Special Issue in honor of John W. Neuberger

Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 255–267.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

NONLOCAL ADVECTION DIFFUSION EQUATIONS AND THE

TWO-SLIT EXPERIMENT IN QUANTUM MECHANICS

GLENN WEBB

Dedicated to the memory of John W. Neuberger

Abstract. We analyze a partial differential equation that models the two-

slit experiment of quantum mechanics. The state variable of the equation

is the probability density function of particle positions. The equation has
a diffusion term corresponding to the random movement of particles, and a

nonlocal advection term corresponding to the movement of particles in the

transverse direction perpendicular to their forward movement. The model is
compared to the Schrödinger equation model of the experiment. The model

supports the ensemble interpretation of quantum mechanics.

1. Introduction

The two-slit experiment demonstrates the fundamental probabilistic nature of
quantum mechanics. In this experiment quantum particles are projected forward
toward a screen with two parallel slits, and then observed on a detection screen
further downstream (Figure 1). An interference diffraction pattern of regularly
spaced intensities is registered on the detection screen. The highest. density occurs
in the center of the detection screen (Figure 2), which is not the sum of the patterns
observed for single slits separately [8]. The observed fringe pattern for two slits is
characteristic of wave phenomena.

In previous work we have investigated mathematical models for the two-slit ex-
periment [9, 10]: the Schrödinger equation model (SE) and the nonlocal advection
diffusion equation model (NLAD). In this work we extend these investigations to
account for higher levels in the fringe patterns images in the experiment. We nu-
merically simulate the SE and NLAD model outputs and compare the simulations
to experimental data.

The organization of this article is as follows: In Section 2 we develop the SE
model, in Section 3 we develop the NLAD model, in Section 4 we provide numerical

2020 Mathematics Subject Classification. 35J10, 35Q40.
Key words and phrases. Nonlocal; advection; diffusion; Schrödinger equation;

two-slit experiment; ensemble interpretation.
©2023 This work is licensed under a CC BY 4.0 license.

Published March 27, 2023.

255



256 G. WEBB EJDE/SI/02

Figure 1. The two slit experiment with electrons.
wikipedia.org/wiki/Double-slit experiment

Figure 2. The interference diffraction pattern of a two slit exper-
iment with high order fringe patterns. wikipedia.org/wiki/Double-
slit experiment

simulations of both models and compare these outputs to experimental images, and
in Section 5 we provide a summary of out work.

2. Schrödinger equation model

The one-dimensional time-dependent complex-valued Schrödinger equation is the
foundational phenomenological model of quantum mechanics:

∂

∂t
ψ(x, t) = i

~
2m

∂2

∂x2
ψ(x, t), t > 0, ψ(x, 0) = ψ0(x), −∞ < x <∞. (2.1)

Here ~ is the reduced Planck’s constant and m is the particle mass, which without
loss of generality can be assumed to satisfy ~/m = 1. The interpretation of the
solution is that

∫ x2

x1
ρ(x, t) dx is the probability of finding a single particle in the

interval (x1, x2) at time t, where ρ(x, t) = |Re ψ(x, t)|2 + | Imψ(x, t)|2, and ρ(x, 0)
is normalized so that

∫∞
−∞ ρ(x, 0) dx = 1. In this formulation, the interpretation of

the state variable ψ at time t is sometimes applied to a single individual quantum
particle. For this interpretation, the following questions arise: What do the real
and imaginary parts of ψ represent for an individual particle? What is time t in
an experiment with randomly separated independent temporal events? What does
the initial condition ψ0 correspond to for single particles emitted one at a time?

The general solution of (2.1) (with scaling ~/2m = 1), is as in [2]:

ψ(x, t) =
1√
2πit

∫ ∞
−∞

e
i(x−y)2

2t ψ0(y)dy, ψ0 ∈ L1((R;C), y2dy) ∩ L2(R;C), (2.2)



EJDE-2023/SI/02 NONLOCAL ADVECTION DIFFUSION EQUATION 257

where

ρ(x, t)

=
1

2πt

((∫ ∞
−∞

cos
( (x− y)2

2t

)
Reψ0(y)− sin

( (x− y)2

2t

)
Imψ0(y)dy

)2
+
(∫ ∞
−∞

sin
( (x− y)2

2t

)
Reψ0(y)dy + cos

( (x− y)2

2t

)
Imψ0(y)dy

)2)
,

(2.3)

with ∫ ∞
−∞

ρ(x, 0) dx =

∫ ∞
−∞
|ψ(x, 0)|2dx = 1,

which implies
∫∞
−∞ ρ(x, t) dx = 1 for t ≥ 0.

