoscillation in (or near) the
classically forbidden region.

Considering the lower diagram in Fig. 6, in the $(Q_{r},K_{c})$ picture, we
see the same role for the reduced quantum potential in driving the flow, but
the wells are significantly deeper in some places, and some of the energy is
stored locally in the form of symmetric kinetic energy $K_{s}$, so the regions
of soft superoscillation where $Q_{r}<0$, or hard superoscillation where
$K_{c}=K_{a}+K_{s}>E_{1}$ are larger than in the standard $(Q,K_{a})$ picture.

###  5.7 The Universality of Fluid Motion

As we have seen in the four prior examples, the quantum potential energy
always smooths out the external potential energy, such that the total
potential energy is completely flat (constant) for energy eigenstates,
$Q(\vec{x})+U(\vec{x})=E$ (and $K_{a}=0$), because the quantum potential
exactly cancels all bumps and barriers in the external potential. For energy
superposition states, the total energy of a fluid particle is no longer
constant, and their motion is guided by one or more moving valleys in the
total potential $Q+U$, separating relatively flat hilltops where the fluid
accumulates.

Remarkably, this picture appears to be universal. Regardless of the external
potential, its shape is canceled by the quantum potential, and the moving
hills and valleys in the quantum potential guide all of the kinetic motion of
the fluid (for static external potentials). To see this, consider Fig. 7,
which shows the flow kinetic energy $K_{a}$ and the total potential energy
$Q+U$ of an elemental particle in the fluid, for the four cases we have
considered of an equal superposition of the ground state and first exited
state, in four different binding potentials $U$, at time $t=T/4$, where $T$ is
the period of oscillation for the given state.

Figure 7: The kinetic energy $K_{a}$ and total potential energy $Q+U$ of an
elemental particle in the fluid, for the four cases we have considered of an
equal superposition of the ground state and first exited state, in four
different binding potentials $U$ (upper left: infinite square well, upper
right: infinite square well with a rectangular barrier, lower left: harmonic
oscillator, lower right: quartic double well), at time $t=T/4$, where $T$ is
the period of oscillation of the corresponding state (see the full animation
in the Supplementary Materials). At this moment, in all four cases, the
fluid’s density is symmetric about $x=0$, it is halfway through flowing from
one side of the well to the other, and its net kinetic energy is maximum.
Also, the shape of $K_{a}$ and $Q+U$ looks essentially the same for all four
cases, even though two of them are in the middle of tunneling through a
barrier, and two of them are open wells. This basic shape is retained as the
dip travels between the nodes in the fluid density, but changes as it
approaches or departs a node. The dip grows deeper as it approaches the
locations where a node forms, and a positive bump begins to appear on its
leading edge. For $Q$, the dip extends instantaneously to $-\infty$ where the
singularity forms, and the bump on the leading edge extends instantaneously to
$+\infty$, while for $K_{a}$, the spike extends instantaneously to $+\infty$.
The fluid density is zero at these events, so no elemental fluid particles
have these infinite energies, but the particles very close to the node do have
arbitrarily large finite energies.

###  5.8 Interference During Reflection

---
Figure 8: Streamline diagrams for a nearly-Gaussian band-limited wave pulse
reflecting off an infinite wall and interfering with itself. The pulse also
spreads significantly during this process. Neighboring streamlines are
separated by equal proportions of fluid, and so they depict the fluid density.
The nodes that appear half-way through the reflection are shown with dots, and
one fluid streamline can be seen whipping around the deep pit in $Q$ that
forms there. As in the simpler examples, we can see that the fluid motion is
guided by a series of moving hills and valleys in the quantum potential which
produce the separation into distinct interference fringes. upper: The lighter
gray shading indicates regions of soft superoscillation, $Q<0$ and the darker
gray shading indicates regions of hard superoscillation $K_{a}>E_{1}-U$.
lower: The reduced quantum potential picture. The darker gray shading
indicates regions of soft superoscillation, $Q_{r}<0$ and the darker gray
shading indicates regions of hard superoscillation $K_{c}>E_{1}-U$.

We now move on to consider a nearly-Gaussian band-limited pulse as it reflects
off an infinite wall. We begin at $t=0$ with a Gaussian of momentum
$25\hbar/L$ centered at $x=0$ within an infinite square well. This initial
wavefunction is expanded into the eigenbasis $\psi_{n}(x)$ of an infinite
square well of width $L$. The coefficients $c_{n}$ in this expansion become
negligible for $n>18$, so we obtain our band limited wave pulse by only using
the first eighteen terms in the expansion, and renormalizing. Fig. 8 shows the
fluid streamlines as this pulse reflects off the wall of the well and
interferes with itself. The nodes are labeled, and shading shows regions of
hard and soft superoscillation.

