Kabir Narayanan

DIMACS REU '22

This is Kabir's research website for the DIMACS REU program during Summer '22.

Contact Info: kabir_narayanan@brown.edu

I am a rising junior at Brown University seeking a B.Sc in Mathematics who is interested in the intersections between math and physics. This summer I am investigating 1-dimensional photon-electron interactions, using a relativistic extension of Bohmian mechanics (based on the multi-time wave function formalism), building on previous work, cited below.

This work was conducted during the DIMACS REU Program.

This project was supervised by Professor A. Shadi Tahvildar-Zadeh

Research conducted in collaboration with Abby Perryman .

Research Log

Week the First

June 1-3

This week, my partner and I met with Prof. Tahvildar-Zadeh to clarify the specific research questions we will be investigating. We were then able to put together a presentation detailing our plans to study the role of momentum and energy in the relativistic Bohmian version of Compton scattering, with the goal of bridging the gap between the classical formulation (p = gamma * mv) and quantum formulations (as operators & expectation values). I began to work on solving the 1-D Dirac equation and Guiding equation for a single free electron as well.

Week 2

June 6-10

After presenting on our project on Monday, my partner and I continued to investigate the single free particle cases. For the photon, using Gaussian initial data we determined a formula for the acceleration and some conditions on the particle's long term behavior (v = +- c). For the electron we continued to work on developing workable code to solve the Cauchy problem, and find particle trajectories. In meetings with Prof. Tahvildar-Zadeh, we continued to discuss concepts in QM and functional analysis (Fourier transforms, self-adjoint operators, spectrum, etc.) and I did some independent review of linear algebra and the relevant chapters of Jean Bricmont's 'Making Sense of Quantum Mechanics.'

Week 3

June 13-17

This week we had success in computing electron trajectories and I began to refine my code to work more efficiently. My partner and I began to gather data, plotting probability density functions, trajectories, and particle momenta to better understand how various physical and control parameters affect the electron behavior. Through looking at our graphs and in conversation with Prof. Tahvildar-Zadeh we became interested in the asymptotic values of particle momentum, which I started to compare with its statistical expected value. We began to investigate what relationship might exist between the momentum operator and the relativistic momentum formula for a Bohmian particle.

Week 4

June 20-24

In addition to momentum, we began to look more seriously at energy. Prof. Tahvildar-Zadeh stressed the importance of sampling trajectories according their initial probability distribution, so we began to generate graphs for large numbers of trajectories at once. While there is no exact match between the average relativistic momentum and the operator value, we still suspect a relationship will hold for 'typical trajectories.' As our time-consuming code ran, I did a little more reading on quantum state operators, the Heisenberg Picture (which we discussed in meetings with our mentor), and the mysterious 'zitterbewegung' of the electron, and started to look at the mean and variance of the electron position distributions.

Week 5

June 27-July 1

This week, near the midpoint of the program marks a turning point in our REU progress, so I will use the opportunity to summarize. We have finally begun to move into the 2-body case, for which it will take some time to develop a workable code in Python, and we are just beginning to sketch out our final report for the REU program. Thus far, our simulations have basically replicated the work of previous REU students and [KLTZ2020], and we will continue to do so. However, we have learned some interested things about energy and momentum! Principally, that while orthodox QM dictates that operators give the probability distributions of 'finding' values of an observable that has been measured for, the statistics of Bohmian trajectories actually agree with particles having these distributions of velocity, energy, momentum, etc. While this is apparent in the non-relativistic case, through numerous numerical computations we have fairly strong evidence for the fact that free electrons asymptotically reach the energy one would expect, and plus or minus the momentum.

