General Information
| Student: | Xiangyue Wang |
|---|---|
| Office: | CoRE 417 |
| School: | Rutgers University |
| E-mail: | xw298@scarletmail.rutgers.edu |
| Project: | Relativistic Quantum Description of Photon-Electron Collision in One Dimension |
Project Description
Among the many significant achievements of modern physics are Special Relativity and Quantum Mechanics. The former describes how space and time behave assuming that the speed of light is constant; the latter describes the world at the subatomic level. Quantum Field Theory is an attempt to reconcile the two. The theory is tremendously successful in terms of making predictions, but it lacks mathematical rigor. Furthermore, nobody has been able to incorporate gravity into Quantum Field Theory. Professor Tahvildar-Zadeh and Professor Kiessling have been exploring an alternative to Quantum Field Theory that is mathematically rigorous, and holds the promise of being reconcilable with General Relativity (Einstein's theory of gravity). In this project, we will use Matlab to examine the results of this alternative, starting from the the equation of motion of photon-electron collision in one dimension.
Weekly Log
- Week 1:
- This week, we started off by examing the wave equation of a single photon in one dimension. Using Matlab and the wave equation provided by Professor Shadi, we were able to come up with a picture of how the probability density and the actual location of the photon change over time. We also prepared the slides for our initial presentation.
- Week 2:
- Adriana and I began this week by tackling the probability density current of a single electron. The case is much more complicated than that of a single photon. Electron, unlike photon, has mass, and the extra mass term in the wave equation means that its solution is no longer simple. After many failed attempts, (and with ample amount of help from Professor Tahvildar-Zadeh) we came up with the picture of electron's probability density current over time.
- Week 3:
- In a similar fashion as that of a photon, we solved the guiding equation of electron to obtain a picture of its trajectories. We varied two parameters, mass and standard deviation of the initial distribution, to see how they influence the trojactory (see graph below)
- Furthermore, we obtained the probability density and trajectories of the case in which both electron and photon are present but not interacting. This involves solving a separate guiding equation for each particle given by one wave function that describes the entire system. The following video shows the time evolution of this two-body system's probability density.
- Week 4:
- We arrived in week 4 with the goal to find out the tragectories of a non-interacting system of two particles. It was precisely what we did. Instead of the single guiding equation, we had two guiding equations, one for each particle. Each guiding equation not only had time and location of its particle as variables, but also time and location of the other. Fortunately, thanks to the Hypersurface Bhom Dirac Thoery (HBD Theory), we were able to choose a "common" time for both particle. We produced the graph, and it looked like what we expected: since the two particles are not interecting, the graph is simply composed of the trajectories of a single electron next to the trajectories of a single photon.
- Week 5:
- Our mentor, Prof. Tahvildar-Zadeh, was leaving for a conference in Europe in the upcoming two weeks. As a result, we spent the week meeting with him everyday to make sure we can tackle the interacting case smoothly. Specifically, we discussed what we meant by "interaction". When a photon collides with an electron, we expect the two particles to not pass right through each other. In other words, we expect the relative velocity between the two particles to become zezo when they approach one another. This is prcisely the boundary condition that we have to add to the non-interacting case to manifest the interaction. Our four-component wave function will vary depends on the initial location and time. Knowing the theoradical background, Adriana and I were ready to tackle the coding in the following weeks. Furthermore, we made significant progress in terms of our final presentation. We finished making the slides that delineate the single electron and single photon cases. We presented what we have so far to Prof. Tahvildar-Zadeh in a small coffee shop in Princeton, during which he helped us to make our language precise.
- Week 6:
- After a few, and a few more attempts, we were able to come up with the graphs of probability current and the physical trajectories of the interacting system of a photon and an electron.
- Week 7:
- This week, we finished our final presentation, and presented it to everyone at DIMACS. We covered everything we had worked on up until this point, which includes the probability density of a system of a single electron, that of a single photon, that of a photon and an electron without and with interactions. We held a mock-presentation in front of Parker, Professor Tahvildar-Zadeh's Graduate student, and the presentation went well.
- Week 8:
- Professor Tahvildar-Zadeh came back from Europe on Monday, and we were able to start working on our final paper. We explored the question, "How can momentum and energy be defined in a system described by an entangled wave function?" This question is perticularly important since we need those two concepts to verify Professor Tahvildar-Zadeh's theory using the results of Compton scattering.
- Week 9:
- We wrapped up our final paper this week. We were not able to come up with any conclusive answer in terms of defining momentum and energy, but we included some possibilities to explore in the paper. We thanked Professor Tahvildar-Zadeh for his patient guidance, and we planned on presenting our results at future conferences
Presentations
Additional Information
- My Mentor
- My Coworker