# A non-hermitean momentum operator for the particle in a box

SKSeyong Kim  ARAlexander Rothkopf  Department of Physics, Sejong University,
143-747, Seoul, South Korea  Faculty of Science and Technology, University of
Stavanger, 4021, Stavanger, Norway

###### Abstract

We construct a discrete non-hermitean momentum operator, which implements
faithfully the non self-adjoint nature of momentum for a particle in a box.
Its eigenfunctions are strictly limited to the interior of the box in the
continuum limit, with the quarter wave as first non-trivial eigenstate. We
show how to construct the corresponding hermitean Hamiltonian for the infinite
well as concrete example to realize unitary dynamics. The resulting Hilbert
space can be decomposed into a physical and unphysical subspace, which are
mutually orthogonal. The physical subspace in the continuum limit reproduces
that of the continuum theory and we give numerical evidence that the correct
probability distributions for momentum and energy are recovered.

Momentum operator, summation-by-parts, finite volume, non-hermitean,

###### keywords:

PreprintFP

##  1 Introduction

###  1.1 Physics motivation

Our understanding of point particle motion rests on the concepts of position
and momentum. In classical mechanics they constitute independent observables
and future motion can be deduced uniquely and deterministically from their
values at present. In quantum mechanics both quantities retain their role as
the fundamental building blocks to describe motion of a point particle. In the
microscopic domain their values however cannot be known simultaneously (at
least not in the same spatial direction) and predictions about future motion
become probabilistic in nature.

Remarkable phenomena occur when microscopic motion is confined between
impenetrable walls. An explicit example are quantum dots, where one can
observe the formation of bound states that are classically forbidden
Schult:1989zz . Another phenomenon is chaotic wave dynamics in quantum
billiard, one example Seba:1990zz  being a triangular domain with a central
point scatterer, which, in classical mechanics avoids chaotic behavior.

It is well known (see e.g. discussion in al2021alternative ) that the standard
concept of momentum in quantum mechanics is unable to consistently describe
motion in a geometry of finite extent. To be concrete, let us take a look at
the one-dimensional infinite well. The conventional momentum operator
$\hat{p}$ has the plane waves as eigenfunctions. If we measure momentum on a
stationary state contained within the infinite well, i.e. we carry out a von
Neumann projective measurement, then we expect to end up in one of the
eigenstates of the momentum operator. These however are extended over the
whole spatial domain and thus are incompatible with the finite extent of the
well. This fact has been interpreted as the momentum measurement imparting an
infinite amount of energy to the state, which is unphysical.

Interestingly the measurement of momentum most commonly is implemented via the
measurement of position after the point particle is allowed to interact with
its environment. Two examples are observing its position after free fall in a
gravitational field or after being forced on a curved trajectory by a magnetic
field. In each case, even though the particle may be confined in a finite
geometry at first, it is released for the purpose of the momentum measurement
(for a more comprehensive discussion see freericks2023measure ).

Given the central role that momentum plays in our understanding of motion and
the rich phenomenology of quantum systems in finite geometries, it is
imperative that a consistent definition of this observable is achieved. In a
recent study al2021alternative  a novel momentum operator was proposed. The
authors construct an operator, with eigenfunctions, which are located exactly
within the finite domain, solving a central shortcoming of the conventional
definition of quantum mechanical momentum. On the other hand this momentum
operator contains terms of infinite value located on the boundary in the form
of delta-function contributions. In addition the Hamilton operator deployed in
that study does not arise naturally from a combination of the momentum
operator and the position operator.

In this study we set out to construct an alternative momentum operator
$\hat{p}$ for the particle in the one-dimensional finite box in order to
address these issues. Its eigenfunctions are also limited to the interior of
the accessible domain. As it is agnostic of the boundary conditions of the
stationary states, the operator is devoid of delta-function like contributions
on the boundary. We show how it can be used to construct a consistent
Hamiltonian, involving only reference to position $\hat{x}$ and momentum
$\hat{p}$. The price we have to pay is that our momentum operator is not
hermitean and does not lend itself to the conventional interpretation of a von
Neumann projective measurement. We will give numerical evidence that our
proposed momentum operator provides the correct statistical predictions of
particle motion, as it reproduces the n-point functions of momentum
$\langle\hat{p}^{n}\rangle$ with respect to the stationary states of the
infinite well.

###  1.2 Observables in Quantum Mechanics

In quantum mechanics observables are conventionally described by normal
operators $\hat{N}\in{\cal N}$ with the defining property
$\hat{N}\hat{N}^{\dagger}=\hat{N}^{\dagger}\hat{N}$. This important class of
operators is not necessarily self-adjoint but is characterized by a purely
real spectrum, i.e. the eigenvalues $\lambda_{i}$ in
$\hat{N}|\lambda_{i}\rangle=\lambda_{i}|\lambda_{i}\rangle$ fulfill
$\lambda_{i}\in{\mathbb{R}}$. This property allows normal operators to
represent measurable properties of quantum systems.

To interpret the measurement process as a von Neumann projective measurement,
one reexpresses the normal operator representing the observable into a set of
projection operators constructed from its eigenstates $|\lambda_{i}\rangle$

$\displaystyle\hat{N}=\sum_{i}\lambda_{i}|\lambda_{i}\rangle\langle\lambda_{i}|=\sum_{i}\lambda_{i}\hat{P}_{i},$
(1)

where $\hat{P}_{i}\hat{P}_{j}=\hat{P}_{i}\delta_{ij}$. This decomposition is
always possible as long as $\hat{N}$ is normal.

When carrying out the measurement of $N$ on a quantum system represented by
the state $|\psi\rangle$, experimental evidence tells us that the state of the
system spontaneously changes into one of the eigenstates of $\hat{N}$:
$|\psi\rangle\to|\lambda_{i}\rangle$ and the measurement apparatus returns the
value $\lambda_{i}$. The probability for this collapse of the wavefunction
into the state $|\lambda_{i}\rangle$ is given by
$|c_{i}|^{2}=|\langle\lambda_{i}|\psi\rangle|^{2}$ if the observable can be
represented by a normal operator $\hat{N}$.

A single measurement however does not allow us to confirm the probability
$|c_{i}|^{2}$, and instead we must repeat the measurement a large number of
times on identically prepared states and record a histogram of outcomes. Only
after we have established knowledge of the probability distribution as a
whole, is the information of an individual $|c_{i}|^{2}$ relevant. (i.e. each
individual measurement will result in a purely random outcome that cannot be
predicted. We can only aposteriori confirm that $\lambda_{i}$ has occurred
with a frequency among our measurements in agreement with $|c_{i}|^{2}$.)

Another way to think about the probabilistic nature of measurements in quantum
mechanics is to consider not the probability distribution of outcomes
$P(\lambda)$ itself, as we did before, but its moments. The lowest moments of
a distribution are well known quantities, its mean $\langle\lambda\rangle=\int
d\lambda\lambda P(\lambda)$ and its spread $\langle\lambda^{2}\rangle=\int
d\lambda\lambda^{2}P(\lambda)$. These moments are straight forwardly related
to the observable through its so called n-point functions
$\langle\psi|\hat{N}^{n}|\psi\rangle$, where $n$ denotes the power of the
operator $\hat{N}^{n}$ inserted in the inner product of the state
$|\psi\rangle$. As discussed in the context of the Hamburger moment problem
shohat1950problem , under well-defined circumstances knowledge of all n-point
functions is equivalent with the knowledge of the probability distribution.
One prerequisite is that the probability distribution exists, which quantum
mechanics guarantees.

In this paper we will study an operator on a discrete grid as candidate for a
momentum operator for the particle in the box. This operator will not be
normal. While it implements various properties of the continuum momentum
operator faithfully, it does not allow a similar straight forward
interpretation of measurement as a von Neumann projective measurement.

On the other hand we will find that it is possible to use our novel momentum
operator to define from it a hermitean Hamilton operator for the infinite
square well. This Hamiltonian encodes as part of its eigenfunction space the
physical energy eigenstates in the continuum limit. Consistency is found when
inspecting the lowest n-point functions of the momentum operator evaluated in
the physical energy eigenstates. Their values reproduce those of the continuum
theory in the continuum limit. In turn we argue that our non-hermitean and
non-normal momentum operator is a viable candidate to represent the physical
property of momentum for a particle in a box.

##  2 Momentum operator for a particle in a box

###  2.1 Defining properties

When a microscopic point particle is restricted to a finite geometry of length
$L$ with physical boundaries, its motion is affected by those boundaries. In
order to describe a point particle in quantum mechanics we must define the
position $\hat{x}$ and momentum $\hat{p}$ operator of the system at hand, from
which the Hamiltonian operator $\hat{H}$ may be constructed. It is the
Hamiltonian, which through functional exponentiation $\hat{U}={\cal T}{\rm
exp}[-i/\hbar\int dt\hat{H}]$ generates the dynamical unitary evolution that
describes the motion of the point particle.