Over the weekend, Prof. Tahvildar-Zadeh showed us a neat way to visualize trajectories using the Bloch Sphere representation. On the Bloch sphere, we clearly see a 'spiraling-in' behavior of typical trajectories to a momentum value corresponding to the expected value of the momentum operator OR (though less likely) to a point on the other side of the sphere corresponding to the opposite momentum. We suspect the relative frequency of convergence to the 'correct' momentum or its opposite is related to the relative weights of the initial spin components, which in 1-D simply correspond to left or right transport. More work needs to be done to test these observations, but we hope that this asymptotic convergence can be described formally/mathematical and that these results for the single-body cases will prove useful for the two-body case (i.e. to extract a 'post-collision' momentum from these particles).

Week 6

July 4-8

I was away this week, but I worked on developing code for the 2-Body problem, sketching out the solutions included in [KLTZ20] and some code to determine trajectories from the derived probability currents. It will likely take some time before this code works however, due to various bugs and issues with computation time and integral convergence. Our mentor recommended we try Monte Carlo methods to increase efficiency for multiple integrals. Additionally, I worked on trying to understand asymptotic convergence in terms of the Cayley-Klein parameters. The plots I generated on the Bloch Sphere showed trajectories accumulating on not-quite antipodal points on the sphere. I made some plots of convergence time and tried to determine whether a plane wave solution existed for all convergence points. I also did some general reading in the hopes of finding a method to demonstrate asymptotic behavior, but learned from Prof. Tahvildar-Zadeh that perturbation methods for stability analysis which are well understood for ODEs are in general more difficult for PDEs.

Week 7

July 11-15

This week was spent trying to make sense of the Bloch Sphere Representation. I tried to look for literature studying the asymptotic behavior of the Dirac equation. I found a calculation dating back to Pauli that uses a WKB-type approximation on the Dirac equation and yields a relativistic Hamilton-Jacobi equation. However, our goal is to analyze behavior that is asymptotic in time and I couldn't find any articles that stated similar results for the free Dirac equation. As much as the discussion on 'semiclassical trajectories' seems to mirror the behavior we are seeing for certain choices of parameters, we are trying to analyze real trajectories, and may need different techniques to achieve that. In the meantime, we began to work on the presentation and continued to make more plots. We made plots in each variable and asymptotically tended to see a line-up with the plane wave solutions, but not exactly: in particular, the trajectories that tended toward -k seemed to have omega values split between +/- pi. Every day, this made less and less sense to me. I made a series of plots of the expected values or derivatives of the Bloch variables along with values obtained from trajectories against initial position and saw quite nicely, as predicted, a point of bifurcation on the initial x-axis which sent trajectories to its right asymptotically to one plane wave solution, and trajectories to its left asymptotically to another. However, the problems with omega (the azimuthal angle persist).

Week 8

July 18-22

The problem has been fixed! During a meeting, we realized there was a division by two error in a calculation, which once resolved shows exact alignment between theta, omega, and the time derivatives of R and phi with the two plane wave solutions we obtained in the Cayley-Klein parameters. Professor Tahvildar-Zadeh recommended we try to determine the rate of convergence with the hopes of making an asymptotic expansion in each variable to show stability around the plane wave solutions. I began to make log-log plots of the difference between the expected value and the values along many trajectories vs. time in order to find a good ansatz for the expansion. Unfortunately, no clean power law seemed to emerge, and the log-log plots looked far from linear. There appeared to be a quick drop, and then no clear continued descent, while particularly for small values of sigma, there were 'straggler' trajectories which did not seem to settle down in a short amount of time. In the mean time, my partner and I worked on our presentation, trying to make it accessible yet mathematically interesting for an audience of REU students who do not know much physics. We made some figures to that end and I worked on an animation of trajectories on the Bloch sphere. We delivered the presentation on Friday.

