










































ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 
 
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. 

PARTICLE  
TRAJECTORIES  
FOR COMPTON  
SCATTERING IN  

ONE SPACE  
DIMENSION 

ADRIANA SCANTEIANU, XIANGYUE WANG 
 
 
 
 

✵ ABSTRACT  

 Using a relativistic extension of Bohmian Me-
chanics known as Multi-Time Wave Function formula-
tion, we examine a two-body, one-dimensional sys-
tem consisting of one photon and one electron that 
interact only upon contact. We investigate the effects 
that various parameters in this theory including mo-
mentum of the incoming photon and mass of the 
electron have on the dynamics of the two interact-
ing bodies with the goal of understanding conser-
vation of momentum and energy in the system. We 
show that the core principles of Compton scattering 
remain when we use this alternative formulation of 
quantum mechanics. Although a complete relativ-
istic theory of Bohmian mechanics has yet to be de-
veloped, our work aims to make the ideas in this the-
ory more accessible to a wider audience. 
 

1 INTRODUCTION  

 Mathematical physics aims to provide logi-
cally consistent and mathematically rigorous expla-
nations for why nature behaves the way it does. Cur-
rently, at the fundamental level, there are two exper-
imentally verified and mathematically rigorous theo-
ries that seem to be in conflict: namely, quantum me-
chanics and general relativity. While quantum field 

theory is, for all practical purposes, highly successful 
at reconciling quantum mechanical principles with 
Einstein’s theory of light (special relativity), it is in-
compatible with Einstein’s theory of gravitation (gen-
eral relativity). It is also mathematically non-rigorous 
due to its inherent infinities[14]. Such incompatibilities 
and lack of rigor have led mathematical physicists to 
continue exploring other theoretical frameworks. 
More than half a century ago, Louis de Broglie and 
David Bohm formulated a non-relativistic theory 
where elementary particles exist regardless of obser-
vations[2,4,10]. De Broglie-Bohm theory, otherwise 
known as Bohmian mechanics, is a formulation of 
quantum mechanics in which subatomic particles in 
a system have definite positions, guided by the wave 
function of the system[10]. In one of the relativistic ex-
tensions of this theory, there is a single multi-time 
wave function originally proposed by Dirac, defined 
on the configuration space of the system that guides 
the motion of all the particles[5]. Some researchers 
believe the photon wave function was never ex-
plored due to historical reasons: Dirac completed 
his quantization of the electromagnetic field prior to 
discovering his relativistic electron wave equation, 
and the world of research acted on the first discovery 
more than the second[1,5,6]. Goldstein and others 
have developed non-relativistic Bohmian mechanics 
in great mathematical detail[9,10]. The remaining chal-
lenge is developing a fully relativistic Bohmian me-
chanical system with a fixed number of particles. 
Some have claimed that this task is downright impos-
sible[15]. Nevertheless, Kiessling and Tahvildar-Zadeh 
successfully developed a relativistic Bohmian theory 
for a single photon and, together with Lienert, they 
have developed an interacting electron-photon the-
ory in one space dimension[11,12]. In this paper, we ex-
plore whether their theory is applicable to the study 
of Compton scattering. 
 Arthur Compton discovered, ninety-six years 
ago, that when X-rays interact with a graphite plate, 
the scattered rays have lower frequencies than the 
incident ones and the difference in the energy of the 
“X-ray quanta,” now called photons, turns into kinetic 
energy of the electrons on the surface of the mate-
rial[3]. With this experiment, Compton provided evi-
dence for Einstein’s 1905 theory, which stated that 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

light behaves as particles. We now know this phe-
nomenon as Compton scattering. Quantum field 
theory provides us with the following narrative about 
what happens during the interaction: just before the 
photon smashes into the electron, it abruptly van-
ishes into nothing. At the same moment, a new pho-
ton with a different frequency emerges next to the 
electron out of thin air, and the two depart as if they 
have just collided. We would like to explore a differ-
ent narrative of a photon-electron collision based on 
Bohmian principles, as provided in the paper by 
Kiessling et al [11]. 
 The novelty of our work lies in the actualiza-
tion of previous theory via our modeling of Compton 
scattering using MATLAB software[11,13]. Further-
more, we are the first to use this novel theoretical 
framework to analyze the effects of changing the 
momentum of the incoming photon on the trajecto-
ries of the scattered photon and electron. We are 
also the first to investigate the conservation of mo-
mentum and applicability of Compton’s scattering 
formula in this system. The framework we use is com-
patible with many experimentally verified theories in 
physics, thus we hope that our work will bring us one 
step closer to a mathematically rigorous understand-
ing of the world around us on the subatomic level[11]. 
 