The probability amplitude ρ(x, t) in (2.3) exhibits a two-phase pattern as t ad-
vances. In the first phase, the initial information ρ(x, 0) evolves to an established
pattern, in which the lower peaks in the fringe pattern lie almost on the x-axis. In
the second phase, this established pattern undergoes a space-time dilation as time
advances. In the second phase the profile of ρ(x, t) is propagated in the spatial
x-direction at a constant speed. In [9] it is proved that for general initial data ψ0 in
(2.3) the probability amplitude ρ(x, t) in (2.3) satisfies the asymptotic space-time
dilation property: uniformly for x ∈ R, T > 0, t ≥ 1,

∣∣ρ(x, tT )− 1

t
ρ(
x

t
, T )
∣∣ ≤ √

2

πtT 2

(∫ ∞
−∞

y2|ψ0(y)|dy
)(∫ ∞

−∞
|ψ0(y)|dy

)
, (2.4)

which implies

lim
t→∞

tρ(x, t) = lim
T→∞

tTρ(x, tT ) =
1

2π

∣∣∣ ∫ ∞
−∞

ψ0(y)dy
∣∣∣2; (2.5)

uniformly for x ∈ R.

2.1. Schrödinger equation with step function initial data. In [10] an exam-
ple for the Schrödinger equation applied to the two-slit experiment was given with
initial data consisting of two rectangular step functions, symmetric about the origin
of the x-axis. In this example, the centers of the two rectangular strips are taken
as s = ±1, which can be applied generally by scaling the spatial variable x. The
probability density ρ(x, t) for this example evolves from the first phase to the sec-
ond phase. In Figure 3, ρ(x, t) is illustrated at t = 1/π, which is approximately the
value of t at which the transition from the first phase to the second phase occurs.

Remark 2.1. In Figure 3 it is seen that the fringe pattern has multiple interference
pattern groupings on either side of a central principal pattern. The diffraction
pattern of ρ(x, t) has infinitely many regularly spaced intervals

. . . , [−30,−20], [−20,−10], [−10, 10], [10, 20], [20, 30], . . . ,

with approximately regularly spaced interior fringes repeated in each of the inter-
vals. This illustration is consistent with the experimental data in Figure 2. The
spacing is determined by s/b = 10.



258 G. WEBB EJDE/SI/02

-30 -20 -10 10 20 30
x

0.05

0.10

0.15

0.20

-1.5-1.0-0.5 0.5 1.0 1.5
x

0.5

1.0

1.5

2.0

2.5

-30 -20 -10 10 20 30
x

10-6

10-5

10-4

0.001

0.010

0.100

Figure 3. ρ(x, 1/π) for ρ(x, (0) = two rectangles with cen-
ters s = ±1, width 2b, b = 0.1, and height scaled so that∫ −s+b
−s−b ρ(x, 0) dx +

∫ s+b
s−b ρ(x, 0) dx = 1. Left: ρ(x, 1/π), the in-

set is the graph of ρ(x, 0). The coordinates x = ±10,±20,±30, . . .
are local maxima for ρ(x, 1/π) and x ≈ ±15,±25,±35, . . . are lo-
cal minima for ρ(x, 1/π). Right: log ρ(x, 1/π).

3. Nonlocal advection diffusion equation model

An alternative to the Schrödinger equation formulation of the two slit experiment
is the nonlocal advection diffusion equation formulation NLAD. This formulation
incorporates the movement at a given spatial x coordinate as dependent on nearby
spatial x coordinates. Such models have been developed in [1] and [10]. The NLAD
model supports the ensemble interpretation of quantum mechanics, which maintains
that mathematical description of particle behavior should correspond to communal
particle behavior, rather than to individual particle behavior [3].