Week 9

July 25-29

After the bug fix finally cleared up confusions about what solutions the trajectories were converging to and the attempts to find a bound on asymptotic convergence in time were unsuccessful, my partner and I re-evaluated a little. We realized that in our desire to understand the well-behaved cases better, we had neglected the pathological instances where our fairly broad claims about asymptotic convergence were less convincing. In particular, we had always know that for small initial expected values of the momentum operator compared to mass, there appeared to be a much wider spread to the momentum distribution between -k and k. This agreed with my own intuition about the Klein-Gordon equation dispersion relation being more dispersive for smaller k, and reducing to simple traveling wave packets in the large k limit. We had even discussed the possibility of shifting reference frames to one in which k was large to see convergence better for general initial data. However, sigma too appeared to be a persistent problem. One of the formal assumptions underlying the well-posedness theorem which Kiessling et al. proved about the 2-body case is that the initial probability distributions for the photon and electron have disjoint compact supports, so that the boundary condition is vacuously satisfied at t=0. In other words, we need the two particles to be initially non-interacting, which means that we need to be able to localize them sufficiently in position space, without having them start ridiculously far off in our unit system. This means that our single particle results should be generalizable for small sigma, but it appears that for small sigma, we need a sufficiently large k to achieve asymptotic behavior with reasonable certainty. A next avenue for inquiry in the single-particle case will be to study the variance of the Bohmian momentum distribution for the electron distribution (by looking at the statistics of trajectories), and determine whether it shrinks slowly in time, or stabilizes to some value if the Bloch sphere variables enter some kind of limit cycle. Moreover, it will be important to make the conditions on asymptotic convergence more precise. On that note, my partner suggested an interesting interpretation of the behavior as a kind of diffraction pattern. The initial standard deviation in position acts as a kind of slit, and since we're in one-dimension, the pattern plays out along the line, thus the diffraction pattern shows up in the probability density function. We're hoping this analogy might be useful in fleshing out when we get more interference effects, and when we can approximate our probability density as two traveling waves at large time. The paragraphs above represent lingering concerns, but the week was spent mostly writing our final report. Further work on the two-particle case has been postponed until after the end of the program (though we have discussed ideas about how to analyze and make sense of plane wave-like behavior in an entangled state). Writing the report has been a little tedious but in general very useful for clarifying our ideas and trying to make a coherent story out of what we've been doing. It's also allowed me to reflect on everything I have learned about quantum mechanics, the De-Broglie Bohm theory, and also general scientific practices over the course of this REU. I am very grateful for my partner and my mentor, and I'm looking forward to continuing this work once the 9 weeks have concluded.

References & Reading

Jean Bricmont, Making sense of quantum mechanics, Springer International Publishing, 2016.

Michael K.-H. Kiessling, Matthias Lienert, and A. Shadi Tahvildar-Zadeh, A lorentz-covariant interacting electron–photon system in one space dimension, Letters in Mathematical Physics 110 (2020), no. 12, 3153–3195.

Arthur H. Compton, A quantum theory of the scattering of x-rays by light elements, Phys. Rev. 21 (1923), 483–502.

Adriana Scanteianu and Xiangyue Wang, Particle trajectories for compton scattering in one space dimension, Aresty Rutgers Undergraduate Research Journal 1 (2021).

New Brunswick Coffee Shop Ratings

Over the 9 weeks, my partner and I decided we needed to explore New Brunswick and find the best and worst places to drink coffee, troubleshoot code, and stare perplexedly into the distance contemplating quantum mechanics. The reviews are in:

Gerlanda's Bakery (Busch Student Center) 3/5

Hot coffee, very serviceable. Noisy location for work, but ideal for professor sightings.

CORE kitchen 2/5

Free coffee, a little cramped but on the plus side you're guaranteed to bump into a fellow REU student...

Efe's Cafe 4/5

Yummy coffee and pastries... very nice vibes. Good place to work if not busy, but bring a sweatshirt!

Gloria Jean's 4/5

Ideal place to sit in a booth for hours and grind on your code... has everything you need (easily accessible outlets included.

Hidden Grounds/Simply Chai 3/5

Good coffee, Hidden Grounds might be a good place to work, but neither the prettiest location nor the best work environment...

Friends Cafe 2/5

Definitely was not my most productive location, but if you're obsessed with Friends (the tv show), love dog-watching, and like to feel like you're in the Twilight Zone, this is the spot!