2 METHODOLOGY  

 In this paper, we treat both the electron and 
the photon as relativistic quantum objects. First, we 
model the trajectories of a single photon in one 
space dimension using the photon wave function 
proposed by Kiessling and Tahvildar-Zadeh[12]. Next, 
we do the same for a single electron using Dirac’s 
relativistic wave equation for the electron[6]. We then 
examine the two-body quantum-mechanical system 
of a photon and an electron, first assuming that the 
two do not interact, and then implementing the pho-
ton-electron interaction through a boundary condi-
tion, as in the paper by Kiessling et al[11]. Afterwards, 
we review the method Compton used to compute 
the formula for the scattering angle in his experi-
ment. We then compare Compton’s predicted value 
for momentum gain of the electron with our com-
puted values for this quantity based on our numeri-
cal solutions of the guiding equations for the photon 

and the electron. We then remark on the obstacles 
that one needs to overcome if one is to rigorously 
derive such a formula from our Bohmian perspec-
tive. Note that our unit of time is the attosecond (1 
attosecond = 10—18 seconds) and our unit of length is 
2.997924 angstroms (1 angstrom = 10—10 meters). 
This allows for the speed of light in a vacuum to be  
1 unit of length over time. 
 

i. Single Photon Theory 
 The wave function of a photon is a rank-2 bi-
spinor field defined on the configuration space- 
time of the photon [12]. In one space dimension this 
means: 

 

where  𝑡𝑡 ∈ ℝ  is a time coordinate and  𝑠𝑠 ∈ ℝ  is a 
space coordinate. Since a photon is a relativistic 
quantum particle, its wave function must satisfy a rel-
ativistic partial differential equation, which was dis-
covered by Kiessling & Tahvildar-Zadeh[12]. The pho-
ton wave equation is a Dirac-type equation, and in 
one-dimension it reads: 

 

Where ℏ is the reduced Planck’s constant, 𝑥𝑥0 = 𝑐𝑐𝑡𝑡 , 

𝑥𝑥1 = 𝑠𝑠 , 𝛾𝛾0 = �0 1
1 0� , and 𝛾𝛾1 = �0 −1

1 0 � ; and re-

peated indices are summed over the range 
 𝜇𝜇 =  0, 1. Here,  𝑐𝑐  is the speed of light in a vacuum. 
We will be working in units where  𝑐𝑐 =  1. The pho-
ton wave equation can be solved given an initial 
wave function 𝛹𝛹𝑝𝑝ℎ(0, 𝑠𝑠) = 𝛹𝛹𝑝𝑝ℎ0 (𝑠𝑠) . Typical initial data 

corresponding to a photon localized in both posi-
tion and momentum space (subject to Uncertainty 
Principle) can be found in FIGURE 1. 

 In our case, the typical initial data are Gauss-
ian distributions in photon position space. Since the 
Fourier transform of a Gaussian is a Gaussian, these 
also have a Gaussian distribution in momentum 
space. Therefore, there are two parameters that we 
can set: the mean  𝑘𝑘0 ∈ ℝ of the momentum distribu-
tion, and the momentum standard deviation 𝛼𝛼𝑝𝑝ℎ.  