In [1] the authors examine an interpretation of the two-slit experiment based
on the nonlocal interaction of a single particle with both slit openings. In their
interpretation a single particle only passes through one slit, but with its corpuscular
motion affected by the other slit. The authors provide a deterministic nonlocal
dynamic equation of motion for the density of particles and relate the behavior
of the solutions to the interference patterns observed in the two-slit experiment.
This interpretation provides an alternative to the classical Schrödinger equation
interpretation of this experiment as an ensemble wave of particles passing through
both slits.

In the NLAD model in [10], the nonlocal advection term represents directed
particle movement due to the influence of nearby particles, and the diffusion term
represents variability of particle movement due to stochastic variation. The nonlocal
advection term involves an integral corresponding to the slit separation widths s in
the ±x-directions. In [10], the NLAD equation analyzed is the following:

∂

∂t
ω(x, t) = α

∂2

∂x2
ω(x, t) +

∂

∂x

∫ s

−s
β0ω(x+ x̂, t)

x̂

|x̂|
dx̂

= α
∂2

∂x2
ω(x, t) + β0

(
ω(x+ s, t)− 2ω(x, t) + ω(x− s, t)

)
,

t > 0, −∞ < x <∞,

(3.1)

ω(x, 0) = ω0(x), −∞ < x <∞, ω0 ∈ L1
+(−∞,∞),

∫ ∞
−∞

ω0(x) dx = 1. (3.2)



EJDE-2023/SI/02 NONLOCAL ADVECTION DIFFUSION EQUATION 259

In equation (3.1) x is the spatial coordinate of particles, t is the downstream
distance perpendicular to the slits openings, α is the diffusion parameter, β0 is
the advection parameter, 2s is the slit width, ω0(x) is the initial data, and ω(x, t)
is the probability density function for the distribution of particle positions. The
values of α and β0 are chosen so that the solutions of (2.1) and (3.1) are similar for
x ∈ [−20, 20] at t = 1/π.

In this work we analyze an extension of equation (3.2) to include higher-order
fringe pattern levels as seen in Figure (2). The extension of (3.2) to include higher
order fringe pattern levels was illustrated in [11] with numerical examples. Equation
(3.2) is modified as follows:

∂

∂t
ω(x, t) = α

∂2

∂x2
ω(x, t) +

∂

∂x

∫ s

−s
β0ω(x+ x̂, t)

x̂

|x̂|
dx̂

+
∂

∂x

∫ 3
2b

− 3
2b

β1ω(x+ x̂, t)
x̂

|x̂|
dx̂+

∂

∂x

∫ 5
2b

− 5
2b

β2ω(x+ x̂, t)
x̂

|x̂|
dx̂

= α
∂2

∂x2
ω(x, t) + β0

(
ω(x+ s, t)− 2ω(x, t) + ω(x− s, t)

)
+ β1

(
ω(x+

3s

2b
, t)− 2ω(x, t) + ω(x− 3s

2b
, t)
)

+ β2

(
ω(x+

5s

2b
, t)− 2ω(x, t) + ω(x− 5s

2b
, t)
)
,

t > 0, −∞ < x <∞,

(3.3)

ω(x, 0) = ω0(x), −∞ < x <∞, ω0 ∈ L1
+(−∞,∞),

∫ ∞
−∞

ω0(x) dx = 1.

Equation (3.3) has two additional fringe pattern levels on either sided of the origin:
[− 4s

b ,−
3s
b ], [− 3s

b ,−
2s
b ], [ 2sb ,

3s
b ], [ 3sb ,

4s
b ]. Additional levels can be added.

