Engineering Quarterly Summer 2024

Engineering Quarterly VOL.VI. . . No.4

SUMMER 2024

W ELCOME !

The Engineering Quarterly is published four times each year by the Hamm School of Engineering at the University of Mary. In each issue, we include articles from each of the eight majors we offer: Electrical Engineering, Mechanical Engineering, Civil Engineering, Environmental Engineering, Computer Science, Computer Engineering, Construction Engineering, and Construction Management. The intended audience of the Engineering Quarterly includes engineers, engineering students, engineering alumni, future engineering students, and people working in related fields such as science, computing, and mathematics.

In this summer issue we describe our student design projects from the spring semester, research our faculty are undertaking, new laboratory modules that we design and construct over the summer to improve the educational experience of our students, fun summer activities in which we participate as faculty, and some of our editorial thoughts on things like Artificial Intelligence and other current and past advancements that have been influenced by clever engineers. As usual, we also include some fun games and problems that you might like to try. Enjoy!

Dr. Terry Pilling

N EW C ONSTRUCTION M ANAGEMENT P ROFESSOR

Michael has been a Construction Manager for many years working across North Dakota, Virginia, and Washington, D.C. He also taught as a professor and program coordinator at the North Dakota State College of Science in Wahpeton, ND. Mike has a bachelor’s degree in Natural Resources Management from the University of Minnesota, Crookston and a Master’s degree in Building Construction Science and Management from Virginia Polytechnic Institute and State University.

Michael grew up in a Christian household that practiced their faith within the Southern Baptist tradition and valued faith, morals, responsibility, commitment, honesty, and fear of the Lord. In 2015, after a strenuous 2-year long spiritual battle and study of the Faith, Michael converted to Catholicism. Michael has been married for 19 years to his bride, Jessica and together they have five children; Timothy (13), Benjamin (10), Paul (8), Hanna (6) and Joseph (4). Michael enjoys spending time with his family and friends and finds joy in most outdoor activities including camping, hiking, hunting and fishing.

“I believe that education is important and it has value. However, without a Catholic tradition and foundation, education in my opinion is nothing more than knowledge. Knowledge in and of itself stops short of valuing and embracing the total human being. As a faithful Catholic and educator, I believe in growing both the mind and the soul. Only through an education that is tempered with Catholic Christian values and morals do we surpass knowledge and encounter wisdom. For this reason, I believe that it is not simply a choice; however, an obligation to support and advance the mission of the University of Mary by embracing and sharing both my Catholic faith and my knowledge. As a faithful Catholic, I welcome the values and I embrace the mission of the University of Mary, not only for myself; but also, for my students, the faculty, the staff and the campus community at-large. I welcome the opportunity and freedom to practice and live out my Catholic faith through service to the University of Mary. Through service to others, I believe we become the face of Jesus to those around us. Service has and always will be one of the most empowering and profound ways to evangelize the people we interact with.”

2024 S ENIOR D ESIGN P ROJECTS

The Hamm School of Engineering welcomes Michael Douglas who joins our faculty as professor of Construction Management and Construction Engineering and will begin teaching courses this fall semester.

The 2024 senior design projects were presented on Monday, April 15, in Casey 128.

These were: Pedestrian Drawbridge Project - Redesign of a pedestrian bridge in Duluth that has been plagued by malfunctioning and expensive repairs.

University of Mary

FREE

University PLA Recycling System - Students designed a filament extruder for recycling PLA.

Autonomous Rover - Students designed a rover style vehicle and a path finding algorithm to attempt to make it autonomous.

South Washington Corridor Reconstruction - Expansion of Bismarck’s South Washington Street including drainage and environmental impact.

Motorized Bowling Aid - A device that attaches to a motorized wheelchair that allows someone with limited mobility to bowl including aiming and putting spin on the ball.

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Smart Bike Rack - A bike rack and electronic lock system that is interfaced through a webpage intended for residential use (such as in an apartment complex or condominium) for people to secure their bikes.