Recall that according to the Uncertainty Principle, 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

𝛼𝛼𝑝𝑝ℎ𝜎𝜎𝑝𝑝ℎ ≥
ℎ
2

, where 𝜎𝜎𝑝𝑝ℎ  is the standard deviation in 

photon position space, and that this inequality is sat-
urated by the Gaussian distribution. Furthermore, 
we recall the Einstein-de Broglie energy and mo-
mentum relations 𝐸𝐸 =  ℎ𝜐𝜐, 𝑝𝑝 =  ℎ𝑘𝑘, and the photon 
dispersion relation 𝐸𝐸 =  |𝑝𝑝|  (recall that 𝑐𝑐 =  1 .) 
These imply that we may identify |𝑘𝑘0| with the mean 
of the initial frequency distribution of the photons. 
 The probability current of detecting a pho-
ton at a point in one space and one time dimension 
is[12]: 

 

where 𝑋𝑋 = (𝑋𝑋0,𝑋𝑋1)  is a constant vector field com-
puted from 𝛹𝛹𝑝𝑝ℎ0 , 𝛾𝛾(𝑋𝑋) ≔ 𝛾𝛾0𝑋𝑋0 + 𝛾𝛾1𝑋𝑋1 , 𝛹𝛹� ≔ 𝛾𝛾0𝛹𝛹†𝛾𝛾0 , 

and Tr denotes matrix trace. The current is  
conserved: 𝜕𝜕𝜇𝜇𝑗𝑗𝑝𝑝ℎ

𝜇𝜇 = 0, future directed (𝑗𝑗0 ≥ 0) , and 

timelike (𝑗𝑗0 ≥ |𝑗𝑗1|).  
 Thus, the probability density of detecting 
the photon at event (𝑡𝑡, 𝑠𝑠) is 𝜌𝜌(𝑡𝑡, 𝑠𝑠)  = 𝑗𝑗𝑝𝑝ℎ0 (𝑡𝑡, 𝑠𝑠). 

 The motion of the photon is guided by its 
wave function through the following system of guid-
ing equations[11]: 

 
Here, 𝑞𝑞(𝑡𝑡) is the position of the photon at time 𝑡𝑡; 𝑞𝑞0 

is the initial position of the photon. All we know 
about the initial position is that it is randomly  
distributed according to the initial probability den-
sity 𝜌𝜌(0, 𝑠𝑠). By solving the system of guiding equa-
tions, we obtain the trajectory of a single photon 
over time. 
 

ii. Single Electron Theory 
 According to Paul Dirac, the wave function 

of a single electron is a spin-1
2
 field defined on the 

configuration space of the electron[6]. In one space 
dimension this means: 

. 

Like in the case of a single photon, the wave function 
of a single electron also satisfies a relativistic equa-
tion. In particular, it satisfies the massive Dirac equa-
tion: 

, 

where 𝑚𝑚𝑒𝑒𝑒𝑒  is the mass of an electron. 
 The probability current of an electron is: 

, 

where 𝛹𝛹� ≔ 𝛹𝛹†𝛾𝛾0  is the Dirac adjoint for rank-one 

FIGURE 1: Initial data for the photon wave equation is split into real and imaginary components. We assume typical initial 
data: the photon starts off as a Gaussian wave packet both in position and in momentum-space. Since the unit of fre-
quency is the reciprocal of time, our unit of frequency is 1018 Hz. 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

bispinors. Furthermore, as in previous research, we 
define the guiding velocity field of an electron in the 
same way as that of a photon[11]: 

. 

Similarly to the photon case, the guiding equation 
for the electron is[11]: 

 

The equation satisfied by the electron wave function 
𝛹𝛹𝑒𝑒𝑒𝑒  has a mass term: 𝜔𝜔 = 𝑚𝑚𝑒𝑒𝑒𝑒𝑐𝑐/ℏ . In addition, the 
Gaussian family of initial distributions has a parame-
ter we can change, namely the standard deviation of 
the probability distribution of electron’s initial posi-
tion: 𝜎𝜎. We will examine the effect of these two pa-
rameters on the joint dynamics. 
 

iii. The Two-Body System 
 Now, we examine the case in which a photon 
and an electron are both present, but do not inter-
act. In this case, we have one wave function that 
guides the motion of each particle through its re-
spective system of guiding equations. The wave 
function, 𝜓𝜓, is a function of four variables, namely the 
time and position of each particle. To get a wave 
function that describes both a photon and an elec-
tron in a non-interacting system, we take the tensor 
product (⨂) of the electron and the photon wave 
functions, giving us a four component object 𝜓𝜓 =
(𝜓𝜓−−,𝜓𝜓−+,𝜓𝜓+−,𝜓𝜓++). The guiding equations for pho-
ton and electron are derived using the Hypersurface 
Bohm-Dirac (HBD) Theory, which allows us to de-
scribe the motion of the photon and electron in a 
common time coordinate[8]. The tensored wave func-
tion satisfies a relativistic wave equation obtained by 
the tensor product of the photon and electron wave 
equations: 