Central properties of operators in quantum mechanics are established through
their behavior in the inner product of the system Hilbert space. When
investigating these properties in a finite geometry, we choose to work in the
position basis, where the scalar product is defined via an integral over
wavefunctions $\langle x|\psi\rangle=\psi(x)$ and the position space matrix
elements of the observable $\langle x|\hat{N}|y\rangle=N$

$\displaystyle\langle\psi|\hat{N}\chi\rangle=\int\,dxdy\;\psi^{*}(y)N(x,y)\chi(x)$
(2)

When being placed in a finite geometry, the position operator $\hat{x}$ is not
affected. By definition, it remains diagonal in the position basis, it is
hermitean, self adjoint and its eigenfunctions remain localized in space.

On the other hand the momentum operator $\hat{p}$ differs from that of a
system without boundaries. The reason lies in the fact that even though the
momentum operator matrix elements in position space
$p(x,y)=-i\hbar\delta(x-y)\frac{d}{dx}$ are local, the presence of the
derivative means that their behavior within the inner product is affected by
the presence of genuine boundaries

$\displaystyle\langle\hat{p}\psi|\chi\rangle$
$\displaystyle=\int_{a}^{b}\,dx\;\big{(}-i\hbar\frac{d}{dx}\psi(x)\big{)}^{*}\chi(x)$
(3)
$\displaystyle=\int_{a}^{b}\,dx\;\psi(x)^{*}\big{(}-i\hbar\frac{d}{dx}\chi(x)\big{)}+\left.i\hbar\psi(x)^{*}\chi(x)\right|_{a}^{b}$
(4)
$\displaystyle=\langle\psi|\hat{p}\chi\rangle+\left.i\hbar\psi(x)^{*}\chi(x)\right|_{a}^{b}=\langle\hat{p}^{\dagger}\psi|\chi\rangle+\left.i\hbar\psi(x)^{*}\chi(x)\right|_{a}^{b}$
(5)

In other words, the momentum operator only remains hermitean, as long as one
imposes periodic boundary conditions or restricts the domain of wavefunctions
to those that vanish on the boundary $\psi(x)^{*}\chi(x)|_{a}^{b}=0$111Note
that the momentum operator is invariant under spatial translations and thus
does not carry explicit knowledge of the absolute coordinates of the geometry
and thus it cannot distinguish between a box extending between $[-L/2,L/2]$
and $[0,L]$..

Naively transplanting the continuum momentum operator into a finite geometry
presents a conceptual challenge. Its eigenfunctions are plane waves, which
extend over the whole of the real axis. In the conventional interpretation of
projective measurement, this entails that a momentum measurement will collapse
the wavefunction of a point particle within the geometry onto a wavefunction
that is extended beyond the boundaries. In the explicit example of an infinite
potential well, this would entail an infinite amount of energy being imparted
on the particle. Such a picture hence must be unphysical.

###  2.2 Summation by parts finite differences

We thus wish to construct a momentum operator, which faithfully implements the
behavior of the conventional momentum operator w.r.t. the scalar product of
eq. 5, but which possesses eigenfunctions which are limited to the accessible
domain.

Note that at this point we do not fix to any specific boundary condition, as
this information must not enter the construction of the momentum operator.
Understood as linear differential operator associated with a boundary value
problem, the momentum operator and boundary conditions constitute two separate
sets of information.

Our strategy is as follows: to avoid technical intricacies of infinite
dimensional vector spaces, we will begin our construction of a momentum
operator on a discrete grid with $N_{x}$ points spaced $\Delta
x=(b-a)/(N_{x}-1)$ apart and thus of total finite extend $L=(N_{x}-1)\Delta x$
as shown in fig. 1.

The guiding principle for our construction is to require that the discrete
operator mimics exactly the non-hermitian behavior of the continuum momentum
operator manifest in eq. 5. In addition we require that the operator becomes
the generator of translations in the continuum limit, and thus must converge
to the standard derivative as $N_{x}\to\infty$ and $\Delta x\to 0$ at constant
$L$.

Figure 1: Sketch of the grid layout with genuine boundaries on which we
formulate a momentum operator

We take inspiration from modern discretization strategies for partial
differential equations and classical actions of dynamical systems, for which
the concept of a summation-by-parts (SBP) finite difference operator has been
developed over the past decades (for reviews see e.g. svard2014review ;
fernandez2014review ; lundquist2014sbp ). Summation-by-parts operators are
designed to exactly mimic integration by parts in the discrete setting, one of
our central requirements.

Let us consider functions discretized on a grid
${\bm{\psi}}=(\psi(a),\psi(a+\Delta x),\psi(a+2\Delta x),\ldots,\psi(b))^{\rm
T}$, where we choose without loss of generality $a=0$ and $b=1$. In order to
mimic integration by parts we need a consistent definition of both integration
and differentiation in the discrete setting.

As first step we thus select a specific quadrature rule for the integral
representing the inner product in eq. 2 via a diagonal and positive definite
matrix $\mathds{H}$. The discrete inner product thus reads
$({\bm{\psi}},{\bm{\psi}}^{\prime})={\bm{\psi}}^{\rm\dagger}\mathds{H}{\bm{\psi}}^{\prime}$,
where the symbol $\dagger$ refers to the combined transpose and complex
conjugation.

A finite difference operator $\mathds{D}$ of order $r$ that mimics
integration-by-parts (IBP) exactly, must obey the following properties:

$\displaystyle\mathds{D}{\bm{x}}^{n}=n{\bm{x}}^{n-1}\quad{\rm for\,n<r}$ (6)
$\displaystyle\mathds{D}=\mathds{H}^{-1}\mathds{Q},\qquad\mathds{Q}^{\rm
T}+\mathds{Q}=\mathds{E}_{N}-\mathds{E}_{1}={\rm diag}[-1,0,\ldots,0,1].$ (7)

Here eq. 6 establishes that $\mathds{D}$ indeed acts as an exact derivative on
polynomials, as long as their degree is lower than $r$. Equation 7 on the
other hand defines the structure of $\mathds{D}$ that, in the interplay with
$\mathds{H}$, allows it to mimic IBP exactly in the discrete setting. The SBP
operator is written as a product of the inverse quadrature matrix
$\mathds{H}^{-1}$ and the matrix $\mathds{Q}$. The latter implements a central
symmetric finite difference stencil in the interior, and stencils close to the
boundary that allow it to fulfill the sumamtion-by-parts property given on the
left of eq. 7. Summation-by-parts requires that when $\mathds{D}$ is moved
within the inner product of the discretized function space, appropriate
boundary terms arise. Such boundary terms make reference only to first and
last entry of the discretized function vectors participating in the inner
product222Note that this summation-by-parts property does not depend on the
specific value of the discretized functions on the boundary or their
functional form.. Here projection operators $\mathds{E}_{1}$ and
$\mathds{E}_{N}$ are used to express the summation-by-parts conditions. The
former refers to a matrix with all entries zero except the top left, where it
contain the value minus one. The latter is a matrix with only a single non-
vanishing entry of value one in the bottom right corner. To be concrete, in
the construction of our momentum operator, we deploy the lowest order diagonal
norm SBP scheme, referred to as SBP21, which is second order in the interior
and reduces to first order on the boundary. With the trapezoidal rule as
choice for integral quadrature, we have

$\mathds{H}=\Delta x\left[\begin{array}[]{ccccc}1/2&&&&\\\ &1&&&\\\
&&\ddots&&\\\ &&&1&\\\
&&&&1/2\end{array}\right],\quad\mathds{D}=\frac{1}{2\Delta
x}\left[\begin{array}[]{ccccc}-2&2&&&\\\ -1&0&1&&\\\ &&\ddots&&\\\ &&-1&0&1\\\
&&&-2&2\end{array}\right].$ (8)

Note that this construction differs from the one of Ref. al2021alternative
and that our derivative operator does not need to be amended by terms specific
to the boundary condition.

Let us also confirm that the finite difference operator $\mathds{D}$ is
antisymmetric under the parity operation ${\cal P}\mathds{D}\to-\mathds{D}$
for ${\cal P}x\to-x$. This property follows from eq. 8, as the only term that
is affected by the change in sign of $x$ is the lattice spacing $\Delta
x\to-\Delta x$.

###  2.3 Spectral structure

In preparation for using $\mathds{D}$ as building block to our momentum
operator, let us investigate its spectrum. We follow here the strategy laid
out in Ref. ruggiu2020eigenvalue . The presence of a different stencil in the
interior and on the boundary in $\mathds{D}$ precludes us from using the
simple ansatz of a single trigonometric function as eigenfunction. Instead, to
determine the eigenvectors and eigenfunctions, one first solves the
eigenvector relation for the interior stencil and exploits the necessary
consistency with the boundary behavior as additional constraint for obtaining
the discrete eigenvalues.