ENR 281: S OPHOMORE D ESIGN

By ANTHONY GARCIA

The Hamm School of Engineering prides itself on student design. We incorporate design projects as part of many of our courses including ones which, traditionally, do not contain design projects at most universities. In addition to those courses, we also have courses, from sophomore through to senior level, that are entirely design-based. This past semester our sophomore ENR 281 students completed the engineering group designs that they began in ENR 280 last semester. Here is a photo montage of the final obstacle course and race where the student designs of their vehicles, static obstacles, and dynamic obstacles were tested in action.

Professor Waldenmaier’s team ended up victorious. Congratulations!

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Their service activities included fabricating canes, building furniture, repairing

By JAMES CARRICO wheelchairs and painting of the sister’s courtThis year Gabriel Caro Sanchez, Sophia Mut- yard. In addition to these service activities, tonen, Christopher Dill, and Dr. Jim Car- trip participants also toured Machu Pichu and rico participated in the service trip to Ayaviri, parts of Lima and Cusco.

Peru.

P ERU 2024

3

years. Several of the light fixtures above the trampolines at SkyZone started experiencing severe flickering after only a few days of operation. Mr. Samardzic has hired several electronics companies through these years to find the root cause of the problem but to no avail.

The service trip to Ayaviri, Peru occurs every year in early May and is open to both students and alumni. Engineering students interested in participating in next year’s trip should email jdcarrico@umary.edu and watch for canvas announcements this coming fall.

He finally reached out to UMary’s electrical engineering students.

The students made several trips to SkyZone to perform diagnosis, testing, troubleshooting, and to propose possible solutions. They also brought some of the LED bulbs back to the electrical engineerign lab at UMary to perform extensive and long-term testing, and then they did testing on site.

The areas considered in the diagnosis were software, power, signal integrity, transmission line length, shielding, and connections.

They assisted health science students and faculty volunteering in the Centro de Rehabilitacion Fisica - San Rafael as well as the religious sisters that hosted the trip participants.

S KYZONE

By FR. ATHANASIUS OWEIS

The Electronics II students, Megan Achbach, Benedict Brophy, Josh Franz, Cole Sebastian, Gannon Steffes and Isaac Trefz, helped SkyZone trampoline park’s owner Mr. Drazen Samardzic who had been facing a problem with the lighting at his facility for about eight

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more points in hands on activities. This included activities from engineering including spaghetti bridge building competition, a math escape room, and an exploration of π calculations in space. There were two other divisions: Science, which offered hands on protein labs, and Cybersecurity, which offered students the opportunity to work on security tasks.

F UN BASH P ROGRAM

By NICHOLAS STANDAGE

After testing all possible areas, the students were able to find out that this was a complex problem and there were multiple causes. They were able to identify each of them and propose solutions to every single cause. They provided these solutions to SkyZone. They then modified and corrected the documentation provided by both the lighting company and the electronics firm who had previously attempted to fix the problem.

F IRST YOUNG E NTREPRENEUR AND STEM TALENT S EARCH (YES!)

By JERIKA HAYES

Over the last year, the Gary Tharaldson School of Business and the Hamm School of Engineering have been planning, preparing, and finally hosted our first Young Entrepreneur and STEM (YES!) Talent Search.

We had 26 students representing schools from the Willison, Minot, Fargo and Grand Forks areas compete in the STEM division of the Talent Search.

The teamwork was amazing, as teams of four to five strategized how to get the most points together. Unfortunately, there could only be three winners. The Semifinals was a Kahoot game based on information shared over the two days of team’s activities. On Wednesday morning, we announced the top seven teams and had them compete in Jeopardy – making it anyone’s game no matter how many points qualified them.

This program was almost a challenge. I was studying with friend, and messing around with Bash and the dictionary, doing things like printing words that contained certain characters or other words (such as names). She asked if Bash could be used to find words that rhyme with a specified word. At the time, I didn’t know how to use AWK, so I took the idea as a challenge. A few weeks later, I successfully figured out the following code at Chesterton’s.

The program, entitled “rhymefinder,” prints out every word in the dictionary that ends in the characters specified by the user.

