Contact info:

Department of Physics

McCardell Bicentennial Hall

Middlebury College

Middlebury, VT 05753

(802) 443-3423

Fax: (802) 443-2072

ngraham [at] middlebury [dot] edu

- A. B., Physics and Mathematics, Harvard, 1994
- Ph. D., Physics, MIT, 1999

- General-audience talk on my research
- General-audience talk on the Higgs boson
- Goals of the Middlebury STEM Innovation program
- Talk on quantum mechanics as linear algebra
- Technical talk on my Casimir research
- Technical talk on my oscillon research
- General relativity traffic jam
- How to succeed in physics with only a reasonable amount of trying
- Ensembles in statistical mechanics

My research centers around applications of quantum mechanics and classical and quantum field theory to a variety of problems in elementary particle physics, physics of solitons and oscillons, and the Casimir effect. I am also interested in applications of physics techniques to applications in computer science. Here are some possible thesis topics on these subjects, or you can read my fascinating papers.

I gratefully acknowledge the National Science Foundation, Research Corporation, Vermont EPSCoR, and Middlebury College for grants supporting my research.

## Spheroidal Functions in Mathematica®:

I have extensively modified the package of Falloon for computing spheroidal wave functions in Mathematica®. The resulting code is faster for common cases, adds series expressions around the spherical limit for the radial functions, and fixes several bugs (particularly in the series expansions); see the readme for details.

These changes were developed with assistance from Pavlo Levkiv.

Spheroidal package for Mathematica 4

Spheroidal package for Mathematica 5

Spheroidal package for Mathematica 6 and later

By downloading or using any of these packages you agree to the following license terms, also which are also given in the license file.

This package is based on previous work of Falloon (see P. E. Falloon, P. C. Abbott, and J. B. Wang, J. Phys. A36 (2003) 5477), which does not indicate any license restrictions or copyright. Modifications are copyright © Noah Graham, 2005-2016, all rights reserved. The modified packages may be used and modified for noncommercial research or educational purposes, provided that citations to the previous work by Falloon et. al. listed above and

T. Emig, N. Graham, R. L. Jaffe and M. Kardar, "Casimir Manipulations: The Orientation Dependence of Fluctuation-Induced Forces," arXiv:0811.1597, Phys. Rev. D77 (2008) 025005

are included in all publications or other products in which the modified package or any programs derived from it was used. Commercial use or inclusion in a commercial product is prohibited. This program and all accompanying materials are provided "AS IS," WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

Update: The package for Version 6 also appears to work in Version 7 (and later) as well. To use it with parallel computation, be sure that the package is loaded in all subkernels. Although the performance of the built-in spheroidal functions is considerably improved in Version 7, the package here still is faster for many common cases and contains additional functionality.

## General-purpose Mathieu Function Routines in Mathematica®:

The following Mathematica® package improves on various limitations of the built-in Mathieu functions. In particular, it supports both first- and second-kind functions, as well as radial and modified functions, and is reliable even for complex parameter and argument, as described in this paper.

Original package written by Elizabeth Noelle Blose '17, Biswash Ghimire '16, Noah Graham, and Jeremy Stratton-Smith '17. Updated July 2016 to include code for series expansions around q=0 by Daniel Esrick '18 and Carrie Owen '18. Further improvements and updates August 2021 by Hyuma Umeda '24.

Mathieu Function Package

By downloading or using this code you agree to the following license terms: All contents are copyright © 2014-21, all rights reserved. These programs may be used and modified for noncommercial research purposes, provided that citation to

Elizabeth Noelle Blose, Biswash Ghimire, Noah Graham, and Jeremy Stratton-Smith, "Edge Corrections to Electromagnetic Casimir Energies From General-Purpose Mathieu Function Routines," arXiv:1411.0734, Phys. Rev. A 91, 012501 (2015)

is included in all publications or other products in which the program or any programs derived from it were used. This program and all accompanying materials are provided "AS IS," WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

## Casimir Energy of a Disk:

English translation of Joseph Meixner, "A Rigorous Theory of the Diffraction of Electromagnetic Waves on a Perfectly Conducting Disk," Z. Naturforschung 3a, 506 (1948). English translation by Nimrod Sadeh '17.5; original paper here.

The following Mathematica® notebook implements the calculation of the Casimir energy for a disk-plane system, as described in this paper.