 

The joint probability current is the following: 

 

Then, the guiding equation for the photon is given 
by: 

 

while the guiding equation for the electron is: 

 

 

iv. The Two-Body System 
 To obtain an interacting system from our 
two-body non-interacting system, it is necessary to 
add a boundary condition that prevents the particles 
from simply going through each other. In this is done 
by adding a boundary condition on the coincidence 
set: they set the relative velocity of photon and elec-
tron to be 0 whenever the two particles are at the 
same space and time point[11]. In other words: 

 

when 

 
 

v. Derivation of Compton Scattering Formula 
 To better explain the formula used to calcu-
late predicted momentum of particles in the system, 
we present a short derivation of the original Comp-
ton Scattering Formula in our one-dimensional set-
ting. Assume we have positive energy plane waves 
going into and coming out of the scattering zone. 
Assume that conservation of momentum and energy 
still hold.  

(2.1) 
 
 
 
 
 
 
(2.2) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
(2.3) 
 
 
 

(2.4) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

Conservation of momentum means:  

 

Going into the rest frame of electron, 𝑘𝑘𝑒𝑒𝑒𝑒𝑖𝑖𝑖𝑖 becomes 
0. Hence: 

 

Suppose the photon is approaching the electron 
from the right. Thus 𝑘𝑘𝑝𝑝ℎ𝑖𝑖𝑖𝑖 > 0 . Conservation of  

energy is given by: 

 

(Here and henceforth, we have chosen units in which 
the speed of light in vacuum is 𝑐𝑐 =  1.) Substituting 
EQUATION 2.6 into EQUATION 2.7 and attempting to solve 
for 𝑘𝑘𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜  we observe that choosing the plus sign in 

the right-hand-side of EQUATION 2.7 yields no solution. 
Thus, the only option is to choose the negative sign, 
which implies that 𝑘𝑘𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜 < 0 since we have assumed 

that the plane waves have positive energy. As a con-
sequence, the direction of motion of the photon has 
changed after the scattering event. 
 Solving for 𝑘𝑘𝑝𝑝ℎ𝑜𝑜𝑜𝑜𝑜𝑜  then yields the scattering 

formula: 

 

We note that this agrees with the three-dimensional 
Compton scattering formula derived in the 1923 pa-
per if the scattering angle in that formula is set to 𝜋𝜋. 
This formula also suggests that the frequency of the 
scattered X-ray will be lower than the frequency of 
the incident ray and that the difference in the mo-
mentum and energy is transferred to the electron. 
More precisely, combining EQUATION 2.8 with  
EQUATION 2.6 and recalling that 𝜔𝜔 = 𝑚𝑚𝑒𝑒𝑒𝑒𝑐𝑐/ℏ  one  
obtains: 

 

3 RESULTS  
 Throughout the results section, it is im-
portant to note that time is measured in attoseconds 
(10—18 seconds), and length is measured in units of 
approximately 3 angstroms (2.997924 × 10—10 me-
ters). In these units the speed of light in vacuum is 
equal to 1 unit of length over time. 
 

i. Single Photon Motion 
The photon probability density looks like this: 

HTTPS://REU.DIMACS.RUTGERS.EDU/~AAS377/PHOTON_PDF.MP4 

Varying the standard deviation of the initial profile 
gives us the following: 

HTTPS://REU.DIMACS.RUTGERS.EDU/~AAS377/MULTIPLE_PHOTON_PDF.MP4 

FIGURE 2 shows multiple trajectories, each with slightly 
different initial positions. 
 

ii. Single Electron Motion 
The evolution of the probability distribution of the 
electron’s position can be found here: 