It is a rather simple program, and doesn’t exactly search for rhymes (e.g. “through” doesn’t rhyme with “bough”, but both would be printed if the user typed “ough” as the ending to search for), but I figured what it prints allows for creative selection. It also has the option to type “count” and print the number of words it finds (in the event that it prints 2,000 words, the user might like to see that, and perhaps make their search a little more specific).

In addition, the user can specify what characters he wants the word to start with by typing “start” after the rhyming characters. rhymefinder

#! /usr/bin/bash rhyme=$1 com=$2 if [[ $# -lt 1 ]]; then echo "Sorry - try typing only the ending of the word you want to rhyme with." exit fi if [[ $# -gt 2 ]]; then echo "Too many arguments - I don’t understand." echo "Just type the ending of the word you want to rhyme with." echo "..and if you want more customization, type the command afterwards.. :)" exit fi

Students stayed on campus, arriving Sunday June 9th and competed until midday on Wednesday June 12th. There were three distinct rounds used to find the top three of the competition. The main component was a team “Capture the Flag” of over 90 science, technology, engineering, and mathematics questions that covered the vast topics of each category. While conquering the game board, students also had opportunities to earn

The winners were Zach in 3rd, winning $1,000, Jacob in 2nd winning $3,000, and Ethan in 1st winning a grand prize of $5,000, all cash prizes. Congratulations to all the participants who were nominated and displayed their own talents fantastically, and particularly to the cash prize winners who showed their talents in groups, individual quizzes, and final Jeopardy.

case $com in [Ss]tart) echo "Type the character(s) you wish the word to begin with" echo "(note capitalization)" read extra grep -E "^$extra.*$rhyme\$" /usr/share/dict/words | column -x echo "Do you want to know how many words I just printed? [yes/no]" read answer case $answer in [Yy]es) grep -E "^$extra.*$rhyme\$" /usr/share/dict/ words | wc -w ;; [Nn]o) exit ;; *) echo "Um, $answer wasn’t an option... sorry." exit esac ;; [Cc]ount) echo "" echo "The number of rhyming words are:" grep -E ".*$rhyme\$" /usr/share/dict/words | wc -w exit ;; *) echo "The words with which that rhymes are:" echo"" grep -E ".*$rhyme\$" /usr/share/dict/words | column -x ;; esac

Here is the man page for rhymefinder written in the groff typesetting language. If you place it in /usr/local/man/man1 then you can use man rhymefinder from a terminal to get the manual page for it.

VOL.VI. . . No.4 rhymefinder.1

.\" DO NOT MODIFY THIS FILE! It was generated by help2man 1.47.3. .TH RHYMEFINDER "1" "January 2024" "GNU coreutils 8.32" "User Commands" .SH NAME rhymefinder \- finds words in the dictionary that might rhyme with the ending you give .SH SYNOPSIS .B rhymefinder ... ending of word ... command .SH DESCRIPTION .\" Add any additional description here .PP Allows you to search the dictionary for words that rhyme with the ending you give. Also offers different customization commands, such as listing how many words are printed or choosing the starting letter of the words printed. .PP Mandatory arguments to long options are mandatory for short options too. .TP "count" lists the number of words printed .TP "start" gives you the option to type what you want the word to start with .SH AUTHOR Written by Nicholas Standage. :) .SH COPYRIGHT Copyright \(co 2024 Free Software Foundation, Inc. This is free software: you are free to change and redistribute it.

There is NO WARRANTY, to the extent permitted by law .

As an example, suppose I wanted to write a poem and needed a word that rhymes with “Engineering”? Here is what rhymefinder gives me: ./rhymefinder eering Start Type the character(s) you wish the word to begin with (note capitalization) s sheering sneering speering steering superdomineering Do you want to know how many words I just printed? [yes/no] yes 5

I like how unbelievably efficient Bash is, and I appreciate its ability to do a repeated task hundreds of times in a few seconds. I especially enjoyed Dr. Pillings lectures and humor in his ENR 210 class and I found the game creation project to be the most enjoyable assignment.