3.1. Analysis of equation (3.3). Let X = L1(−∞,∞), the space of integrable
functions on (−∞,∞) with norm ‖f‖ =

∫∞
−∞ |f(x)|dx. For σ > 0 let

(Tσ(t)f)(x) =
1

2
√
σt

∫ ∞
−∞

exp
(
− (x− y)2

4σt

)
f(y) dy, (3.4)

for f ∈ X, t > 0, −∞ < x < ∞. Tσ(t) for t ≥ 0 is a strongly continuous
holomorphic semigroup of positive linear operators inX with infinitesimal generator
(Aσf)(x) = σd2f(x)/dx2 satisfying |Tσ(t)| ≤ 1, t ≥ 0 ([5]). Further, (Tσ(t)f)(x) is
the strong solution in X to the diffusion equation

∂

∂x
u(x, t) = σ

∂2

∂x2
u(x, t), u(x, 0) = f(x), t > 0, −∞ < x <∞, f ∈ X. (3.5)

and [12] ∫ ∞
−∞

(Tσ(t)f)(x) dx =

∫ ∞
−∞

f(x) dxf ∈ X. (3.6)

For a bounded linear operator B in X define the exponential of B as

exp(tB)f =

∞∑
n=0

(tB)n

n!
f, f ∈ X, t ≥ 0.



260 G. WEBB EJDE/SI/02

Define the bounded linear operators B0,±, B1,±, B2,± in X as follows: for f ∈ X,
−∞ < x <∞,

(B0,±f)(x) = β0f(x± s),

(B1,±f)(x) = β1f(x± 3s

2b
),

(B2,±f)(x) = β2f(x± 5s

2b
).

Define the bounded linear operators B0, B1, B2 in X as follows:

(B0f)(x) = β0

(
f(x+ s)− 2f(x) + f(x− s)

)
,

(B1f)(x) = β1

(
f(x+

3s

2b
)− 2f(x) + f(x− 3s

2b
))
)
,

(B2f)(x) = β2

(
f(x+ s)− 2f(x) + f(x− s)

)
.

Since B0,+, B0,−, B1,+, B1,−, B2,+, and B2,− commute, we have

exp(tB0) = e−2β0t exp(tB0) = e−2β0t exp(tB0,+) exp(tB0,−),

exp(tB1) = e−2β1t exp(tB1) = e−2β1t exp(tB1,+) exp(tB1,−),

exp(tB2) = e−2β0t exp(tB2) = e−2β2t exp(tB2,+) exp(tB2,−).

Theorem 3.1. Let X = L1(−∞,∞), let α, s, b, β0, β1, β2 > 0, Let Tα(t), t ≥ 0
be the semigroup of linear operators as in equation (3.4). The unique generalized
solution of equation (3.3) is given by the strongly continuous semigroup of linear
operators T (t), t ≥ 0 in X with

T (t)ω0 = Tα(t) exp(tB0) exp(tB1) exp(tB2)ω0, t ≥ 0, ω0 ∈ X. (3.7)

Further, if

ω0(x) ≥ 0 a.e. on (−∞,∞) and

∫ ∞
−∞

ω0(x) dx = 1, (3.8)

then

(T (t)ω0)(x) ≥ 0 a.e. on (−∞,∞) and

∫ ∞
−∞

(T (t)ω0)(x) dx = 1. (3.9)

Proof. For ω0 ∈ X, ω0(x) ≥ 0 a.e. on (−∞,∞),

(B0,+ω0)(x) = β0ω0(x+ s) ≥ 0 a.e. on (−∞,∞),

(B2
0,+ω0)(x) = B0,+(β0(ω0(x+ s)) = β2

0ω0(x+ 2s) ≥ 0 a.e. on (−∞,∞),

(B3
0,+ω0)(x) = B2

0,+(β0(ω0(x+ s)) = β3
0ω0(x+ 3s) ≥ 0 a.e. on (−∞,∞) . . . .

Thus,

exp(tB0,+)ω0 =

∞∑
n=0

tnBn0,+
n!

ω0 ≥ 0 a.e. on (−∞,∞).

Similarly, exp(tB0,−)ω0, exp(tB1,+)ω0, exp(tB1,−)ω0, exp(tB2,+)ω0, exp(tB2,−)ω0

≥ 0 a.e. on (−∞,∞). From [5] Tα(t)ω0(x) ≥ 0 a.e. on (−∞,∞). Thus, for t ≥ 0,

T (t)ω0 = Tα(t) exp(tB0) exp(tB1) exp(tB2) ≥ 0.