Without loss of generality, let $\Delta x=1$ for the remainder of this
subsection. In ruggiu2020eigenvalue  the authors considered an SBP operator,
which was amended by a so called simultaneous-approximation term (SAT). For
this type of SBP-SAT operators there exists a theorem that guarantees that
their spectrum is located on the imaginary axis. Our operator $\mathds{D}$
does not fulfill the same property. At the same time, we find from a heuristic
numerical survey of the spectrum of $\mathds{D}$ for various choices of
$N_{x}$ that its eigenvalues too are purely imaginary. For even values of
$N_{x}$ one finds two zero eigenvalue, while for odd values of $N_{x}$ there
exist three. This however does not represent a problem, as the associated
eigenspace remains one-dimensional and, as expected, is spanned solely by the
constant function.

We thus start our analysis using the same ansatz ${\bm{\psi}}_{k}=r^{k}$ as in
ruggiu2020eigenvalue  to solve the internal stencil relation for the k-th
entry of an eigenvector $\bm{\psi}$ with a purely imaginary eigenvalue $i\xi$
with $\xi\in\mathbb{R}$

$\displaystyle\frac{1}{2}{\bm{\psi}}_{k+1}-\frac{1}{2}{\bm{\psi}}_{k-1}=i\xi{\bm{\psi}}_{k}\qquad\Rightarrow\qquad
r^{k+1}-r^{k-1}=2i\xi r^{k}.$ (9)

We see that there are two possible solutions for the values of
$r_{1,2}=i\xi\pm\sqrt{1-\xi^{2}}$, which imply $r_{1}r_{2}=-1$.

Assuming that $r_{1}\neq r_{2}$, i.e. $\xi\neq 1$ we are led to the following
ansatz for the eigenvector as linear combination of two independent
contributions

$\displaystyle{\bm{\psi}}_{k}(\xi)=c_{1}(\xi)r_{1}^{k}(\xi)+c_{2}(\xi)r^{k}_{2}(\xi).$
(10)

By inserting eq. 10 into the boundary stencil relation at the left boundary
involving ${\bm{\psi}}_{0}$ and ${\bm{\psi}}_{1}$ one finds that $c_{1}$ and
$c_{2}$ can be expressed solely in terms of ${\bm{\psi}}_{0}$ and $\xi$ by
proxy of $r_{1}$ and $r_{2}$

$\displaystyle c^{\rm
L}_{1}/{\bm{\psi}}_{0}=-\frac{2\xi^{2}+i\xi(r_{2}-2)+r_{2}-1}{r_{1}(r_{1}-r_{2})},\quad
c^{\rm
L}_{2}/{\bm{\psi}}_{0}=\frac{2\xi^{2}+i\xi(r_{1}-2)+r_{1}-1}{r_{2}(r_{1}-r_{2})}.$
(11)

Writing $c_{1}$ and $c_{2}$ in the above fashion we exclude the case of a zero
eigenvalue for which $r_{1}=r_{2}=0$. I.e. we will not recover $\xi=0$ from
this analysis. This missing information is obtained from the construction of
$\mathds{D}$, which tells us that the eigenvector with zero eigenvalue is
simply the constant function.

Similarly we can insert the ansatz eq. 10 into the right boundary relation to
obtain

$\displaystyle c^{\rm
R}_{1}/{\bm{\psi}}_{N_{x}}=\frac{2r_{2}\xi^{2}+i\xi(2r_{2}-1)-r_{2}+1}{r_{1}^{Nx-2}(r_{1}-r_{2})},$
$\displaystyle c^{\rm
R}_{2}/{\bm{\psi}}_{N_{x}}=\frac{2r_{1}\xi^{2}+i\xi(2r_{1}-1)-r_{1}+1}{r_{2}^{Nx-2}(r_{2}-r_{1})}.$
(12)

In order for us to obtain a consistent solution to the eigenvector relation
both in the interior and on the boundary the solutions for ${\bm{c}}^{\rm
L,R}=(c^{\rm L,R}_{1},c^{\rm L,R}_{2})^{T}$ must represent the same vector,
i.e. ${\bm{c}}^{\rm L}$ and ${\bm{c}}^{\rm R}$ must be linearly dependent.
This requirement is most efficiently expressed in that the determinant of the
$2\times 2$ matrix constructed of ${\bm{c}}^{\rm L}$ and ${\bm{c}}^{\rm R}$
must vanish

$\displaystyle\left|\begin{array}[]{cc}c^{\rm L}_{1}/{\bm{\psi}}_{0}&c^{\rm
L}_{2}/{\bm{\psi}}_{0}\\\ c^{\rm R}_{1}/{\bm{\psi}}_{N_{x}}&c^{\rm
R}_{2}/{\bm{\psi}}_{N_{x}}\end{array}\right|=0.$ (15)

Using the Mathematica software333Accompanying computations for this paper can
be found at rothkopf_2024_10843823 ., we solve the above equation and find by
comparison to ruggiu2020eigenvalue  that for even values of $N_{x}$ the
allowed $N_{x}-2$ eigenvalues $\xi\neq 0$ for $\mathds{D}$ are given by

$\displaystyle\xi_{l}={\rm
sin}\Big{[}\big{(}\frac{l}{N_{x}-1}+\frac{1}{2}\big{)}\pi\Big{]}={\rm
cos}\Big{[}\big{(}\frac{l}{N_{x}-1}\big{)}\pi\Big{]},\qquad
l\in[1,\ldots,N_{x}-2].$ (16)

Once the eigenvalues are known, we can construct the eigenfunctions by
inserting the definition of the coefficients $c_{1,2}$ from e.g. eq. 11 into
eq. 10. A thorough discussion of the structure of the eigenfunctions will be
presented in the next section where $\mathds{D}$ is used to construct a
candidate momentum operator.

###  2.4 Momentum operator candidate and its properties

Having introduced the summation-by-parts finite difference operator
$\mathds{D}$ in the previous section, we proceed to construct from it a
momentum operator $\hat{p}$. For finite grid spacing we will refer to its
discrete form as $\mathds{P}$, from which the physical momentum operator
$\hat{p}$ emerges in the continuum limit

$\displaystyle\mathds{P}=-i\hbar\mathds{D},\qquad{\rm
where}\qquad\hat{p}=\lim_{\tiny\begin{array}[]{c}\Delta x\to 0\\\
N_{x}\to\infty\\\ L=(N_{x}-1)\Delta x\end{array}}\mathds{P}.$ (20)

Due to the fact that $\mathds{D}$ is odd under parity, also $\mathds{P}$
retains this property. Note that it is even under combined ${\cal PT}$
symmetry, as ${\cal T}$ sends $i\to=-i$.

Let us inspect its spectral properties. From the study of the spectrum of
$\mathds{D}$, it follows via eq. 20 that the discrete eigenvalues $p_{l}$ of
$\mathds{P}$ are purely real and on a grid with even $N_{x}$ take on the
following values

$\displaystyle p_{l}=\left\\{\begin{array}[]{lc}\frac{1}{\Delta x}{\rm
sin}\Big{[}\big{(}\frac{l}{N_{x}-1}+\frac{1}{2}\big{)}\pi\Big{]}=\frac{1}{\Delta
x}{\rm sin}\Big{[}\big{(}\frac{l\Delta
x}{L}+\frac{1}{2}\big{)}\pi\Big{]}&l\in[1,\ldots,N_{x}-2]\\\
0&l=N_{x}-1,N_{x}.\end{array}\right.$ (23)

Let us consider what eq. 23 tells us about the spectrum of $\mathds{P}$. Due
to the antisymmetry of the ${\rm sin}$ function around $\pi$, all the
eigenvalues $p_{l}$ come in pairs of equal magnitude and opposite sign. The
smallest non-vanishing magnitude of the momentum is obtained for $l=N_{x}/2$
and $l=N_{x}/2-1$. We find

$\displaystyle p_{N_{x}/2}=$ $\displaystyle\frac{1}{\Delta x}{\rm
sin}\Big{[}\big{(}\frac{N_{x}\Delta
x}{2L}+\frac{1}{2}\big{)}\pi\Big{]}=\frac{1}{\Delta x}{\rm
sin}\Big{[}\big{(}\frac{(N_{x}-1)\Delta x}{2L}+\frac{\Delta
x}{2L}+\frac{1}{2}\big{)}\pi\Big{]}$ $\displaystyle=$
$\displaystyle\frac{1}{\Delta x}{\rm sin}\Big{[}\big{(}\frac{\Delta
x}{2L}+1\big{)}\pi\Big{]}\overset{\Delta x/L\ll
1}{\approx}-\frac{\pi}{2L}=-\frac{2\pi}{4L}.$ (24)

Interestingly it turns out that the smallest momentum eigenvalue corresponds
to a wavelength of $\lambda=4L$, which indicates that the associated
eigenvector resembles a quarter wave, in contrast to the half-wave that has
been obtained in previous studies, such as in al2021alternative . As one lets
the grid spacing in eq. 23 tend towards zero the largest accessible momentum
value increases, until in the continuum limit the whole real axis of momenta
is covered.