Nicholas is a mechanical engineering major originally from LA County, California, USA.

C ONVOLUTION , D IFFUSION , AND Q UANTUM M ECHANICS

By TERRY PILLING

In general conversation the word “convolution” is usually taken to refer to something that is complex, intricate, or twisted, like a convoluted argument. In this article I am going to do exactly that. I will make a convoluted argument that quantum mechanics may in fact arise from the possibility that we live in a universe where “time” is actually a complex, rather than a real, number. The real part of that complex number leads to phenomena that we perceive as diffusion or heat conduction whereas the imaginary part of that complex number leads to phenomena that we perceive as quantum mechanical. In fact, it may be that the reverse implication is the actual one and that “time” is, itself, nothing but our perception of those two processes: diffusion (e.g. thermodynamics) and quantum mechanics (e.g. chemistry).

In mathematics and engineering “convolution” refers to mathematically combining or intertwining elements to produce a new, often more complicated, result. For example in the mathematics of integral transforms the convolution is an integral that expresses the amount of overlap of one function as it is shifted over

Engineering Quarterly SUMMER 2024 another function. It is a fundamental operation in signal processing and the theory of linear systems. Convolution is used to apply filters to signals, such as in image and audio processing. It involves sliding a kernel (a small matrix) over the signal (or image) to produce a new transformed signal (or image). In the field of machine learning and artificial intelligence, convolution is a key operation in convolutional neural networks (CNNs), which are used primarily for image recognition tasks. It involves applying a convolution operation to the input data to extract features.

We all recall, from elementary mechanics, that if we would like to know whether or not a particle which is moving at a constant speed will arrive at a position x in time t we simply take its position now x0 and add its velocity v multiplied by the time. If x = x0 + vt then the particle will indeed be at position x at time t. Suppose that instead of a single particle we look at a swirling mixture of particles and we would like to know how many of those particles will be located at the position x at a given future time? If we know exactly where every particle is and exactly what each velocity is we could simply figure out which ones will be located at x at that future time by solving x = x0 + vt for all the particles as we did with the single particle above. However, in the case of real mixtures, there are far too many particles for us to do this deterministic calculation. Instead, we assume that the particles in the mixture are distributed according to some probability distribution (perhaps coming from the theory of thermodynamics, fluid dynamics, random diffusion, etc.). In that case, the particle positions and velocities become random variables whose values are taken from the probability distribution rather than being known in advance.

When dealing with random variables we cannot simply add and multiply them as we did with deterministic variables x and v above. Instead, to find the sum of the two random positions, we use a convolution to find the probability distribution of their sum.

Suppose X and Y are two independent random variables with probability density functions (PDFs) pX (x) and pY (y) respectively. Recall that the PDF is used to calculate the probability that the value of X is between X = a and X = b as follows Z b P (a ≤ X ≤ b) = pX (x)dx a

To add random variables X and Y means that we want to find the probability density function of the random variable Z = X + Y . The PDF of Z, denoted pZ (z), is found by the convolution of pX (x) and pY (y). Mathematically, this is expressed as: Z ∞ pZ (z) = pX (x)pY (z − x)dx −∞

where the integrand, pX (x)pY (z − x) is the probability that X is near x AND Y is near z − x. So for some fixed value of z we are integrating over all possible allowable values of x to arrive at pZ (z), the probability density function of Z as the sum of X and Y .

The convolution integral effectively “shifts” one distribution over the other and calculates the weighted overlap for each point