Casimir energy of disk and plane

By downloading or using this code you agree to the following license terms:

All contents of these notebooks are copyright © Noah Graham, 2016, all rights reserved. These programs may be used and modified for noncommercial research purposes, provided that citation to

N. Graham, "Exact Electromagnetic Casimir Energy of a Disk Opposite a Plane," arXiv:1606.01090, Phys. Rev. A 94, 032509 (2016)

is included in all publications or other products in which the program or any programs derived from it were used. This program and all accompanying materials are provided "AS IS," WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

## Parallel C++ classical field theory simulation of electroweak SU(2)xU(1) model:

Here is some code that does a lattice simulation of oscillons in the electroweak Standard Model, as shown in this paper and this paper. It includes SU(2)xU(1) gauge fields and a fundamental Higgs field and allows for parallel processing using MPI, threads, or both. It has been adapted to a number of other situations, including SU(2) adjoint gauge fields, abelian Higgs models, and expanding universe backgrounds -- please contact me if you are interested in the details. By downloading or using this code you agree to the following license terms:

All contents of this package are copyright © Noah Graham, 2006-2007, all rights reserved. This program may be used and modified for noncommercial research purposes, provided that citation to N. Graham, hep-th/0610267, "An Electroweak Oscillon," Phys. Rev. Lett. 98 (2007) 101801 and/or N. Graham, arXiv:0706.4125 [hep-th], "Numerical Simulation of an Electroweak Oscillon," Phys. Rev. D 76 (2007) 085017 is included in all publications or other products in which the program or any programs derived from it were used. This program and all accompanying materials are provided "AS IS," WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

## Variable Phase S-Matrix Calculations for Asymmetric Potentials and Dielectrics:

The following Mathematica® notebooks implement the variable phase method for potentials and dielectrics, as described in this paper.

Variable phase calculation, scalar case

Variable phase calculation, vector Helmholtz case

Variable phase calculation, electromagnetic case

By downloading or using this code you agree to the following license terms, which are also included in the packages themselves.

All contents of this notebook are copyright © Aden Forrow and Noah Graham, 2012, all rights reserved. These programs may be used and modified for noncommercial research purposes, provided that citation to

A. Forrow and N. Graham, "Variable Phase S-Matrix Calculations for Asymmetric Potentials and Dielectrics," arXiv:1210.0777, Phys. Rev. A 86 (2012) 062715

is included in all publications or other products in which the program or any programs derived from it were used. This program and all accompanying materials are provided "AS IS," WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

## Variable Phase S-Matrix Calculations for Periodic Dielectric Gratings:

The following Mathematica® notebook implements the variable phase method for dielectric gratings, as described in this paper.

Variable phase calculation, electromagnetic case

By downloading or using this code you agree to the following license terms:

All contents of these notebooks are copyright © Noah Graham, 2014, all rights reserved. These programs may be used and modified for noncommercial research purposes, provided that citation to

N. Graham, "Casimir Energies of Periodic Dielectric Gratings," arXiv:1407.4642, Phys. Rev. A 90 (2014) 032507

is included in all publications or other products in which the program or any programs derived from it were used. This program and all accompanying materials are provided "AS IS," WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESSED OR IMPLIED, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.

Below are lessons on quantum mechanics I have developed for the course "Quantum Mechanics From a Linear Algebra Point of View." They assume knowledge of linear algebra (at the level of Strang's book, for example) and basic (high school or introductory college level) familiarity with introductory mechanics and electromagnetism. Rather than the usual wave-mechanics approach used in most textbooks and quantum mechanics courses (such as our PHYS 0202 and PHYS 0401), they use the more physically abstract but mathematically simpler picture of finite dimensional matrices. My hope is they can provide a complement to standard undergraduate quantum mechanics references such as Gasiorowicz, Griffiths, and Liboff. This approach is also more directly applicable to problems in quantum computing.

All materials are copyright © 2002-2010, Noah Graham. These materials may be used for noncommercial purposes with proper attribution, including this notice.

Lesson 1

Lesson 2

Lesson 3

Lesson 4

Lesson 5

Lesson 6

Lesson 7

Lesson 8

Here are slides from a talk introducing this approach, aimed at an audience familiar with introductory linear algebra (no physics background is assumed).

Having worked in industry doing research in speech recognition, I am also interested in applications of scientific computing, both to physics and to subjects like speech and vision. (See also work by my brother, Middlebury class of '01.) Below are the first three projects from PH120, Computers in the Physical Sciences. These are designed to provide introductions to the applications of an object-oriented approach using Mathematica© and C++ to problems in the physical sciences. Other examples of computational assignments from PH301, Intermediate Electrodynamics, PH350, Statistical Mechanics, and PH401, Quantum Mechanics, are below as well.

All materials are copyright © 2002-2007, Noah Graham. These materials may be used for noncommercial purposes with proper attribution, including this notice.

Project 1

Project 2

Project 3

PH301 Project

PH350 Project

PH401 Project