Let ω0 ∈ X, ω0(x) ≥ 0 a.e. on (−∞,∞), and
∫∞
−∞ ω0(x) dx = 1. Then∫ ∞

−∞
(B0ω0)(x) dx = β0

(∫ ∞
−∞

(
ω0(x+ s)− 2ω0(x) + ω0(x− s)

)
dx = 0,



EJDE-2023/SI/02 NONLOCAL ADVECTION DIFFUSION EQUATION 261∫ ∞
−∞

(B2
0ω0)(x) dx = β0

(∫ ∞
−∞

(
ω0(x+ 2s)− 2ω0(x+ s) + ω0(x)

)
− 2
(
ω0(x+ s)− 2ω0(x) + ω0(x− s)

)
+
(
ω0(x− 2s)− 2ω0(x− s) + ω0(x)

))
dx = 0,∫ ∞

−∞
(B3

0ω0)(x) dx = 0, . . . .

Similarly,∫ ∞
−∞

(Bn1 ω0)(x) dx = 0,

∫ ∞
−∞

(Bn2 ω0)(x) dx = 0, n = 1, 2, 3, . . . .

Then∫ ∞
−∞

(exp(tB0)ω0)(x) dx =

∞∑
n=0

tn

n!

∫ ∞
−∞

(Bn0 ω0)(x) dx =

∫ ∞
−∞

ω0(x) dx = 1.

Similarly, ∫ ∞
−∞

(exp(tB1)ω0)(x) dx = 1,

∫ ∞
−∞

(exp(tB2)ω0)(x) dx = 1.

From (3.6) ∫ ∞
−∞

(Tα(t)ω0)(x) dx = 1, t ≥ 0.

Thus, for t ≥ 0,∫ ∞
−∞

T (t)ω0dx =

∫ ∞
−∞

Tα(t) exp(tB0) exp(tB1) exp(tB2) dx = 1

. �

4. Numerical simulations of the models

In this section we provide numerical simulations of the solutions ρ(x, t) of the SE
model (2.3) and the solutions ω(x, t) of the NLAD model (3.3). The SE solutions
and the NLAD solutions are similar, depending on the values of parameters and
the values of t. There are two significant differences in the model outputs, which
we will demonstrate in the simulations.

One difference is in the spacing of the local minima and maxima in the fringe
patterns in the first phase. This spacing is very regular in the solutions of the
NLAD model, but irregular in the solutions of the SE model. Another difference is
in the second phase, where the solutions of the SE model demonstrate the space-
dilation property, but the solutions of the NLAD model elevate above the x-axis
and demonstrate a dispersion property of the fringe pattern peaks.

In Figures (4), (5), (6), and (7), we present simulations for the first phase of
the SE and NLAD models at four different time values. The spacing of peaks
for the NLAD model is very regular, whereas the spacing of peaks for the SE
model is very irregular. In Figure (7) the spacing of the local minima of ω(x, 1/π)
occurs regularly at x ≈ .5, 1.5, 2.5, 3.5, . . . . The spacing of local minima of ρ(x, 1/π)
occurs irregularly at x ≈ .5, 1.5, 2.5, 3.5, . . . , 7.5, 8.5, 9.25, 10.0, 10.75, 11.5, 12.5, . . . ,
17.5, 18.5, 19.25, 20.0, 20.75, 21.5, . . . .



262 G. WEBB EJDE/SI/02

-30 -20 -10 10 20 30
x

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-2 -1 1 2
x

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
x

0.002

0.004

0.006

0.008

0.010

Figure 4. The interference diffraction pattern of the solution
ρ(x, t) of SE (blue) in equation (2.3) and the solution ω(x, t)
of NLAD (red) in equation (3.3). α = 1/π3, s = 1, b = .1,
β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t = .1/π. The
bottom graphs are log(ρ(x, t)) (blue) and log(ω(x, t)) (red).