5

x. Another way of looking at it is this: The quantity pY (z−x) is the probability that a particle at position z − x in the previous time step and pX (x) is the probability of a particle being at a given value of x. If we integrate over all possible x we are computing a sum of all of the particles that could arrive at z weighted by the probability that they will. This results in the new probability distribution, one time step later, for the point Z. This can be made explicit by using mechanics as we did above. Let x(t) = x0 + v0 t give the position of a particle at time t if it started at position x0 and traveled at a constant speed v0 . Now instead of deterministic variables, let X0 and V0 be random variables that come from probability distributions p1 (y) and p2 (v0) then our mechanical equation is saying that the position of the particle at time t is given by x(t) = X0 + V0 t and the probability density function for x(t) can be found by using the convolution. First, let Wt = V0 t be the scaled version of V0 so that (by properties of probability distributions under scaling) 1 w pWt (w) = p2 t t and now x(t) = X0 + Wt and so the convolution is Z ∞ p1 (y)pWt (x − y)dy p3 (x) = −∞

which is Z ∞

1 p1 (y) p2 p3 (x) = t −∞

x−y t

dy

This shows that x(t) is a random variable whose probability density function (PDF) is the convolution of the PDF of the initial position X0 and the scaled PDF of the initial velocity V0 multiplied by t. This last equation says that the probability that x(t) is near x at time t is given by the joint probability that X0 is near y and that V0 t is near x − y for all possible y scaled by 1/t. Note that in the case that the time step is taken to be 1 this reduces to the simple convolution of two random variables we looked at above.

Diffusion In the context of diffusion, we use the convolution of probability distributions to compute the concentration of particles (number of particles per unit volume) at a given position x and time t. The diffusion equation in one dimension for constant diffusivity ∂ 2 C(x, t) ∂C(x, t) =D ∂t ∂x2 where C(x, t) is the concentration of particles at position x and time t, and D is the diffusivity or diffusion coefficient. We will solve this using the method of Green’s functions.

Consider a linear differential operator L acting on a function C(x) such that LC(x) = S(x) The Green’s function G(x, ξ) is defined as the solution to the differential equation with S(x) a delta function source at ξ LG(x, ξ) = δ(x − ξ) The steps for finding the Green’s function G(x, ξ) are:

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• Solve the homogeneous equation This is a first-order linear ordinary differential The diffusion kernel, K(x, t), for time t is exLG(x, ξ) = 0 for regions away from equation in t. For t ̸= 0, the homogeneous actly the Green’s function that we found in the source. part of this equation is: solving the diffusion equation for the delta function source: • Apply the boundary conditions. b t) ∂ G(k, b t) = 0 x2 + Dk 2 G(k, 1 • Ensure the Green’s function satisfies the G(x, t) ≡ K(x, t) = √ e− 4Dt ∂t 4πDt jump conditions imposed by the delta with general solution function source.

The concentration at position x and time t, −Dk2 t b C(x, t), is obtained by convolving the initial G(k, t) = A(k)e Once the Green’s function G(x, ξ) is known, concentration C(x, 0) with the diffusion kerthe solution to the original inhomogeneous To find A(k), we need to consider the effect nel K(x, t): differential equation is given by: of the delta function source at t = 0. InteZ Z ∞ grating both sides over an infinitesimal interC(x) = G(x, ξ)S(ξ) dξ C(ξ, 0)K(x − ξ, t) dξ C(x, t) = val around t = 0 we get −∞ # Z 0+ Z 0+ " b as can be seen by direct substitution: The C(ξ, 0) is the initial concentration distri∂G 2 b dt = Z

δ(t) dt + Dk G bution at ξ and K(x − ξ, t) is the probability ∂t 0− 0− LC(x) = L G(x, ξ)S(ξ) dξ that a particle was at position x − ξ and difZ The integral of the delta function is 1 and so2 fuses to position x. Inserting the initial Gaussian distribution we have = LG(x, ξ)S(ξ) dξ + h i 0 Z 2 b t) Z ∞ − 2σξ 2 − (x−ξ)2 G(k, =1 − = δ(x − ξ)S(ξ) dξ e 0 e 4Dt 0 p √ dξ C(x, t) = 2 4πDt 2πσ −∞ which implies a jump discontinuity at t = 0 0 = S(x) R b 0+ ) − G(k, b 0− ) = 1 G(k, The integral C(x) = G(x, ξ)S(ξ) dξ represents the superposition of the responses to b t) = 0 since the Now for t < 0 we have G(k, point sources distributed according to S(x).