HTTP://REU.DIMACS.RUTGERS.EDU/~AAS377/ELECTRON_NEW.MP4 

FIGURE 3 shows the trajectories of many electrons, each 
in a single-body system guided by its single wave 
function through its guiding equation, with different 
initial conditions for each electron. Unlike the trajec-
tories of the photon, the trajectories of the electron 
exhibit oscillations and travel slower than the speed 
of light. 
 

iii. Non-Interacting Two-Body System 
Since tensor products preserve probability distribu-
tions, “multiplying” the probability density movies of 
the photon and electron gives the joint probability 
density for the non-interacting system. Since there is 
no interaction, we expect the joint probability den-
sity to be simply the product of the probability den-
sity of a single electron and that of a single photon. 
The numerical results confirm our expectation. 

HTTPS://REU.DIMACS.RUTGERS.EDU/~AAS377/NON_INT_PDF_MESH_NEW.MP4 

By solving the system of guiding equations, we ob-
tain FIGURE 4, which shows the trajectories of a non-in-
teracting system of one electron and one photon. 

(2.6) 
 
 
 
 
 
 

(2.7) 

 

 

 

 

 

 

 

 

 

 

(2.8) 

 

 

 

 

 

 

 

 

 

(2.9) 

https://reu.dimacs.rutgers.edu/%7Eaas377/photon_pdf.mp4
https://reu.dimacs.rutgers.edu/%7Eaas377/multiple_photon_pdf.mp4
http://reu.dimacs.rutgers.edu/%7Eaas377/electron_new.mp4
https://reu.dimacs.rutgers.edu/%7Eaas377/non_int_pdf_mesh_new.mp4


ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 
 

TOP  
FIGURE 2: A free photon in one dimension would move  
either left or right at the speed of light. The linear trajectories 
in the diagram validates this intuition. Trajectories with differ-
ent initial conditions are differentiated by color. 

MIDDLE 
FIGURE 3: A free electron in one dimension moves according 
to its wave function at a velocity slower than the speed of 
light. Trajectories with different initial conditions are differen-
tiated by color. 

BOTTOM  
FIGURE 4: Non-interacting photon (LEFT) and electron (RIGHT) 
trajectories. Since the two particles do not interact, they pass 
through one another as the diagram indicates. Trajectories 
with different initial conditions are differentiated by color. 

BELOW  
FIGURE 5: Interacting and non-interacting photon (RED) and 
electron (BLUE) trajectories. The interaction resembles a colli-
sion, after which the two particles bounce away from each 
other. Dotted lines show the non-interacting case, and the 
solid lines show the interacting case. 

 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

[a]For reasons having to do with numerical computations, the value we use for ω is numerically equal to the mass of the electron but is not the 
actual physical value. The actual mass of an electron in our units is 776.343694 units of mass, but we are working on the order of 1 unit of mass. 
That is to say that we are working with an electron mass that is 700 times smaller.  

 

iv. Interacting Two-Body System 
Adding the boundary condition to the wave function 
gives us a modified probability density function: 

HTTP://REU.DIMACS.RUTGERS.EDU/~AAS377/INT_PDF_MESH_NEW.MP4 

FIGURE 5 is the contour map. The shape of the interact-
ing probability density is delineated by lines from 
the perspective of someone looking down from 
above: 

HTTP://REU.DIMACS.RUTGERS.EDU/~AAS377/INT_PDF_MESH_NEW.MP4 

These are the trajectories of the interacting electron-
photon system with the boundary condition added 
in. The solid lines show the interacting system trajec-
tories and the dotted lines show the non-interacting 
trajectories.  

Plotting 100 overlaid trajectories of the inter-
acting system gives FIGURE 6. To make this clearer, 
here is a video of the collision trajectories being plot-
ted individually on the same set of axes. 