Note that the doubly degenerate zero mode is associated with a one-dimensional
eigenspace, spanned by the constant function and thus does not pose a
conceptual problem. We have confirmed by numerical determination of the
spectrum, that for an odd number of grid points a third degenerate zero
eigenvalue ensues. The space spanned by the associated eigenfunctions however
still remains one-dimensional, as each zero eigenvalue is associated with the
same function.

The appearance of multiple zero modes may be understood intuitively as
follows. The spectra of $\mathds{P}$ and $\mathds{P}^{\rm T}$ are necessarily
the same, as they are based on the same characteristic polynomial. However,
since $\mathds{P}$ is non hermitean, its right and left eigenvectors do not
necessarily agree. In particular it is known from the study of the underlying
SBP finite difference operator in Rothkopf:2022zfb  that the right zero-mode
is the constant function and the left zero-mode is the highly oscillatory
$\pi$-mode. In order to accommodate such two different zero modes in the
common spectrum, it needs to feature at least two zero eigenvalues.

In case that we have an odd number of grid points, this $N_{x}$ determines the
overall number of eigenvalues of $\mathds{P}$. Since all eigenvalues have to
come in positive-negative pairs, in addition to the two zero modes, one is
left with one remaining eigenvalue to fill. The only number that is its own
negative is zero and thus one naturally expects to find another zero mode in
case that $N_{x}$ is odd.

We can now visually inspect the eigenfunctions $\bm{\psi}$ of the operator
$\mathds{P}{\bm{\psi}}_{l}=p_{l}{\bm{\psi}}_{l}$, which follow from inserting
the definition of the coefficients $c_{1,2}$ from e.g. eq. 11 into eq. 10 and
using the values just obtained in eq. 23 as the eigenvalues.

Figure 2: Example of two non-constant eigenvectors of $\mathds{P}$ in position
space for $N_{x}=32$. Real-part (blue circles) and imaginary-part (mustard
triangles) of the normalized eigenfunction to $l=N_{x}/2-1$ (top) and to
$l=N_{x}/2-2$ (bottom). Note that on the boundary either the value, or the
derivative of the functions is zero.

In fig. 2 we plot the real- (red circles) and imaginary part (blue triangles)
of the lowest non-trivial normalized eigenfunction, corresponding to the
eigenvalue $l=N_{x}/2-1$ (top) and for the second to lowest non-trivial
eigenfunction $l=N_{x}/2-2$ (bottom).

While the boundary behavior entails that these functions are not exactly plane
waves, they exhibit very similar oscillatory features in the interior of the
domain. As predicted by eq. 24, we find that the lowest non-trivial
eigenfunction of the operator $\mathds{P}$ corresponds to the quarter wave.
Note that this structure is the smallest fraction of a standing wave that can
be consistently placed within a bounded domain.

The condition that needs to be fulfilled by such an eigenfunction is that no
probability leaks from the finite domain. We will now show that in the
continuum limit this property is indeed fulfilled. Note that the real- and
imaginary part of the eigenfunctions in fig. 2 either takes on the value zero
at the boundary or, crucially, exhibits a zero derivative. This is most
clearly seen in the lower panel of fig. 2 where the first two red circles on
the left and the last two blue triangles on the right take on exactly the same
value leading to a vanishing of the derivative there.

We can check the probability flux through the boundary defined as

$\displaystyle j(x)=$
$\displaystyle\frac{1}{2mi}\Big{(}\psi^{*}(x)\frac{\partial}{\partial
x}\psi(x)-\frac{\partial}{\partial x}\psi^{*}(x)\psi(x)\Big{)}$ (25)
$\displaystyle\approx$
$\displaystyle\frac{1}{2mi}\Big{(}{\bm{\psi}}^{*}\circ{\mathds{D}}{\bm{\psi}}-{\mathds{D}}{\bm{\psi}}^{*}\circ{\bm{\psi}}\Big{)},$
(26)

where in the second line the symbol $\circ$ refers to component-wise
multiplication. Boundary leakage is related to a non-zero value of $j$ at the
boundaries of the domain. Since the reference to $m$ is made for purely
dimensional reasons, we will set $m=1$ at this point.

Figure 3: Disappearance of boundary leakage under grid refinement by the three
lowest lying non-constant eigenfunctions of the operator $\mathds{P}$.

In fig. 3 we show the remnant boundary leakage for the lowest three non-
trivial eigenfunctions of $\mathds{P}$. One finds that even on relatively
coarse grids with $N_{x}=16$, the right-most data points, the values we obtain
a small. More importantly as one refines the grid, going to smaller values of
$\Delta x$, the remnant boundary leakage diminishes with a powerlaw, according
to $\Delta x^{\nu}$ with $\nu\approx 2$. It is reassuring that even with the
lowest order SBP21 finite difference operator a quadratic scaling of this
crucial property towards the continuum limit is achieved.

By now we have confirmed two milestones in the construction of a viable
candidate momentum operator for the particle in the box. Our discrete operator
$\mathds{P}$ features eigenfunctions, which in the continuum limit are exactly
constrained to the accessible domain. In addition the operator is defined
without reference to any boundary conditions that may be enforced on the
stationary states of the system and thus is devoid of the infinite constant
boundary terms necessary in other constructions of alternative momentum
operators (see e.g. al2021alternative ).

One may ask, how our operator will recover the well-known eigenfunctions of
the momentum operator in an unbounded domain? If we naively let the size of
the domain increase, by increasing the number of grid points, then, due to the
normalization of the eigenfunctions, the quarter wave will be increasingly
pulled apart and eventually become the zero function. The same would happen to
all the other eigenfunctions with extent of a finite half-integer multiple of
the wavelength. We know however that the (unnormalizable) eigenfunctions of
the conventional momentum operator are the finite plane waves associated with
a continuum of eigenvalues.

This conundrum can be resolved by recognizing that there is a qualitative
difference between the setup of finite extent with genuine boundaries and the
infinite extend without explicit boundary terms. This difference formally
manifests itself in eq. 15, where the presence of the explicit boundaries
provides us with a quantization conditions for the eigenvalues of the SBP
finite difference operator and thus for our proposed momentum operator. In
case that the domain is infinite we have no explicit boundary terms and thus
eq. 15 is not applicable. Indeed for finite lattice spacing $\Delta x$ but
$N_{x}$ taken to infinity $\mathds{D}$ would only contain the interior
symmetric stencil, considered in eq. 9. The corresponding eigenvalue relation
for the momentum operator, in the absence of boundary terms is solved by

$\displaystyle\mathds{P}{\bm{\psi}}=p^{N_{x}=\infty}{\bm{\psi}}.\qquad{\bm{\psi}}_{k}={\rm
exp}[i\kappa\Delta xk],\qquad p^{N_{x}=\infty}=\frac{1}{\Delta x}{\rm
sin}[\kappa\Delta x],$ (27)

for continuous values of $-\frac{\pi}{\Delta x}<\kappa<\frac{\pi}{\Delta x}$.

These are exactly the unnormalizable plane waves that arise for the
conventional momentum operator on the unbounded domain. Thus our novel
momentum operator not only produces eigenfunctions of finite extent and
without boundary leakage in the presence of genuine boundaries but also
reproduces the known spectrum as the extend of the domain is let go to
infinity.

Now we must confirm that the operator defined in this section can be used to
consistently construct a quantum theory of point particle motion undergoing
unitary time evolution, which is the focus of the next section.

##  3 Hamiltonian for a particle in a box

The defining properties of microscopic motion are encoded in the position
$\hat{x}$ and momentum operator $\hat{p}$, which fulfill the canonical
commutation relation

$\displaystyle[\hat{p},\hat{x}]=-i\hbar\hat{I}.$ (28)

From these fundamental building blocks a hermitean or at least ${\cal PT}$
symmetric Hamiltonian is constructed, which as part of the time evolution
operator generates unitary time evolution.