In the case of the diffusion equation we system is initially undisturbed have b 0− ) = 0 G(k,

∂ ∂2 − D 2 C(x, t) = S(x, t) thus ∂t ∂x b 0+ ) = 1 G(k, which identifies the linear operator L. We can −Dk2 t b , we solve it by using a delta function source at the and plugging in to G(k, t) = A(k)e origin S(x, t) = δ(x)δ(t) ≡ δ(x, t) and find- find the constant A(k) is 1 and so ing the Green’s function. b t) = e−Dk2 t G(k,

∂ ∂2 − D 2 G(x, t) = δ(x, t) To find G(x, t), we take the inverse ∂t ∂x b t) Fourier transform of G(k, First we take the Fourier transform of both n o 2 sides of the equation with respect to x. The G(x, t) = Fx−1 e−Dk t Fourier transform of G(x, t) with respect to x is defined as: The inverse Fourier transform of a Gaussian Z ∞ in k-space is another Gaussian in x-space b t) = G(k, G(x, t)e−ikx dx Z ∞ −∞ 1 2 e−Dk t eikx dk G(x, t) = 2π −∞ The Fourier transform of the delta function is Z ∞ This integral is known to be a Gaussian funcb t) = δ(k, δ(x)δ(t)e−ikx dx tion3 . −∞

= e−ik·0 δ(t) = δ(t) 1 x2 G(x, t) = √ exp − 4Dt 4πDt Now we apply the Fourier transform, denoted Fx , to both sides of the differential equation Hence we find the Green’s function G(x, t)

2 for the diffusion equation with a delta func∂ G(x, t) ∂G(x, t) Fx −D = F {δ(x, t)} x tion source. ∂t ∂x2 Suppose we examine a bunch of particles Using linearity of the Fourier transform we with some initial distribution and we want to have find the concentration of particles at a given

position x at some later time t due to difb t) ∂ 2 G(x, t) ∂ G(k, − DFx = δ(t) fusion. In that case, since we know that ∂t ∂x2 the Green’s function for the diffusion equaThe Fourier transform for the second deriva- tion, we simply convolve the initial concentration distribution with the diffusion equation tive1 with respect to x Green’s function, or “kernel”.

2 ∂ G(x, t) 2b Let us assume our initial concentration of = −k G(k, t) Fx ∂x2 particles C(x, 0) as a Gaussian distribution centered at x = 0 with some initial spread σ0 gives then 2 − x2 1 b t) 2σ0 ∂ G(k, p C(x, 0) = e 2b + Dk G(k, t) = δ(t) 2πσ02 ∂t

The exponents can be combined by completing the square as follows.

Suppose we have a Gaussian integral of the form Z ∞ 2 e−αx +βx dx I= −∞

where α is a positive constant and β is a constant. The exponent −αx2 + βx can be rewritten by rewriting it in the form −α(x − h)2 + k

β 2 2 −αx + βx = −α x − x α

β 2 to get 2α

now add and subtract β x − x= α 2

2 2 β β x− − 2α 2α

and the integral now becomes Z ∞ β 2 β2 I= e−α(x− 2α ) + 4α dx −∞

and we factor out the constant term: Z ∞ β2 β 2 4α I=e e−α(x− 2α ) dx −∞ β Now let u = x − 2α so that du = dx, and the limits of integration remain −∞ to ∞ Z ∞ β2 2 I = e 4α e−αu du −∞

R∞

2

The integral A = −∞ e−αu du is a standard Gaussian integral which can be solved by squaring it and switching to polar coordinates: Z ∞ 2 2 2 A = e−α(u +v ) du dv −∞ Z 2π Z ∞ 2 = dθ e−αr r dr 0 Z ∞ 0 2 = 2π e−αr r dr Z0 ∞ dy = 2π e−y 2α 0 π = α

R b t) = ∞ G(x, t) ∂ 22 e−ikx dx and integrating by parts twice.

This can be derived by using −k 2 G(k, ∂x R 0+−∞ 2 R 0+ b 2 b b 0+ Note that on the left hand side the integral 0− Dk G dt → 0 since the integration interval vanishes and the integral 0− ∂∂tG dt = G| 0− gives our result.