HTTP://REU.DIMACS.RUTGERS.EDU/~AAS377/INT_TRAJS_SLOWMO.MP4 

 
 

v. Varying the Electron Parameters 
Here, we sought to perform numerical ex-

periments using our model. The units used in calcu-
lating the values of our parameters are chosen such 
that Planck’s Constant ℏ = 1 unit of mass × length 

squared over time: ℏ = 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 × 𝐿𝐿𝑒𝑒𝑖𝑖𝐿𝐿𝑜𝑜ℎ2

𝑇𝑇𝑖𝑖𝑇𝑇𝑒𝑒
. In our units, 

one unit of mass is 0.117337 × 10—32 kg[a]. The param-

eter 𝜔𝜔 = 𝑇𝑇𝑒𝑒𝑒𝑒𝑐𝑐
ℏ

= 1
𝜆𝜆
 , where 𝜆𝜆 is the reduced Compton 

Wavelength of the electron and equal to approxi-
mately 0.001288 units of length. Note that change in 
𝜔𝜔 corresponds to a change in mass of electron 𝑚𝑚𝑒𝑒𝑒𝑒  
since the units are set such that 𝑐𝑐  and ℏ are set to 
equal 1. Since 𝜔𝜔 is the reciprocal of wavelength, we 

TOP  
FIGURE 6: 100 Overlaid interacting photon (RED) and  
electron (BLUE) trajectories. 

BOTTOM  
FIGURE 7: Looking across each row, we can see variation 
in standard deviation of initial distribution of the elec-
tron’s position, σ, while the frequency of the electron, ω, 
is kept constant across each row. Looking down each 
column, we can see variation in the frequency of the 
electron, ω!, as the standard deviation of the initial dis-
tribution of the electron’s position, σ, is kept constant. 

http://reu.dimacs.rutgers.edu/%7Eaas377/int_pdf_mesh_new.mp4
http://reu.dimacs.rutgers.edu/%7Eaas377/int_pdf_mesh_new.mp4
http://reu.dimacs.rutgers.edu/%7Eaas377/int_trajs_slowmo.mp4


ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

[b]We are of course aware that electron’s rest mass is fixed. The point here is to do a mathematical study of how the trajectories would look like 
for a hypothetical particle with the same characteristics of the electron but a different rest mass. 

 
 

 

can refer to 𝜔𝜔 as electron frequency. Change in 𝜎𝜎 re-
fers to change in the standard deviation of the initial 
probability distribution of the electron’s position in 
one-dimension, i.e. the standard deviation of the 
Gaussian distribution representing initial data. The 
units of 𝜎𝜎 are the same as the units for position: ap-
proximately 3 angstroms. The changes in these pa-
rameters are visible in FIGURE 7[b].  
 

vi. Varying the Phase Angle of Interaction 
In the boundary condition that provides the 

interaction, it is possible to specify a phase angle. It 
seems there are no changes as a result of variation in 
this parameter; however, future research is neces-
sary to elucidate whether this parameter truly has no 
effect on trajectories. It seems to be that the three 
graphs in FIGURE 8 are the same since they look like 
one set of trajectories instead of three when overlaid. 

 
vii. Varying Momentum of Incoming Photon 

Changing the momentum of the incoming 
photon gives us a different picture and lays the 
groundwork for analyzing Compton Scattering in 
our setting. The units of momentum are mass ×  
velocity. Our unit of mass is 0.117337 × 10—32 kg, and 
our unit of velocity is 2.997924 × 10–8 m/s (the speed 
of light in a vacuum). Hence, one unit of momentum 
is 0.351767 × 10–24 kg m/s. 

In order to get a more detailed picture, we 
varied the mean momentum of the incoming photon 
between 0 and 10 with mesh of 0.5. The result is  
FIGURE 10. 

 
viii. Compton Scattering in Our Model 

We have found some evidence of the ap-
plicability of Compton’s Scattering Formula to our 
model, as seen in FIGURE 11. The figure corresponds to 
the parameter values 𝜔𝜔 =  2  and  𝜎𝜎𝑝𝑝ℎ = 𝜎𝜎𝑒𝑒𝑒𝑒 = 0.1 , 

while the initial photon mean frequency is increased 
from 𝑘𝑘0 = 1  to 𝑘𝑘0 = 10 , and the resulting electron 
trajectories are distinguished by color. Again, the 
unit of momentum is 0.351767 × 10–24 kg m/s. 

  

TOP  
FIGURE 8: The left-most plot shows an angle of interaction 
of 0. The middle plot shows an angle of interaction of π/2 
and the right-most plot shows an angle of interaction of  
π /4. We can see that as the angle of interaction varies, no 
change occurs in the particle trajectories. Non-interacting 
trajectories are visible in dotted lines, and all other param-
eters are kept constant. 