In our discrete setting the counterpart to $\hat{x}$ is $\mathds{X}={\rm
diag}[a,a+\Delta x,\ldots,b]$. Together with our definition of $\mathds{P}$,
we find that

$\displaystyle\Big{(}\mathds{P}\mathds{X}-\mathds{X}\mathds{P}\Big{)}{\bm{\psi}}$
$\displaystyle=-i\hbar(\psi_{1},\frac{1}{2}(\psi_{0}+\psi_{2}),\frac{1}{2}(\psi_{1}+\psi_{3}),\ldots,\psi_{N_{x}-1})$
(29)
$\displaystyle=-i\hbar(\psi_{0},\psi_{1},\psi_{2},\ldots,\psi_{N_{x}})+{\cal{O}}(\Delta
x).$ (30)

That is, the commutation relation between $\hat{x}$ and $\hat{p}$ is recovered
in the continuum limit. And while convergence overall is of linear order, one
converges to the correct result quadratically in the interior.

The operator $\mathds{P}$ features a purely real spectrum, even though is is
not described by a hermitean matrix $\mathds{P}^{\dagger}\neq\mathds{P}$. This
requires care when using our momentum operator to define the corresponding
point particle Hamiltonian. We suggest the following generalized construction

$\displaystyle\mathds{H}=\frac{\mathds{P}^{\dagger}\mathds{P}}{2m}+V(\mathds{X}),\qquad{\rm
where}\qquad\hat{H}=\frac{\hat{p}^{\dagger}\hat{p}}{2m}+V(\hat{x})=\lim_{\Delta
x\to 0}\mathds{H},$ (31)

which of course agrees with the conventional definition of the Hamiltonian in
case that the momentum operator is hermitean. Note that we only make reference
to the two operators $\mathds{X}$ and $\mathds{P}$ and do not need to
introduce a separate Hamiltonian in addition to these (c.f. Ref.
al2021alternative ).

The non-standard Hamiltonian of eq. 31, by construction, is hermitean and thus
leads to unitary time evolution.

###  3.1 Particle in the infinite well

Let us focus on a model system, in which the finite extend of the spatial
domain is vital: the infinite square well. This system is defined by the
potential function

$\displaystyle V(x)=\left\\{\begin{array}[]{lc}0&a<x<b\\\ \infty&{\rm
otherwise}\end{array}\right.\qquad{\rm approx.}\qquad V({\mathds{X}})={\rm
diag}[10^{7},0,\ldots,0,10^{7}],$ (34)

which in the discrete setting is approximated with large but finite values on
the boundary. In order to accommodate such a large dynamic range of values we
use in the following the high-precision arithmetic of the Mathematica
software, when investigating the spectral structure of $\mathds{H}$.

It is the presence of the potential function $V$, which enforces the
finiteness of the domain in this case. In an analytic computation, if we
require the wavefunction of our system to remain continuous, the presence of
an impenetrable potential wall forces us to adopt boundary conditions where
the wavefunction vanishes at the boundary. This continuous behavior is implied
when solving for the eigenvectors based on the Hamilton operator $\mathds{H}$
defined above. I.e. as the value of the approximate potential function $V$ are
increased the resulting eigenvectors approach the value zero on the boundary
more and more closely.

Figure 4: Examples of the two lowest lying physical (colored symbols) and
unphysical (gray cross) energy eigenfunctions of the Hamilton operator
$\mathds{H}$ built from our non-hermitean momentum operator $\mathds{P}$ on a
$N_{x}=32$ grid. The continuum ground state (red) and first excited (blue)
stationary state wavefunctions are provided as solid lines.

When inspecting the spectrum of $\mathds{H}$, we find several peculiar
properties. The first is that the spectrum $E_{n}$ is double degenerate,
similar to the behavior observed in al2021alternative . We can understand this
phenomenon from recent insight into the spectral structure of summation-by-
parts operators Rothkopf:2022zfb . The fact that $\mathds{D}$ is not anti-
symmetric means that its left and right eigenvectors are not necessarily the
same. And indeed while the right eigenvector with zero eigenvalue is the
physical constant zero mode, it turns out that the left eigenvector associated
with a zero eigenvalue is the maximally oscillating function that can be
resolved on a finite grid of $N_{x}$ points, the so called $\pi$-mode. The
occurrence of such artificial oscillatory zero-modes is well known from the
study of symmetric discretization schemes in lattice field theory and known
there as the doubler-problem.

And since $\mathds{H}$ contains both $\mathds{P}$ and $\mathds{P}^{\dagger}$
its eigenfunctions $\mathds{H}|\psi^{n}\rangle=E_{n}|\psi^{n}\rangle$ are
affected by the presence of such oscillatory modes. In fig. 4 we plot two of
the lowest lying physical stationary states for $N_{x}=32$. The ground state
as red circles and the first excited state as blue triangles. The continuum
solutions for this system are provided as colored solid lines. We find that
the physical stationary states are already well reproduced in the discrete
setting.

In addition to the physical modes, we also plot as gray crosses one of the
unphysical eigenfunctions, the one with the same energy eigenvalue as the
physical ground state. This eigenfunction is exactly the point-wise product of
the physical ground state wavefunction and the $\pi$-mode.

While in non-linear quantum field theory doubler-modes represent a formidable
challenge, in linear quantum mechanics they are less malign. Indeed, since
$\mathds{H}$ is hermitean, the physical and unphysical eigenstates are exactly
orthogonal. In turn the Hilbert space of stationary states decomposes into a
direct sum of a physical and unphysical Hilbert space. Importantly, unitary
time evolution will not mix between the two subspaces, i.e. if the initial
conditions of the time evolution are prepared from physical states, our
$\mathds{H}$ will produce only physical states at later time.

Figure 5: Deviation $\Delta E^{n}$ of the lowest four $n=1,\ldots,4$ energy
eigenvalues of $\mathds{H}$ from the continuum values of the infinite square
well for different lattice spacings $\Delta x$.

Let us collect numerical evidence that the unconventional construction of
$\mathds{H}$ reproduces the known results for the infinite well in the
continuum limit. To this end we plot in fig. 5 the difference $\Delta E^{n}$
between the eigenvalue of the n-th stationary state obtained numerically from
$\mathds{H}$ and the analytic continuum expression. As one sees, this
difference diminishes consistently under refinement of the grid $\Delta x\to
0$. The powerlaw dependence of $\Delta E^{n}\propto\Delta x^{\nu}$ with
$\nu\approx 1$ tells us that here the influence of the boundary remains
relevant, since it is only there that the $\mathds{P}$ converges linearly to
the continuum limit. (Note that since we use an approximate potential
function, at some point deviations from the continuum result for the infinite
potential well will appear as $\Delta x$ is made smaller than the values
considered here.)

We find that our $\mathds{H}$, constructed from $\mathds{P}$, reproduces,
within the physical subspace of its Hilbert space, the correct stationary
states in the continuum limit. This further supports $\mathds{P}$ as candidate
for an appropriate momentum operator for the particle in a finite box.

###  3.2 Momentum and Energy in the finite domain

We now turn to a discussion of momentum measurement in the infinite potential
well, carried out on a initially prepared stationary state.

Since our momentum operator is not normal, due to its structure at the
boundary, its eigenfunctions are not mutually orthogonal nor complete. While
one can always orthogonalize the set of eigenvectors, the fact that the sum
over their outer product does not reproduce the identity operator means that
$\mathds{P}$ does not lend itself to the conventional interpretation of a von
Neumann projective measurement. I.e. a stationary state cannot be uniquely
decomposed into the set of eigenstates of the momentum operator. Can we still
make predictions about momentum measurements using $\mathds{P}$?

As alluded to in the introduction, we may either make predictions on the
outcome of experiments by citing the probabilities of individual measurements,
or we may refer to the set of n-point functions of the operator encoding the
observable of interest. Each provide us with complementary insight on the
probabilistic nature of measurement.

Note however that only expectation values are deterministic predictions of
quantum mechanics. I.e. what can in principle be predicted without uncertainty
is the mean of outcomes of a large number of experiments carried out on an
ensemble of identically prepared states, as the number of measurements is
taken to infinity. Thus knowledge of the n-point functions of the momentum
operator constitutes all possible deterministic predictions quantum mechanics
can make about momentum of the point particle.

Interestingly the evaluation of $\langle\psi|\hat{p}|\psi\rangle$ only makes
reference to the local properties of the wavefunction ${\psi}$, as the
momentum operator in position space is a derivative. Since we are guaranteed
that $\mathds{D}$ converges to the correct derivative $d/dx$ in the continuum
limit, we expect that we are able to reproduce the continuum momentum
expectations values by use of $\mathds{P}$.

Figure 6: Diminishing of $\Delta p^{2}$ the deviation of the momentum two-
point function from its continuum value evaluated in the lowest four energy
eigenstates $n=1,\ldots,4$. Figure 7: Diminishing of $\Delta p^{4}$ the
deviation of the momentum four-point function from its continuum value
evaluated in the lowest four energy eigenstates $n=1,\ldots,4$.