3 See derivation by completing the square later in this article 1

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and so

r A=

I=e

r

π α

So, the value of the Gaussian integral is: r Z ∞ π β2 2 e 4α e−αx +βx dx = α −∞ We can now use this result to write our Gaussian integral for the concentration of particles at position x at time t as −

C(x, t) = p

solve the Schrödinger equation parallels the use of the diffusion kernel to solve the diffusion equation. Both involve convolving an initial state with a kernel that encapsulates the dynamics of the system. So we find that the propagators, or kernels, for evolving both For a free particle (i.e., V (x) = 0), the equaa classical diffusion system and a quantum tion simplifies to: free particle system are identical if you sim2 2 ply Wick rotate the time and redefine the dif∂ψ(x, t) ℏ ∂ ψ(x, t) iℏ =− fusion coefficient as we will show later. How∂t 2m ∂x2 ever, the concentration, C(x, t) that we deNotice that if we divide both sides by iℏ we rived from the diffusion equation get x2 − 2 +2Dt) ℏ ∂ 2 ψ(x, t) ∂ψ(x, t) 2(σ0 e =− C(x, t) = p ∂t 2mi ∂x2 2π(σ02 + 2Dt) Now compare this with the diffusion equation: is not as simply related to the quantum wave∂C(x, t) ∂ 2 C(x, t) function ψ(x, t). However, if we square the =D 2 complex wave function to find its squared ∂t ∂x modulus we find both expressions are identiwhich shows that the Schrödinger equation cal in the limit of ℏ → 0 and D → 0. could be thought of as the diffusion equaFood for Thought tion with imaginary diffusion coefficient D = iℏ with the concentration being interpreted 2m We have seen that, although the classical as the wave function4 or ‘probability ampliprocess of diffusion and the particle propagatude’. However, we will later interpret it tion in quantum mechanics may seem worlds ℏ with a real diffusion coefficient D = 2m apart, they share a striking mathematical parand use an imaginary time instead. For a allel through the concepts of kernels and propfree particle, the propagator is found from the agators revealing how convolution of probaSchrödinger equation using the same Green’s bility distributions plays a pivotal role in both function method we used above for the diffucontexts. In diffusion, the kernel describes sion equation above to be how particle concentrations spread out over r time and in quantum mechanics, the propm im(x−x0 )2 e 2ℏt K(x, t; x0 , 0) = agator describes how probability amplitudes 2πiℏt evolve in time.

Notice that if we start with the kernel (or We can now solve the time-dependent propagator) governing the diffusion equation Schrödinger equation using the propagator in the same way that we did with the diffusion (or heat equation) and we rotate (Wick rotaequation, by convolving the initial wave func- tion) to imaginary time via t → it then we get exactly the Schrödinger equation if we idention ψ(x0 , 0) with the propagator: tify D = ℏ/(2m). Why is this? Is it possible Z ∞ that we live in a universe with a complex time ψ(x, t) = K(x, t; x0 , 0)ψ(x0 , 0) dx0 −∞ but that we only perceive the real projection Let’s consider an initial wave packet that is of it? So that the real part appears to us as difGaussian just as we did in the diffusion case5 fusion and the imaginary part appears to us as quantum mechanics? Quantum effects such as 1/4