###  5.9 A 1D Mach-Zehnder Interferometer

---
Figure 9: Streamline diagrams for a nearly-Gaussian band-limited wave pulse
dividing at beam splitter, reflecting off an infinite wall and recombining at
the beam splitter due to interference, effecting a 1D Mach Zehnder
Interferometer. Neighboring streamlines are separated by equal proportions of
fluid, and so they depict the fluid density. As in the simpler examples, we
can see that the fluid motion is guided by a series of moving hills and
valleys in the quantum potential which produce the separation into distinct
interference fringes. upper: The lighter gray shading indicates regions of
soft superoscillation, $Q<0$ and the darker gray shading indicates regions of
hard superoscillation $K_{a}>E_{1}-U$. lower: The reduced quantum potential
picture. The darker gray shading indicates regions of soft superoscillation,
$Q_{r}<0$ and the darker gray shading indicates regions of hard
superoscillation $K_{c}>E_{1}-U$.

So far, the examples we have considered have been simple cases at relatively
low energy, which help to reveal the underlying mechanisms at work in the
fluid interpretation, but it is also illustrative to consider a familiar
higher-energy scenario, which is a 1D Mach-Zehnder interferometer (MZI). This
is an infinite square well of width $4L$ with a thin barrier of width $L/50$
and height $U_{0}$ in the center at $x=0$. We again construct a Gaussian pulse
with an initial positive momentum $60\hbar/L$, which begins centered at
$x=-L$, and expand this into the energy eigenbasis of the infinite square
well. Then, noting that the coefficients $c_{n}$ become negligible for $n>89$,
we get a band-limited pulse by using only the first 89 terms in the expansion,
and renormalizing. The barrier height $U_{0}$ was then tuned so that exactly
half of the band-limited pulse is reflected and half is transmitted, resulting
in a 50/50 beam splitter, although one which, due to the barrier width,
slightly distorts the wave packet. Fig. 9 shows the packet as it is divided,
reflected, and then recombined (almost perfectly), and thus we effectively
have a MZI in a 1D cavity, with shading showing the regions of soft/hard
superoscillation.

Two zoomed-in animations of a band limited Gaussian pulse incident on
different finite barriers tuned as a 50/50 beam splitter are given in the
Supplementary Materials. The parameters are chosen differently than in the 1D
MZI shown here in order to better illustrate the details of tunneling and
interference in this model.

###  5.10 Degenerate Superposition in the 2D Infinite Square Well

Figure 10: The momentum density and streamlines of the degenerate superposition state of Eq. 11 in the 2D infinite square well, with side lengths $L$. The momentum streamlines start separated by equal proportions of fluid along the vertical axis of symmetry, and then closed loops are generated by following the momentum flow from those starting points. The arrows show the local direction and magnitude of the momentum density. In the $(Q,K_{a})$ picture, we have a stable vortex with the fluid orbiting with kinetic energy $K_{a}$ around a central singularity in $Q$. The region shaded a darker red has $Q<0$, which shows superoscillation in the $(Q,K_{a})$ picture, and the region shared the lighter red has $Q_{r}<0$ also shows superoscillation in the $(Q_{r},K_{c})$ picture. Figure 11: The quantum potential energy $Q$ in which the fluid particles orbit (the external potential is $U=0$ inside the well. In the $(Q,K_{a})$ picture, these are essentially classical orbits with speed $v_{a}=(\vec{\nabla}S)/m$ an a force $-\vec{\nabla}Q$, and near the center of the well, they become uniform circular orbits, and the quantum potential goes as $-1/r^{2}$, where $r$ is the distance from the center. Regions where $Q$ is below the translucent gray plane at $Q=0$ are superoscillatory, where the orbital speeds within the vortex approach infinity. In the $(q_{r},k_{c})$ picture, some of the $Q$ seen here is understood as symmetric kinetic energy $K_{s}$ which properly belongs to the total classical kinetic energy $K_{c}$, but it is still the total $Q=Q_{r}+K_{s}$ that guides the bulk flow with energy $K_{a}$. Quantum Potential, $q$ |  Average Kinetic, $k_{a}$
---|---
|
Reduced Quantum Potential, $q_{r}$ |  Classical Kinetic, $k_{c}$
|
Fluid Density, $\rho\equiv R^{2}\equiv|\psi|^{2}$ |  Symmetric Kinetic, $k_{s}$
|
Figure 12: Fluid and energy densities for the state of Eq. 11 in the 2D
infinite square well.