And indeed, as shown in fig. 6 and fig. 7, we find numerical evidence that the
correct n-point functions ensue. In fig. 6 we plot the difference $\Delta
p^{2}=\langle\mathds{P}^{2}\rangle-\langle\hat{p}^{2}\rangle$ between the
value of $\langle\mathds{P}^{2}\rangle$ in the four lowest lying stationary
states (ground state data as open circles) obtained from $\mathds{H}$ and the
continuum expectation value $\langle\hat{p}^{2}\rangle$ for the infinite well.
The same comparison is made in fig. 7 for the four-point function $\Delta
p^{4}=\langle\mathds{P}^{4}\rangle-\langle\hat{p}^{4}\rangle$.

While the absolute differences on coarse grids may at first sight appear
relatively large, they consistently diminish towards zero as the grid is
refined $\Delta x\to 0$. We find for both $\langle\Delta p^{2}\rangle$ and
$\langle\Delta p^{4}\rangle$ a powerlaw $\propto\Delta x^{\nu}$ with
$\nu\approx 1$. The odd n-point function are, as required by symmetry, exactly
zero.

The lowest four n-point functions contain a wealth of phenomenologically
relevant information about the observable of interest and figs. 6 and 7
suggest that we are able to reproduce them reliably in our setup based on
$\mathds{P}$ and $\mathds{H}$.

There is another unusual feature in the relation of the momentum operator and
the Hamiltonian in a finite domain: they do not commute. In al2021alternative
this fact has been interpreted in the context of the von Neumann projective
measurement. In that context a measurement of $\hat{p}$ will project the
stationary state of the well into one of its eigenstates. These eigenstates
however are not simultaneous eigenstates of the Hamiltonian and actually
constitute linear combinations from both physical and unphysical parts of the
Hilbert space. Since the unphysical states are associated with infinite energy
solutions, the momentum measurement was interpreted to endow the point
particle with an infinite amount of energy.

As we do not have a similar interpretation of the measurement as a von Neumann
projective measurement in our setup, we would like to provide a different
interpretation of the fact that $[\mathds{H},\mathds{P}]\neq 0$. Making only
reference to the operators themselves, we may inspect Heisenberg’s equations
of motion, which states

$\displaystyle\frac{d}{dt}\mathds{P}=\frac{1}{i\hbar}[\mathds{H},\mathds{P}]\neq
0.$ (35)

In an unrestricted domain, where the system is spatially translationally
invariant, momentum must be preserved, i.e. the time derivative of
$\mathds{P}$ must vanish. This is the case for the conventional momentum
operator and the Hamiltonian in the absence of a potential. In the presence of
impenetrable walls, or more generally in the presence of a $\hat{x}$ dependent
potential term, translational invariance is lost. An encounter with the walls
of the well reverses the momentum of the particle. For an infinite well this
finite change in momentum happens instantaneously.

Thus in Heisenberg’s equation of motion the commutator between $\mathds{P}$
and $\mathds{H}$ necessarily must lead to a nonzero value and since we are
considering the infinite well it actually must lead to an infinite value due
to the instantaneous nature of the reversal of momentum at the boundary. Thus
we consider the non-commutativity of momentum and Hamiltonian as a necessary
consequence of the absence of translation symmetry.

##  4 Summary

We have presented a concrete realization of a momentum operator for a point
particle confined to a finite domain. Starting from a discretized setting, we
use a summation-by-parts finite difference operator $\mathds{D}$ to construct
a corresponding momentum operator $\mathds{P}$, which mimics exactly the
boundary effects associated with integration by parts in the inner product of
wavefunctions.

We investigated the spectral properties of this momentum operator confirming
that its eigenfunctions are indeed confined to the interior of the domain in
the continuum limit, where no residual boundary leakage occurs. To connect to
the known spectrum of the momentum operator on the infinite domain, we
discussed how the absence of genuine boundaries leads to an eigenvalue
relation that produces the correct plane wave solutions.

The fact that $\mathds{P}$ is described by a non-hermitean matrix, required us
to resort to a non-standard construction of the Hamiltonian for the point
particle involving $\mathds{P}$ and $\mathds{P}^{\dagger}$. We showed that the
spectrum of the Hamiltonian contains the physical stationary states and we
reproduce the correct energies of the infinite well in the continuum limit.

The symmetric stencil used in the interior of the operator $\mathds{D}$ leads
to the occurrence of oscillatory doubler-modes in the spectrum of the
Hamiltonian. These however are orthogonal to the physical states and unitary
time evolution does not mix between the two sectors.

Finally we confirmed that the four lowest n-point functions of the operator
$\mathds{P}$ evaluated in the low-lying eigenstates of $\mathds{H}$ reproduce
the continuum values as the lattice spacing is reduced $\Delta x\to 0$.

And while relevant statistical information about momentum measurement is
contained within the correctly reproduced n-point functions we cannot use
$\mathds{P}$ to make statements about individual projective measurements, as
it is not normal, and thus it cannot be uniquely decomposed in projection
operators made from its eigenvectors.

Last but not least, we relate the fact that Hamiltonian and momentum do not
commute to the loss of translation symmetry and the encounter of the point
particle with the impenetrable wall, which leads to an instantaneous reversal
of its momentum.

###  4.1 Further venues for investigation

The spectrum of the momentum operator $\mathds{P}$ differs from previously
suggested candidate operators in that the first non-trivial eigenfunction is
the quarter wave. It would be interesting to see, whether different momentum
concepts can be distinguished in experiment according to the lowest possible
momentum value to be observed. It has to be kept in mind though that (at
least) on the level of the lowest four expectation values, $\mathds{P}$ makes
exactly the same predictions as the continuum $\hat{p}$.

One may wish to also revisit the question of how to consistently combine
solutions of the time independent Schrödinger equation if they are considered
to arise in separated domains connected by interfaces, similar to the study in
al2021canonical . The solution of the finite well, e.g. proceeds by dividing
space up into three domains, in each of which one would have to deploy a
momentum operator similar to $\mathds{P}$, adapted to the existence of a
genuine boundary.

In addition it remains to be determined whether and how a path integral could
be formulated based on the concept of a non-hermitean momentum operator. This
question is directly related to the use of summation-by-parts finite
difference operators for the discretization of the system action in the
functional approach to quantum mechanics and quantum field theory.

We believe that our proposed candidate for a momentum operator on a finite
domain $\mathds{P}$ together with a non-standard construction of an associated
hermitean Hamiltonian $\mathds{H}$ provides a new viewpoint to this active
field of research and hope that it will contribute to the eventual resolution
of this fundamental question in quantum mechanics.

## Acknowledgements

A. R. acknowledges fruitful discussions with David Ploog. This research
project was initiated as part of the Korea-Nordic R&D networking grant
2022K2A9A2A23000252. S. K is supported by the National Research Foundation of
Korea, under grant NRF-2021R1A2C1092701.

## Competing interests

The authors declare that they have no competing interests.

## Author’s contributions

  * •

S. Kim: literature review, infinite volume limit, discussions, editing, grant
management

  * •

A. Rothkopf: construction of the momentum operator, computation of spectrum,
numerical implementation, writing, editing

## References

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  * (2) Seba, P.: Wave chaos in singular quantum billiard. Phys. Rev. Lett. 64, 1855–1858 (1990). doi:10.1103/PhysRevLett.64.1855
  * (3) Al-Hashimi, M., Wiese, U.-J.: Alternative momentum concept for a quantum mechanical particle in a box. Physical review research 3(4), 042008 (2021)
  * (4) Freericks, J.: How to measure the momentum of single quanta. arXiv preprint arXiv:2302.12303 (2023)
  * (5) Shohat, J.A., Tamarkin, J.D.: The Problem of Moments vol. 1. American Mathematical Society (RI), ??? (1950)
  * (6) Svärd, M., Nordström, J.: Review of summation-by-parts schemes for initial–boundary-value problems. Journal of Computational Physics 268, 17–38 (2014)
  * (7) Fernández, D.C.D.R., Hicken, J.E., Zingg, D.W.: Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations. Computers & Fluids 95, 171–196 (2014)
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  * (9) Ruggiu, A.A., Nordström, J.: Eigenvalue analysis for summation-by-parts finite difference time discretizations. SIAM Journal on Numerical Analysis 58(2), 907–928 (2020)
  * (10) Rothkopf, A.: Accompanying computations for ”A non-hermitean momentum operator for the particle in a box”. Zenodo (2024). doi:10.5281/zenodo.10843823. https://doi.org/10.5281/zenodo.10843823
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in measuring the neutron star mass-radius relation by LIGO-Virgo with
gravitational waves, and with NASA’s X-ray telescope NICER; the results are
challenging our understanding of QCD. The list goes on and on.