2 superposition and interference all come from x0 1 exp − ψ(x0 , 0) = the fact that amplitudes are complex numbers 2πσ02 4σ02 which are added together as complex numand convolving this with the free particle bers before projecting to their real parts when a measurement is done. propagator we get The imaginary time transformation we are ψ(x, t) = describing is known as a “Wick rotation”, a Z ∞s x2 im(x−x0 )2 method used in theoretical physics where time − 02 m 2ℏt 4σ0 p e dx 0 is replaced by imaginary time via the substi2πiℏt 2πσ02 −∞ tution t → it. This transformation turns the As before we combine exponents and com- exponential factors in solutions of differenplete the square to simplify the integral and tial equations from oscillatory (trigonometric find the resulting integral to be a Gaussian that functions) into decaying exponentials, which are often easier to handle mathematically, parcan be evaluated to give: ticularly in the context of quantum field thex2 − ory and statistical mechanics. Starting with 2 e 4(σ0 +iℏt/2m) the classical diffusion kernel ψ(x, t) = p 4 2π(σ02 + iℏt/2m) (x−x0 )2 1 e− 4Dt K(x, t; x0 , 0) = √ 4πDt The initial Gaussian wave packet spreads out ℏ over time due to the uncertainty principle. The and letting t → it while letting D → 2m gives width of the wave packet increases, and the (x−x0 )2 − 1 ℏ τ −4i 2m peak decreases, similar to the diffusion proK(x, iτ ; x0 , 0) = q e ℏ cess. The propagator K(x, t; x0 , 0) describes −4πi 2m τ how the wave function evolves from the inir m im(x−x0 )2 tial state ψ(x0 , 0) to the state ψ(x, t) at time = e 2ℏt 2πiℏt t. This process of using the propagator to Schrödinger equation for a particle in one dimension with a potential V (x) is given by:

ℏ2 ∂ 2 ∂ψ(x, t) = − + V (x) ψ(x, t) iℏ ∂t 2m ∂x2

π α

and we have β2 4α

7

x2 2

e 2(σ0 +2Dt) 2π(σ02 + 2Dt)

which shows that the initial Gaussian distribution spreads out over time, with p the width of the distribution increasing as σ02 + 2Dt. The peak of the distribution decreases, maintaining the total number of particles constant.

This is a direct consequence of the convolution of the initial distribution with the diffusion kernel. In other words, this is showing us that the concentration of particles that we can expect at a given location x at future time t is found by adding up all of the particles that can possibly get there weighted by their probability of doing so.

Quantum Mechanics In quantum theory, the evolution of a particle’s wave function can be described using the path integral formalism introduced by Richard Feynman. Here, the propagator (or kernel) plays a central role, similar to how the diffusion kernel functions in the diffusion equation. The propagator K(x, t; x0 , t0 ) in quantum mechanics is the probability amplitude for a particle to travel from position x0 at time t0 to position x at time t. It is given by: i

K(x, t; x0 , t0 ) = ⟨x|e− ℏ H(t−t0 ) |x0 ⟩ where H is the Hamiltonian of the system.

The propagator can also be expressed as a sum over all possible paths from x0 to x: Z i K(x, t; x0 , t0 ) = D[x(τ )]e ℏ S[x(τ )] where S[x(τ )] is the action along the path x(τ ), and D[x(τ )] represents the integration over all paths.

In both the diffusion and quantum mechanical contexts, the kernel (propagator) represents the system’s evolution over time. In diffusion, it shows how particle concentration spreads out, while in quantum mechanics, it describes how the probability amplitude evolves. By using the convolution (or path integral) approach, we can see how the initial distribution transforms over time, providing a deep connection between classical diffusion processes and quantum mechanical evolution.

We can solve the Schrödinger equation of quantum mechanics in a similar way to how we solved the diffusion equation above by finding the propagator. The time-dependent

4 Note that the wavefunction cannot be identified with the concentration since it is the modulus squared that is observable in the case of the wavefunction rather than the wavefunction itself.

R∞ 5 We have chosen the numerical factors so that the wavefunction is normalized correctly: −∞ |ψ(x0 , 0)|2 dx0 = 1.

VOL.VI. . . No.4

Engineering Quarterly SUMMER 2024

which is identical to the quantum mechanical inherently probabilistic nature of phenomena, the wave-particle duality, and non-locality – propagator. could potentially be understood in a more If the time dimension of our universe were classical way if we had access to or could percomplex rather than real then it might be pos- ceive these additional complex dimensions. sible that what we observe as quantum mechanical phenomena are actually just manifestations of underlying processes that are more diffusion-like in a higher-dimensional complex space. This viewpoint would suggest that quantum mechanical “weirdness” – the

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