| Name: | Hallie Byatt |
|---|---|
| Email: | hallie.byatt@gatech.edu |
| Home Institution: | Georgia Institute of Technology |
| Project: | Mathematical Adventures in One-Dimensional Physics |
| Mentor: | Professor A. Shadi Tahvildar-Zadeh |
About My Project
I will be analyzing the interaction between an electron with $N$ photons in a one dimensional system; first by looking at the case with one electron surrounded by two photos (i.e. an electron with a photon on either side). Then, I will formulate a multi-time wave function for the entire system by taking the tensor product of the three individual wave functions, establish a boundary condition for the system, and finally solve the IVBP for the the three-time, three-body wave functions.
Research Log
Week 1: May 25 - May 29
I met with Professor A. Shadi Tahvildar-Zadeh twice this week. We figured out what I was interested in and which project to pursue. I then met with the Mathematical Physics research group at Rutgers and found someone who is also interested in the two photon one electron case.
Week 2: May 30 - June 4
I met with Professor A. Shadi Tahvildar-Zadeh once this week. We went over the different appearance of the wave function and trajectories for massive vs massless particles. Massless particles, photons, move along a constant and straight trajectory whereas massive particles, electrons, have a more jittery motion. Additionally, there is smearing involved in the graph of an electron's wave function which is not present for photons. For the rest of the week, I worked on my conceptual understanding of the project. The case with two electrons and one photon has already been done by Rutgers' students, and I spent a lot of time going over their paper and watching their seminar to further understand what I will need to accomplish for my own configuration.
Week 3: June 6 - June 11
This week I focused specifically on the non-interacting case, i.e. I found solutions for when the electron and photons are very very far away from one another. In total, I will be looking at three cases: non-interacting (or far configurations), near configurations (where one photon interacts with the electron), and super-near configurations (where both photons interact with the electron). Since I was only focused on the non-interacting case this week I didn't have to deal with boundary conditions, which made the problem significantly easier. Getting a solid understanding of the problem in its most basic form was my goal.
Week 4: June 13 - June 18
This week I moved onto the solutions for the near configuration, i.e. when only one of the photons is interacting with the electron. I completed these solutions, but it took me a while to understand what was going on conceptually. I eventually got the concepts, so I began typing up my solutions on LaTex and write some MATLAB code. For the code, I started working on the probability density plots of a single photon and a single electron. I am still working on the electron plot as I haven't been able to properly apply the bessel functions in the code.
Week 5: June 20 - June 25
This week I worked on the probability density plots for one particle and two particles. I only focused on the non-interacting version for the two particle case, so no boundary conditions were implemented. I created a heat map and a 3-D plot for the 2 particle case and the outcome was what should have been expected. After working in MATLAB, I went back to my solutions for the near configurations and I put them in terms of the initial data and put these solutions into LaTex. I did this so that way when I put these into code it would be easier for me to break up the values and do it correctly.
Week 6: June 27 - July 2
This week I completed the code for the probability density plots of the 2 particle non-interacting case. Then I went to work on the solutions for the super near configurations of the 3 particle case with an electron surrounded by two photons. This was incredibly complicated for me to understand conceptually, and working through my conceptual understanding of the math took me a long time. I feel confident in my intuition for the solutions, but I still need to finalize the equations and express them in terms of initial conditions. I also started working on the code for the 2 particle interacting case. This code is much more complex than my previous work since I have to implement the boundary conditions and the equations themselves are much longer.
Week 7: July 4 - July 9
This week I finalized my solutions for the super-near configurations of the 3 particles. It turned out to be much more simple than I thought it would be since it became a "far-near" configuration as soon as the first photon was dealt with. I am continuing to work on the code for 2 particles interacting, but it is proving to be much more difficult than I anticipated.
Week 8: July 11 - July 16
This week I have still been working on the code for 2 particles interacting. This has been taking me a very long time as I keep having to change small things to try and get the code to run. I believe I have all of the concepts down correctly in MATLAB, but I think there's just a small bug in my code that needs to be fixed. As soon as I find this bug I should be able to quickly change this 2 particle scenario into a 3 particle scenario. I am rushing to get this done before the end of next week, I really hope I will be able to figure everything out. I've also been working on my paper in LaTeX, but I haven't been able to include any graphical MATLAB results in yet since my code isn't working.