Reciprocally, particle physics gives invaluable contributions to Astrophysics
and Cosmology. Two of the three themes prioritized by the 2020 Astro Decadal
Survey are strongly aligned with the goals articulated in the Cosmic Frontier:
“New Messengers and New Physics”, with particular stress on multi-messenger
astrophysics which will dramatically enhance the utility of cosmic particle
and gravitational wave probes, as well as “Cosmic Ecosystems”, aimed at
improving our understanding of the astrophysical aspects of the evolution of
the universe.

The underlying scientific motivation to study common topics is often different
in the HEP and astrophysics communities, with astronomers interested for
instance in questions like understanding the properties of galaxies and how
they have evolved, and the menagerie of objects they contain, or the nature of
the sources of the highest energy cosmic rays, whereas particle physicists are
more focused on how the nature of the dark matter particle can be inferred
from features imprinted on astrophysical systems, or distinguishing a
potential dark matter annihilation signal from astrophysical background, or on
using ultrahigh-energy cosmic particles to probe BSM physics far beyond
accelerator energies. Since an essential part of discovering new phenomena
through astrophysics and cosmology is understanding the baseline expectations
of the standard model of cosmology and the many other aspects of astrophysics
that produce foregrounds and backgrounds to some possible BSM signal,
physicists and astrophysicists have a common purpose in these explorations.

Technologically, there is also a strong synergy between particle physics and
astrophysics. A major triumph of the past decade in astrophysics was the
discovery of gravitational waves by Advanced LIGO, largely funded by the NSF,
which made essential use of developments spearheaded by HEP in vacuum systems,
alignment challenges, distributed control, and data acquisition systems. A
proposed future much larger gravitational wave observatory would further
benefit from HEP expertise in km-scale facility placement, design, and
execution, as well as the operation of multi-billion-dollar facilities. The
scientific rewards of such an instrument for particle physics span from an
independent, accurate measurement of the Hubble constant, to the determination
of the QCD equation of state in neutron stars, to strong constraints on
theories of modified gravity, to signatures of phase transitions in the Early
Universe, and much more. Another triumph has been the detection of ultra-high-
energy astrophysical neutrinos with the IceCube detector at the South Pole.

Presently coming to fruition, the Rubin Observatory was the highest priority
ground-based project of the 2010 Astro Decadal survey. Among its goals are
mapping the structure of the Universe and its expansion history in much
greater detail and precision than at present, which is of key importance for
particle physics because of what it can tell us about inflation and dark
energy, and possible tensions within $\Lambda$CDM - as well as a multitude of
applications in high energy and other parts of astrophysics. Its remarkable
camera, LSSTCam, has been built at SLAC with DOE funding and expertise.
LSSTCam presented a number of technical challenges that had not been
encountered with other astronomical cameras: large modular arrays of 4-side
buttable detectors with extremely tight positional tolerances; fast readout
necessitating a high degree of parallelization with several thousand channels
of readout electronics; finally, the positioning of the camera within the beam
exiting the secondary mirror implied a severely constrained envelope to work
in, with tight thermal requirements. These features are more characteristic of
the inner detectors associated with collider experiments than they are of
typical ground-based astronomical instrumentation. The team assembled to
design and build the camera (largely at SLAC, BNL, and the IN2P3 labs in
France) mostly involved technical people from the HEP community who had
experience with such issues.

A current example of astro-EPP technical synergy is in building CMB-S4, one of
the key projects identified by the last P5, which will have unprecedented
sensitivity to the physics of the Early Universe. The CMB-S4 transition-edge-
sensor bolometric detectors, the superconducting circuity to make them dual-
polarization and dual-band sensitive, and the low-noise SQUID-based
multiplexed readout were all advanced by DOE-funded research. The laboratories
involved include SLAC, LBNL, ANL, and FNAL. Another crucial HEP contribution
has been the capability of LBNL NERSC computing to support the large data sets
and the computation of the large sets of simulations needed to properly
conduct the analysis.

These examples hardly scratch the surface of the multitude of ways that prior
and current investments in particle physics R&D provide tools and expertise
for astrophysics and other areas of science.

##### High Energy Physics and Nuclear Physics

The Nuclear Physics community is currently developing a new Long Range Plan
(LRP) for Nuclear Science, aimed to be released in 2023. The fact that it
coincides with the Snowmass and P5 process is an opportunity to further
develop synergies, define common strategies when necessary, and maximize the
scientific and societal impact of both fields.

Heavy ion measurements at the LHC are performed with the ATLAS and CMS
detectors, as well as a dedicated detector ALICE. This complementarity and
collaboration between the scientists working on high-energy colliders will
grow with the advent of the Electron-Ion Collider (EIC). EIC, the top priority
new facility construction in the 2015 Long Range Plan for Nuclear Science,
will be the premier facility for precision studies of nuclear and nucleon
structure, including the polarization and angular momentum observables, and
the transition between the perturbative and non-perturbative regimes of QCD.
The fundamental interest in the understanding of quantum chromodynamics at all
regimes is a common thread between HEP and NP. Precision studies of the QCD,
measurements of the parton distribution functions with high precision, and
computational techniques that would be tested by the EIC could inform future
measurements at the Energy Frontier. Conversely, nuclear collider experiments,
including at RHIC and at the EIC, benefit from the tools and techniques
developed by HEP, in particular jet reconstruction, precision tracking, heavy
flavor tagging and reconstruction, calorimetry, particle ID, and others.

The overlap of scientific interest between nuclear and high-energy physics is
probably greatest in the Neutrino Frontier. Snowmass has discussed in detail
the many overlapping interests of neutrino scientists, and the desire of the
scientific community to find ways to enhance the collaboration in neutrino
physics while respecting the specific interests of the various stakeholders.
The most pressing questions – the fundamental nature of the neutrino and the
violation (or conservation) of lepton number, the number of neutrino species,
the absolute scale of the neutrino mass and its fundamental electromagnetic
properties, as well as precision studies of the ways neutrinos interact with
matter – are of great interest to both communities. In fact, studies of
neutrino oscillations are performed by both communities and supported by both
OHEP and ONP, depending on the neutrino sources; some experiments, including
SNO and KamLAND, have been supported by both OHEP and ONP. These are just a
few examples of constructive and fruitful collaboration between the two
fields.

Neutrinoless double beta ($0\nu\beta\beta$) decay remains an important topic
for both HEP and NP. The next-generation (“ton-scale”) experiments will be
stewarded by the Office of Nuclear Physics, as recommended by the 2015 LRP. It
is also well recognized by the community and the funding agencies that the
quest for, or perhaps precision measurements of, $0\nu\beta\beta$ decay must
continue beyond these experiments, independently of whether the discovery is
made at the ton scale. The Snowmass process has showcased a plethora of novel
ideas and experimental techniques aimed at developing experiments with
sensitivity beyond the parameter space defined by the inverted ordering of
light neutrino masses. At this scale, it would be especially important to
develop multi-purpose detectors and maximize the scientific return on
investment. Synergies in technologies between $0\nu\beta\beta$ and dark matter
experiments, $0\nu\beta\beta$ and CE$\nu$NS experiments, as well as
$0\nu\beta\beta$ and the long-baseline neutrino program, were discussed at
Snowmass. Deployment of such multi-purpose detectors may require cooperation
between multiple communities and funding agencies.

Measurements of other neutrino properties such as the absolute neutrino mass
or the neutrino magnetic moment, and new technological developments that could
be pursued by both fields to enhance sensitivity, have also been discussed at
Snowmass. Searches for sterile neutrinos – with different sources and
detection techniques but in overlapping parameter space – are done by both HEP
and NP. Measurements of neutrino cross sections, and of other processes that
could improve the precision of neutrino reconstruction at DUNE, are examples
where NP experiments could provide crucial input to a flagship HEP program.
Examples of such HEP/NP collaborative experiments include electron-nucleus
scattering experiments at Jefferson Lab, as well as neutrino scattering
experiments at Oak Ridge National Laboratory’s Spallation Neutron Source (a
DOE Basic Energy Sciences facility), which produces very high-quality, high-
intensity neutrinos.

Precision tests of the Standard Model is another area where nuclear and high
energy physicists complement each other, and in some cases directly
collaborate. The Muon $g-2$ experiment is a great example of a project funded
primarily by HEP and operated at a HEP facility in which NP-funded scientists
played a crucial role, brought specific expertise, and made substantial
contributions. Precision measurements of electroweak couplings of electrons
and quarks through parity-violating electron scattering at Jefferson Lab and
at the EIC will provide an important window to search for new physics and will
complement electroweak observables at the LHC and at the $Z$ pole.
Measurements of the CKM parameters, in particular those connected to the first
generation, are often done in nuclear processes and are of great significance
for the tests of the unitarity of the CKM matrix. Searches for Lepton Number
and Baryon Number violation, EDMs, tests of lepton universality, and searches
for dark sector bosons are other areas of joint interest. Some of the
experiments with HEP funding are performed at NP-funded facilities (e.g. dark
sector searches at Jefferson Lab) and therefore require collaboration between
the offices. Searches for EDMs, enhanced by the advent of new atomic
techniques and the use of molecules and deformed, often unstable, nuclei could
be performed at nuclear facilities, such as FRIB. Conversely, the Advanced
Muon Facility proposed at Fermilab could enable measurements that would
benefit nuclear physics experiments directly, e.g. through precision
measurements of muon capture.