In Figures (8), (9), (10), and (11), we present simulations for the second phase of
the SE and NLAD models at four different time values t2 = 2/π, t3 = 3/π, t4 = 4/π,
t6 = 6/π. In these simulations for the second phase, the solutions ω(x, t) of the
nonlocal advection-diffusion equation (3.3) are space-time dilated according to the
formulas ω̂(x, t2) = 1

2ω(x2 , t2), ω̂(x, t3) = 1
3ω(x3 , t3), ω̂(x, t4) = 1

4ω(x4 , t4), ω̂(x, t6) =
1
6ω(x6 , t6). This dilation of ω(x, t) to ω̂(x, t) in the second phase corresponds to
an extension of the fringe pattern established in the first phase, with increasing
distance of the detection plate. The simulations ρ(x, t) of the SE model exhibit
the space-time dilation property, with local minima remaining on the x-axis. The
simulations ω(x, t) of the NLAD model exhibit an elevation of local minima above
the x-axis and a dissipation of the fringe pattern peaks.

5. Summary

We have developed a nonlocal advection-diffusion model NLAD for the two-
slit experiment of quantum mechanics, in which quantum particles are projected



EJDE-2023/SI/02 NONLOCAL ADVECTION DIFFUSION EQUATION 263

-30 -20 -10 10 20 30
x

0.2

0.4

0.6

0.8

-2 -1 1 2
x

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
x

0.002

0.004

0.006

0.008

0.010

Figure 5. The interference diffraction pattern of the solution
ρ(x, t) of SE (blue) in equation (2.3) and the solution ω(x, t)
of NLAD (red) in equation (3.3). α = 1/π3, s = 1, b = .1,
β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t = .25/π. The
bottom graphs are log(ρ(x, t)) (blue) and log(ω(x, t)) (red).

forward, one at a time, through two slits, and detected downstream on a detection
surface. Our work here is an extension of the models in [10] and [11], which allowed
higher order fringe pattern levels observed in experiments (Figure 2). We compare
the NLAD equation model to the Schrödinger equation model SE with initial data
consisting of two rectangular steps.

In both formulations NLAD and SE, there is a two-phase development of the
multi-level fringe pattern. In the first phase the initial data transitions to an estab-
lished multi-level fringe pattern with local minima located approximately on the
x-axis. This transition is very simple for the NLAD model, but very complex for
the SE model.

In the second phase the multi-level fringe pattern established in the first phase
evolves in a space-time dilation in both models. In the second phase of the SE
model, the pattern established in the first phase is preserved almost perfectly in the
space-time dilation with constant speed. In the second phase of the NLAD model,



264 G. WEBB EJDE/SI/02

-30 -20 -10 10 20 30
x

0.1

0.2

0.3

0.4

0.5

0.6

-2 -1 1 2
x

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
x

0.002

0.004

0.006

0.008

0.010

Figure 6. The interference diffraction pattern of the solution
ρ(x, t) of SE (blue) in equation (2.3) and the solution ω(x, t)
of NLAD (red) in equation (3.3). α = 1/π3, s = 1, b = .1,
β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t = .5/π. The
bottom graphs are log(ρ(x, t)) (blue) and log(ω(x, t)) (red).

the pattern established in the first phase dissipates, with local minima rising above
the x-axis and with magnitude of the fringe pattern oscillations decreasing.

The SE formulation and the NLAD formulation of the 2-slit experiment have very
different interpretations. The interpretation of the SE model is that an individual
quantum particle exists as a wave moving forward in space. The interpretation
of the NLAD model is that an individual quantum particle exists as a component
of an ensemble, with its forward movement influenced by nonlocal reaction to its
environment within a sensing radius of its spatial position.

Scientific understanding of the two-slit experiment, which is one of the most fun-
damental experiments in science, requires mathematical formulations, which pro-
vide descriptive connection to experiments, and interpretation of physical processes.
The two formulations SE and NLAD can be compared for these purposes.



EJDE-2023/SI/02 NONLOCAL ADVECTION DIFFUSION EQUATION 265

-30 -20 -10 10 20 30
x

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-2 -1 1 2
x

0.5

1.0

1.5

2.0

2.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
x

0.002

0.004

0.006

0.008

0.010

Figure 7. The interference diffraction pattern of the solution
ρ(x, t) of SE (blue) in equation (2.3) and the solution ω(x, t)
of NLAD (red) in equation (3.3). α = 1/π3, s = 1, b = .1,
β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t = 1.0/π. The
bottom graphs are log(ρ(x, t)) (blue) and log(ω(x, t)) (red).

