This web page describes an activity within the Department of Mathematics at Ohio University, but is not an official university web page.
If you have difficulty accessing these materials due to visual impairment, please email me at mohlenka@ohio.edu; an alternative format may be available.

Toward Direct Numerical Solution of the Multiparticle Schrodinger Equation

Abstract

The multiparticle Schrodinger equation is the basic governing equation in quantum mechanics, and thus the foundation for much of chemistry and physics. A tremendous amount of effort, over several decades, has gone into approximating its solutions, but high quality numerical solutions have proved elusive. Recently developed mathematical tools for computing in high dimensions have revealed a new path toward such solutions, which this project will explore. Such solutions will enable accurate computation of chemical and physical properties of molecules and materials, and thus have an impact on many problems in chemistry, physics, biology, and materials science.

This research will occur in the Department of Mathematics at Ohio University, a state university located in the Appalachian region of southeastern Ohio, with a renewed commitment to strengthening its research at all levels. The research group will include both undergraduate and graduate students, who will contribute to this project and develop valuable skills in computational mathematics. The students will also learn about the research process and develop their skills at communicating mathematics.

Acknowledgments

This work is supported by the National Science Foundation under Grant No. 0545895 from August 1, 2006 to July 31, 2011, with principal investigator Martin J. Mohlenkamp. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Goals

Scientific Goal

The primary scientific goal is to develop methods to enable the computation of wavefunctions to whatever accuracy is desired. This ability will allow chemists, physicists. etc. to compute quantities of physical interest, and thus increase our understanding of the physical word and enable advances in technology.

Educational Goal

By participating in this research project, students will both learn specific mathematical and computational skills and also develop an understanding of how science and research works. Since the students will range from advanced doctoral students to undergraduates, the desired outcome will vary with the individual. Students at all levels will develop the ability to express mathematical ideas, both in writing and orally.

Process Goal

Both science and education are rather messy, imperfect processes. By keeping aware of the process we use when pursuing the scientific and educational goals, we will attempt to continually improve this process.

Opportunities to Participate

This project has many facets, from purely theoretical mathematical considerations, to very concrete programming tasks. The interests of individual students will be accommodated whenever possible. There are several levels at which you can participate, from exploratory to full-time research.

Exploratory Participation

Anyone who wishes to can join the project for one quarter. You will learn more about the project, as well as develop valuable skills in mathematical writing, searching the literature, reading a research paper, presenting a seminar talk, and programming. I will learn about your strengths and weaknesses, and we both will learn how well you would fit into the research effort.

Each week we will have a group meeting where we will discuss what was accomplished on the project so far and what the next steps should be. As exploratory participants you will have a small task to complete each week. A common task each week is to write a journal entry about what you did. A sample weekly plan of other tasks for the quarter is as follows:

Write your mathematical autobiography, using LaTeX.
Write up something mathematical (like a homework assignment for a class) using LaTeX.
Read the introduction to the latest research paper from this project. Write a short summary in your own words of the topic, and some questions you had.
Program something very simple in Python.
Create a design for a subroutine to compute something related to this project, and write about it.
Implement the subroutine you designed, and test it.
(Re-implement the subroutine if needed.) Locate and get copies of two papers referenced by this project.
Read the two papers and write a paragraph summary of the topic of each. Choose your favorite.
Present your paper. Critique the presentations of others.
Present your paper if you have not already. Critique the presentations of others.

Research Assistants

For graduate students in the Mathematics department, a limited number of paid research assistantships are available. In general, a student must participate at the exploratory level for one quarter before being considered for a paid assistantship. Research assistantships range from 5 hours per week to 15 hours per week (equivalent to a teaching assistantship). You will be paid at an hourly rate similar to that you would receive as a teaching assistant; at minimum, this is $1600 for 5 hours per week for one quarter. Graduate students already working as teaching assistants can also work (at most) 5 hours per week as a research assistant.

For undergraduate students, paid research assistantships are available for 5-10 hours per week, at an hourly rate somewhat less than for graduate students.

Each participant's research project will be determined to meet their interests and level. To help keep them on track and progressing toward the overall project's three goals, there are the following specific expectations:

Meet weekly with the research group.
Each week submit a journal of what they did the previous week, questions or difficulties they encountered, and what they plan to do next.
Each week update a summary document of what they have learned so far that quarter. This document contains research results and other information that should be preserved. At the end of the quarter this becomes their final report.
Be prepared each week to informally present to the group what they have done. Give a formal presentation at least once per quarter.
Assist in the direction of more junior participants.
Give feedback and suggestions on how the group's research process could be improved.

Participant Resources

LaTeX and Python links

Participants

Winter 2007

Adrian Gusa: M.S. student in the Computational Track; Mathematical Autobiography; Journal.
Aleksandra Orlova
: M.S. student in the Applied Track; Journal.

Fall 2006

Tejaswi Gottipati: M.S. student; Mathematical Autobiography; Journal.
Adrian Gusa: M.S. student in the Computational Track; Mathematical Autobiography; Journal.
Jyothsna Jakka: M.S. student in the Computational Track; Mathematical Autobiography; Journal.
Venu Kotha: M.S. student in the Computational Track; Mathematical Autobiography; Journal.
Srikanth Madasu: M.S. student in the Computational Track; Mathematical Autobiography; Journal.
Aleksandra Orlova
: M.S. student in the Applied Track; Journal.
Benigno Parra: PhD student in Mathematics; Mathematical Autobiography.
Aravind Srirangam: M.S. student; Mathematical Autobiography; Journal.

Martin Mohlenkamp

Last modified: Thu Aug 30 15:42:41 EDT 2012

Toward Direct Numerical Solution of the Multiparticle Schrodinger Equation

Abstract

Acknowledgments

Goals

Scientific Goal

Educational Goal

Process Goal

Opportunities to Participate

Exploratory Participation

Research Assistants

Participant Resources

Participants

Spring 2011

Winter 2011

Fall 2010

Spring 2010

Winter 2010

Fall 2009

Summer 2009

Spring 2009

Winter 2009

Fall 2008

Summer 2008

Spring 2008

Winter 2008

Fall 2007

Summer 2007

Spring 2007

Winter 2007

Fall 2006