The U.S. Nuclear Physics community (in particular the DOE Office of Nuclear
Physics) operates multiple accelerator facilities. Among the larger ones are
the Relativistic Heavy Ion Collider (RHIC) at BNL, the Continuous Electron
Beam Accelerator Facility (CEBAF) at Jefferson Lab, the Facility for Rare
Isotope Beams (FRIB) at Michigan State University, and a score of smaller
accelerator facilities at national labs and universities. The highest priority
for new facility construction in the 2015 LRP is the Electron-Ion Collider
(EIC), to be built and operated at BNL by the beginning of the next decade.
This facility is the only particle collider expected to be operated in the
U.S. in the 2030s. It will require significant innovations in accelerator
physics and technology and will employ a substantial fraction of the
accelerator physics workforce in the U.S. The developments stimulated by HEP –
high-field magnets, accelerating structures, power delivery systems, polarized
sources, final focus systems, beam instrumentation, and precision control of
the beam dynamics – will be deployed, further developed, and thoroughly tested
by the EIC project. EIC will play an essential role in recruiting, training,
and supporting the future accelerator frontier workforce. This workforce will
be essential for any U.S. involvement in future large-scale accelerator
projects for HEP and other fields.

###  6 Summary

The aspirations of the U.S. HEP community as informed by the Snowmass 2021
process can be succinctly summarized as follows:

> Lead the exploration of the fundamental nature of matter, energy, space and
> time, by using ground-breaking theoretical, observational, and experimental
> methods; developing state-of-the-art technology for fundamental science and
> for the benefit of society; training and employing a diverse and world-class
> workforce of physicists, engineers, technicians, and computer scientists
> from universities and laboratories across the nation; collaborating closely
> with our global partners and with colleagues in adjacent areas of science;
> and probing the boundaries of the Standard Model of particle physics to
> illuminate the exciting terrain beyond, and to address the deepest mysteries
> in the Universe.

This volume presents the main conclusions of Snowmass 2021, whose goal was to
examine the status and scientific goals of U.S. HEP and to propose essential
scientific programs to pursue, and to thereby provide community input to the
2023 DOE/NSF “Particle Physics Project Prioritization Panel” (P5) charged with
updating the strategic plan for U.S. HEP for the coming decade and beyond. The
work included herein was the output of ten Snowmass Frontiers and their
Topical Groups, which comprised a broad array of ground-breaking scientific
research topics and the underlying technology and infrastructure needed to
execute them, as well as a forum to examine how the U.S. HEP community can
become more representative of and responsive to all members of our community
and can engage with society as a whole.

The consensus of the Snowmass 2021 community was that the five science Drivers
formulated by the 2014 P5 panel continue to be appropriate for the next
decade. Specifically, the questions that should guide the planning for the
next decade and beyond include:

  1. 1.

Use the Higgs Boson as a Tool for Discovery,

  2. 2.

Pursue the Physics Associated with Neutrino Mass,

  3. 3.

Identify the New Physics of Dark Matter,

  4. 4.

Understand Cosmic Acceleration: Dark Energy and Inflation,

  5. 5.

Explore the Unknown: New Particles, Interactions, and Physical Principals.

As a result of the breadth of flavor physics, its potential for discovering
BSM physics, and hints of possible new physics in the current data, the Rare
Processes and Precision Measurement Frontier proposes that the upcoming P5
adds a new science Driver: flavor physics as a tool for discovery.

For the future U.S. HEP experimental program, an important general conclusion
of Snowmass was that the completion, construction, and operation of currently
approved projects, especially those prioritized by P5 in 2014, such as the HL-
LHC, LBNF/DUNE, LSST/Rubin Observatory, and Mu2e programs, is essential.
Furthermore, as the construction of approved large experimental projects come
to a completion, U.S. leadership in a broad and complementary set of new large
experimental programs covering the general scientific areas of the Energy,
Neutrino, Cosmic, and Rare Process and Precision Measurements Frontiers is
needed to provide opportunities for incisive new discoveries. The specific
projects considered are listed in section 1 of this document (without regard
to cost or priority, since that is to be done by P5), and described in greater
detail in the Frontier Reports in this volume as well as in the Topical Group
Reports and white paper references contained therein. In addition, the
inclusion in the U.S. HEP experimental portfolio of a broad and complementary
set of mid- and small-scale experiments is vital to maintaining the diversity
of physics topics investigated and of timescales to completion that are
essential for a vibrant program. Medium- and smaller-scale projects are
discussed more fully in section 2 and in the reports in this volume.

A healthy HEP program requires more than projects, however, and the Snowmass
community emphasized the need for robust support for physics research programs
at universities and national laboratories. This consideration includes the
strong and continued support for all aspects of particle theory, comprising
the interconnected themes of fundamental theory, phenomenology, and
computational theory. Also needed is greater support for the infrastructure
and enabling technologies, namely accelerators, instrumentation and detectors,
and computation — both targeted support for specific future projects and
generic support to develop new enabling technologies.

Input received during the Snowmass process highlighted that HEP projects and
research benefit from collaboration with adjacent scientific disciplines, such
as Nuclear Physics, Astronomy and Astrophysics, Gravitational Physics, Atomic
and Molecular Physics, Quantum Information Science and Sensing, and Materials
Science, and contributes to them in return. Opportunities to strengthen and
expand such collaborations are mutually beneficial and should be pursued.

Of particular importance in considering the future of the U.S. HEP program is
to understand and support the needs of its early career physicists. Snowmass
participants recommended strengthening the connections between early career
and senior researchers, fostering the professional success of early career
physicists both within and outside of academia, and ensuring that the voices
of younger scientists are well-represented in physics and community planning.

Mindful of the continuing lack of diversity in particle physics, a strong
consensus exists in the HEP community demanding the development of a cohesive,
strategic approach to promoting diversity, equity, and inclusion in high-
energy physics, in collaboration with funding agencies, universities, adjacent
scientific disciplines, APS/DPF and others. Specifically, the community should
institute a broad array of practices and programs to reach and retain the
diverse talent pool needed for success in achieving our scientific vision and
to address the persistent under-representation of women scientists, LGBT+
scientists, scientists who are Black, Indigenous, and people of color (BIPOC),
and scientists with disabilities.

The U.S. HEP community is generously supported by and embedded in the nation
as a whole. The HEP community must therefore engage in a coordinated way with
five other interrelated communities: academia, the K-postdoc education
community, private industry, government policymakers, and broader society.
Snowmass participants identified the need for a structure for formulating a
coordinated approach to achieve these goals which is provided with the
resources needed for success.

The Snowmass 2021 process highlighted the breadth, depth, and effectiveness of
the U.S. HEP program in exploring the fundamental nature of matter, energy,
space, and time. The remarkable progress that has been made in probing the
boundaries of the Standard Model of particle physics since 2013 was analyzed,
and different and complementary methods to move forward in uncovering the
mysteries of nature were examined. Highlights and opportunities that were
identified are described in detail in the reports in this volume, as well as
in the numerous topical group reports and white papers summarizing the work
done for Snowmass. The members of the U.S. high-energy physics community left
the Seattle Community Summer Study workshop with an appreciation of the great
opportunities present in each Frontier, the interconnections between the
Frontiers, and the connections to programs in the rest of the world. Through
the Snowmass process, the U.S. HEP community has created an integrated vision
to make progress in the coming decades to advance our understanding of the
Universe.

### Acknowledgements

Several thousand people participated in the Snowmass 2021 process and provided
their scientific input. Their enthusiasm and dedication is what made Snowmass
2021 a success! We also acknowledge the crucial contributions of the host
institutions of all the virtual and in-person preparatory meetings. The
Snowmass Steering Group, Advisory Group, and Frontier Conveners Group provided
overall coordination of Snowmass 2021. The Community Summer Study and Workshop
was held at the University of Washington in Seattle. We thank the University
and the CSS Scientific Program Committee and Local Organizing Committee for
producing a well-organized and successful ten-day-long meeting. The names of
the members of all these groups are listed in the acknowledgments at the end
of the full report. Finally, we thank the four units of the APS whose work is
closely related to HEP, namely Astrophysics, Nuclear Physics, Gravitational
Physics, and Physics of Beams, for their many contributions.

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