References

[1] Y. Aharonov, E. Cohen, F. Columbo, J. Tollaksen; Finally making sense of the double-slit

experiment, Proc. Nat. Acad. Sci., 114.25 (2017), 6480–6485.
[2] J. Goldstein; Semigroups of Linear Operators and Applications, Oxford University Press,

Oxford, 1985.

[3] A. Einstein, B. Podolsky, N. Rosen; Can quantum-mechanical description of physical reality
be considered complete?, Phys. Rev., 41 (1935), 777–780.

[4] A. Friedman; Partial Differential Equations, Holt, Rinehart and Winston, New York and

London, 1969.
[5] T. Kato; Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.

[6] John W. Neuberger; Analyticity and quasi-analyticity for one-parameter semigroups, Proc.

Amer. Math. Soc., 25 (1970), 488–494.
[7] John W. Neuberger; Quasi-analyticity and semigroups, Bull. Amer. Math. Soc., 78(6) (1972),

909-922.
[8] A. Tonomura; The Quantum World Unveiled by Electron Waves, World Scientific, Singapore,

1998.



266 G. WEBB EJDE/SI/02

-60 -40 -20 20 40 60
x

0.02

0.04

0.06

0.08

0.10

-2 -1 1 2
x

0.5

1.0

1.5

2.0

2.5

Figure 8. The interference diffraction pattern of the solution
ρ(x, t2) of SE (blue) in equation (2.3) and the dilated solution
ω̂(x, t2) (red) in the NLAD equation (3.3). α = 1/π3, s = 1,
b = .1, β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t2 =
2.0/π.

-50 50
x

0.01

0.02

0.03

0.04

0.05

0.06

-2 -1 1 2

0.5

1.0

1.5

2.0

2.5

Figure 9. The interference diffraction pattern of the solution
ρ(x, t3) of SE (blue) in equation (2.3) and the dilated solution
ω̂(x, t3) (red) in the NLAD equation (3.3). α = 1/π3, s = 1,
b = .1, β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t3 =
3.0/π.

[9] G. F. Webb; Event based interpretation of Schrödinger’s equation for the two-slit experiment,
Int. J. Theor. Phys., 50(11) (2011), 3571–3601.

[10] G. F. Webb; The ensemble interpretation of quantum mechanics and the two-slit experiment,
Comp. Meth. Appl. Sci., 47 (2018), 433–452.

[11] G. F. Webb; The Schrödinger equation and the two-slit experiment of quantum mechanics,
to appear.

[12] K. Yosida; Functional Analysis, Springer-Verlag, New York, 1968.



EJDE-2023/SI/02 NONLOCAL ADVECTION DIFFUSION EQUATION 267

Out[359]=

-100 -50 50 100
x

0.01

0.02

0.03

0.04

0.05

-2 -1 1 2

0.5

1.0

1.5

2.0

2.5

Figure 10. Interference diffraction pattern of the solution ρ(x, t4)
of SE (blue) in equation (2.3) and the dilated solution ω̂(x, t4)
(red) in the NLAD equation (3.3). α = 1/π3, s = 1, b = .1,
β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t4 = 4.0/π.

-150 -100 -50 50 100 150
x

0.005

0.010

0.015

0.020

0.025

0.030

0.035

-2 -1 1 2
x

0.5

1.0

1.5

2.0

2.5

Figure 11. The interference diffraction pattern of the solution
ρ(x, t6) of SE (blue) in equation (2.3) and the dilated solution
ω̂(x, t6) (red) in the NLAD equation (3.3). α = 1/π3, s = 1,
b = .1, β0 = 1/(8 b2), β1 = π/(2 b 152), β2 = π/(2 b 252), t6 =
6.0/π.

Glenn Webb

Mathematics Department, Vanderbilt University, Nashville, TN 37212, USA
Email address: glenn.f.webb@vanderbilt.edu


	1. Introduction
	2. Schrödinger equation model
	2.1. Schrödinger equation with step function initial data

	3. Nonlocal advection diffusion equation model
	3.1. Analysis of equation (3.3)

	4. Numerical simulations of the models
	5. Summary
	References