BOTTOM  
FIGURE 9: Interacting photon (RED) electron (BLUE) trajecto-
ries with varying photon momentum. In the left-most plot, 
the momentum of the incoming photon is 0.1; in the mid-
dle plot, the momentum of the incoming photon is 1, and 
in the right-most plot, the momentum of the incoming pho-
ton is 10. Thus, we see that a higher energy photon—a pho-
ton with higher momentum—gets the electron to bounce 
away. Again, non-interacting trajectories are visible in dot-
ted lines. 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 
 

TOP  
FIGURE 10: Each pair of trajectories depicts a variation in 
the momentum of the incoming photon. 

BOTTOM  
FIGURE 11:  
LEFT: Change in Bohmian trajectories of the two particles 
due to the increase in incoming photon frequency.  
RIGHT: The corresponding change in electron’s post-scat-
tering momentum, computed from those trajectories. 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 
 

TOP  
FIGURE 12: Change in electron momentum due to scattering (detail). 

BOTTOM  
FIGURE 13:  
LEFT: Dotted lines show the non-interacting case, and solid lines show interacting cases at various incoming photon frequencies.   
RIGHT: Solid lines refer to momentum of the outgoing electron at various frequencies of incoming photon. The dotted line shows 
the momentum of the electron in the non-interacting case, as the two particles “go through” each other. 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

If we zoom in on the interaction region, we 
clearly see that the gain in electron momentum in-
creases linearly with 𝑘𝑘0 , in qualitative agreement 
with EQUATION 2.9. See FIGURE 12. In these figures we 
have calculated the electron’s “post-collision” mo-
mentum by 

 

where 𝑣𝑣𝑒𝑒𝑒𝑒 is the Bohmian velocity of the electron cal-
culated from the wave function, as in EQUATION 2.2, and 
evaluated at the actual positions of the two particles 
that have been calculated by solving the two differ-
ential equations EQUATION 2.1 and EQUATION 2.2. The 
graph on the right of FIGURE 13 shows that even in the 
non-interacting case, the momentum of the electron 
computed from trajectories is not constant because 
its Bohmian velocity is not constant; instead, it fluctu-
ates in time.  

Finally, we compared our computed elec-
tron momentum to the theoretical values computed 
with Compton’s Formula. Our results show that 
some frequencies of the incoming photon allow for 
a better match than others between Compton’s For-
mula and computations based on De Broglie-Bohm 
theory. Further research in this area would be useful 
to understand which regime of parameters allow for 
a better fit, and why. 
 

4 DISCUSSION  
 In this research, we studied probability den-
sity functions and Bohmian trajectories for a single 
electron and a single photon guided by their respec-
tive wave functions in one space dimension[10,12]. We 
then analyzed the probability density functions and 
Bohmian trajectories of both a non-interacting and 
interacting two-body system consisting of a photon  
and an electron, in one space dimension, where 
both particles were guided by the same two-body 
multi-time wave function.  
 Dirac introduced the concept of multi-time 
wave functions, but the idea that the wave function 
guided the motion of particles in the system did not 
arise until Bohmian Mechanics was developed[2,7]. 
The idea that multi-time wave functions can guide 
the motion of relativistic particles is far more recent[8]. 

 We studied how the dynamics of the two 
particles depended on several key parameters in the 
problem, such as the frequency of the incoming 
photon, and the rest mass of the electron. We did a 
preliminary analysis of Compton scattering, and 
found qualitative agreement between Compton’s 
quasi-classically derived formula for the gain in elec-
tron’s momentum, and the corresponding values 
computed from the trajectories. However, there is 
much work that remains to be done.  
 In our modeling of the two-body system, the 
trajectories of the non-interacting system were 
shown to simply be a superposition of the trajecto-
ries of a single photon and a single electron. This was 
no longer the case once the boundary condition was 
added. The wave function governing the system in-
dicates that post-interaction, the two particles be-
come entangled even if we initially have a pure prod-
uct state.  
 In varying the parameters of mass and stand-
ard deviation of the initial distribution for the elec-
tron, we saw that more massive electrons were likely 
to have a straighter trajectory, indicating slower 