Another important example can only be seen in 2D or 3D, and this is a complex
superposition of degenerate eigenstates. We consider the 2D infinite square
well with side lengths $L$, and the degenerate eigenvalues are
$(n_{x}=1,n_{y}=2)$ and $(n_{x}=2,n_{y}=1)$ superposition,

$\psi(x,y,t)=\frac{\sqrt{2}}{L}e^{-\frac{iEt}{\hbar}}\bigg{(}\sin\Big{(}\frac{\pi
x}{L}\Big{)}\sin\Big{(}\frac{2\pi y}{L}\Big{)}+i\sin\Big{(}\frac{2\pi
x}{L}\Big{)}\sin\Big{(}\frac{\pi y}{L}\Big{)}\bigg{)},$ (11)

which has energy $E=\frac{5\pi^{2}\hbar^{2}}{2mL^{2}}$. Because this state has
definite energy, there is no difference here between hard and soft
superoscillation (i.e., $-\frac{\partial S}{\partial t}=E$). However, because
the superposition is complex, it is not the case that $\vec{\nabla}S=0$, which
means the fluid is in motion, even though the fluid density
$|\psi|^{2}\equiv\rho\equiv R^{2}$ is constant, so the state is still
‘stationary’ in some sense. In this case, the fluid moves in stable orbits
around the center of the well, where a node in the density results in a
negative singularity in the quantum potential, maintaining a symmetric vortex
distribution. Near the singularity, the orbits are essentially circular, which
means that the quantum potential energy $Q$ must be proportional to
$-1/r^{2}$, with $r$ the distance from the center, as discussed above. The
momentum density and streamlines are shown in Fig. 10, and regions of
$(q,k_{a})$-picture superoscillation, $Q<0$. are shown in darker shaded red,
and regions of $(q_{r},k_{c})$-picture superoscillation, $Q_{r}<0$ are shown
in lighter red.

The fluid density $\rho$ of the stable vortex, and the five energy densities
of interest are shown in Fig. 12. It is worth noting that $k_{a}$ is not zero
at the node in $\rho$, and is in fact at a local maximum there. In our other
examples, this was seen for $k_{s}$, but $k_{a}=0$ at the nodes. The
similarity between this case for $k_{a}$ and the more general case for $k_{s}$
further motivates the interpretation of $k_{s}$ as describing orbital motion
about axis $\hat{v}_{s}(\vec{x},t)$ at speed $|\vec{v}_{s}(\vec{x},t)|$ in
infinitesimal potential wells with nearly infinite central constraint forces,
at each event $(\vec{x},t)$ in spacetime.

##  6 Conclusions

The quantum fluid picture, where the quantum potential $Q$ is treated as
another classical potential energy in the fluid, provides an intuitive and
natural way to think about quantum mechanics. The total potential $Q+U$ seen
by the fluid particles, and their resulting motion, tends to look very similar
for superpositions with similar quantum numbers, regardless of the shape of
the external potential $U$.

Including the quantum potential provides a natural explanation for tunneling
and barrier penetration, because $Q$ tends to cancel out the barrier in $U$ so
that the fluid flows through unobstructed. It also explains the appearance of
interference fringes as a series of broad hills and narrow valleys, where the
fluid flows quickly across the valleys, and then slows down and accumulates on
the hilltops.

The local energy $E_{p}$ of particles in the fluid also leads to refined
notions of when a particle is in a classically forbidden region, and when a
particle’s kinetic energy is high enough to constitute (soft)
superoscillation, and this notion extends Berry’s original $Q<0$ bound to
general superposition states. For some purposes, the global band limit $E_{+}$
is of more interest, so we have also discussed the case of hard
superoscillation.

We also explored the above concepts in the alternative $(q_{r},k_{c})$
picture, where the terms have been reorganized to give a different
interpretation, where the fluid contains local symmetric kinetic energy
$k_{s}$ that does not relate to its net flow, and a corresponding reduced
quantum potential. Both the $(q_{r},k_{c})$ picture, which is probably more
physically correct, and the standard $(q,k_{a})$ picture, which gives an
intuitive explanation of the fluid motion, have their conceptual advantages,
so it is useful to consider situations from both perspectives.

We hope that the extensive set of examples provided here will help the reader
to build a good physical intuition for the fluid picture and how the quantum
potential gives rise to all of the seemingly nonclassical aspects of the fluid
dynamics - namely tunneling, interference, and superoscillation.

This work fits into a larger program aimed at developing a fully local
interpretation of quantum mechanics in spacetime [29, 20] (at the cost of
having many local worlds defined in one spacetime). In this model, the
standard universal quantum wavefunction is entirely replaced by a collection
of fluids in spacetime, with elemental particles in the fluids moving along
time/light-like world-lines, and no nonlocal connections of any kind. All of
the physics in this model of quantum physics occurs in spacetime, which may be
a critical step toward integrating quantum physics and general relativity.

In the shorter term, we aim to develop the relativistic generalization of this
particle treatment, for both massive and massless particles, and for both
fermions and bosons. We hope this will resolve the issues with infinite
velocity in the nonrelativistic treatment, and that we will obtain a picture
where can understand the Lorentz transform of a generalized quantum potential.

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