FIGURE 14: The red line shows theoretical values of momen-
tum of the outgoing electron computed using Compton’s 
Formula. The blue circles indicate the results of our numer-
ical investigation of the outgoing electron momentum 
computed from Bohmian trajectories. 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

movement and less likelihood to recoil away. Fur-
thermore, varying the standard deviation of the ini-
tial distribution of the electron’s position seemed to 
alter the trajectories and the nature of the interac-
tion: lower standard deviation values resulted in the 
electron and photon staying together for longer in-
stead of bouncing off immediately. However, more 
investigation is required to better understand the in-
terplay between these parameters. In varying the 
phase angle of interaction, we found evidence that 
the dynamics of the interaction is independent of 
that phase angle. Our work in varying the momen-
tum of the incoming photon shows that shooting a 
higher energy photon at the electron will propor-
tionally increase the electron’s post-collision mo-
mentum in a manner consistent with Compton’s  
formula. 
 

i. Analysis of Compton Scattering Results 
 It should be noted that we cannot expect the 
actual electron momenta to be uniformly close to the 
values predicted by EQUATION 2.9, since this formula is 
of a classical nature (apart from using Einstein’s sug-
gested values for the momentum and energy of the 
photon). It has been derived using the rules of clas-
sical mechanics, such as exact conservation of a well-
defined energy and momentum for both particles. 
For instance, in the example above, the initial elec-
tron distribution has a standard deviation of 0.1; 
therefore, by the Uncertainty Principle, the standard 
deviation of its momentum distribution will be at 
least 5~. If we randomize the initial position of the 
electron according to the initial distribution, as we 
are supposed to do in a quantum theory, we cannot 
expect the electron momentum at any later time to 
be close to any one particular value. It is only ex-
pected to fluctuate around some mean. This would 
be equally true for a non-interacting electron. It is a 
hallmark of the quantum nature of elementary  
particles. 
 We would nevertheless like to be able to de-
rive a quantum-mechanical (i.e. appropriately statis-
tical) version of formula EQUATION 2.9 within our frame-
work. There are, however, at least two major obsta-
cles on the way to defining a notion of momentum 
for a quantum particle, particularly for a photon. 

Firstly, our theory does not admit plane waves since 
these are not square-integrable, and therefore do 
not belong to our Hilbert space. Instead, we work 
with Gaussian wave packets. Secondly, after interac-
tion, the system is entangled. We may start with a 
wave function that is a pure product, but we no 
longer have a pure product after the photon and 
electron interact. Thus, even if we could start with 
plane waves going in, we would not have plane 
waves coming out, which violates the underlying as-
sumption used to derive EQUATION 2.9. Developing a 
workable notion of momentum for the particles in 
our system will be part of our future work. 
 

ii. Future Research 
 We plan to investigate ways of defining a no-
tion of momentum in our framework: one that would 
allow us to derive the one-dimensional analog of the 
Compton scattering formula in some appropriate 
classical or semi-classical limit. Defining momentum 
is easy for a single particle using Fourier transform. 
There are, however, many difficulties in isolating 
photonic and electronic properties from the two-
body wave function of the system. A potential way to 
address this might be to use techniques from Linear 
Algebra to decide whether the wave function of the 
system becomes locally approximately a pure prod-
uct far away from the interaction zone. 
 Naturally, we also plan to expand our analy-
sis to three space dimensions. We hope to eventu-
ally arrive at an N-body, relativistic, and quantum 
mechanical system in three space dimensions from 
which Maxwell’s equations can be derived in the 
limit as the number of photons goes to infinity∎ 
 

 

5 ACKNOWLEDGEMENTS  
 We would like to thank the Rutgers  
University DIMACS REU program and the Rutgers 
University Math Department for their generous sup-
port. We are grateful to Dr. Shadi Tahvildar-Zadeh 
for his patient guidance, and Parker Hund for our val-
uable discussions. We thank Erika Melder for her 
technical support. 



ARESTY  RUTGERS UNDERGRADUATE RESEARCH JOURNAL, VOLUME I, ISSUE II 
 
 
 

 

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