This repository contain lecture slides, programs, exercises and projects for a more advanced course in computational physics, with an emphasis on quantum mechanical problems with many interacting particles. The applications and the computational methods are relevant for research problems in such diverse areas as nuclear, atomic, molecular and solid-state physics, quantum chemistry and materials science.

A theoretical understanding of the behavior of quantum-mechanical many-body systems - that is, systems containing many interacting particles - is a considerable challenge in that no exact solution can be found; instead, reliable methods are needed for approximate but accurate simulations of such systems on modern computers. New insights and a better understanding of complicated quantum mechanical systems can only be obtained via large-scale simulations. The capability to study such systems is of high relevance for both fundamental research and industrial and technological advances.

The aim of this course is to present applications of, through various computational projects, some of the most widely used many-body methods with pertinent algorithms and high-performance computing topics such as advanced parallelization techniques and object orientation. Furthermore, elements of Machine Learning and quantum computing may be presented if of interest. In particular we will here employ deep learning techniques based on neural networks and so-called Boltzmann machines. The methods and algorithms that will be studied may vary from year to year depending on the interests of the participants, but the main focus will be on systems from computational material science, solid-state physics, atomic and molecular physics, nuclear physics and quantum chemistry. The most relevant algorithms and methods are microscopic mean-field theories (Hartree-Fock and Kohn-Sham theories and density functional theories), large-scale diagonalization methods, coupled-cluster theory, similarity renormalization methods, and quantum Monte Carlo like Variational Monte Carlo and Diffusion Monte Carlo approaches. Methods to study phase transitions for both fermionic and bosonic systems can also be studied.

The course introduces a variety of central algorithms and methods for professional studies of quantum mechanical systems, with relevance for several problems in physics, materials science and quantum chemistry. The course is project based and through the various projects, normally two, the participants will be exposed to fundamental research problems in these fields, with the aim to reproduce state of the art scientific results. The students will learn to develop and structure large codes for studying these systems, get aquainted with supercomputing facilities and learn to handle large scientific projects. A good scientific and ethical conduct is emphasized throughout the course.

The course is also a continuation of FYS3150 – Computational Physics, and it will give a further treatment of several of the numerical methods given there.

Basic knowledge in programming and mathematics, with an emphasis on linear algebra. Knowledge of Python or/and C++ as programming languages is strongly recommended and experience with Jupiter notebook is recommended. Required courses are the equivalents to the University of Oslo mathematics courses MAT1100, MAT1110, MAT1120 and at least one of the corresponding computing and programming courses INF1000/INF1110 or MAT-INF1100/MAT-INF1100L/BIOS1100/KJM-INF1100. Most universities offer nowadays a basic programming course (often compulsory) where Python is the recurring programming language.

We recommend also that you have some background in quantum mechanics, typically at the level of FYS2140 and/or FYS3110.

Computational aspects play a central role and you are expected to work on numerical examples and projects which illustrate the theory and varous algorithms discussed during the lectures. We recommend strongly to form small project groups of 2-3 participants, if possible.

• Name: Morten Hjorth-Jensen
• Email: [email protected]
• Phone: +47-48257387
• Office: Department of Physics, University of Oslo, Eastern wing, room FØ470
• Office hours: Anytime! For the spring Semester 2021 (SS21), as a rule of thumb office hours are planned via computer or telephone. Individual or group office hours will be performed via zoom. Feel free to send an email for planning. In person meetings may also be possible if allowed by the University of Oslo's COVID-19 instructions (see below for links).

• Name: Øyvind Sigmundson Schøyen,
• Email: [email protected]
• Office: Department of Physics, University of Oslo, Eastern wing, room FØ452
• Office hours: For the spring Semester 2021 (SS21), as a rule of thumb office hours are planned via computer or telephone. Individual or group office hours will be performed via zoom. Feel free to send an email for planning. In person meetings may also be possible if allowed by the University of Oslo's COVID-19 instructions (see below for links).

This course will be delivered in a hybrid mode, with online and on site lectures and on site and online laboratory sessions. The first week (January 11-15) is fully digital and the computer lab and the lectures will be done via zoom.

1. Two lectures per week, spring semester, 10 ECTS. The lectures will be fully online. The lectures will be recorded and linked to this site and the official University of Oslo website for the course;
2. Thre hours of laboratory sessions for work on computational projects and exercises for each group. Due to social distancing, at most 15 participants can attend. There will also be fully digital laboratory sessions for those who cannot attend;
3. Two projects which are graded and count 1/2 each of the final grade;
4. The course is offered as a FYS4411 (Master of Science level) and a FYS9411 (PhD level) course;
5. Videos of teaching material are available via the links at https://www.uio.no/studier/emner/matnat/fys/FYS4411/v21/timeplan/index.html#FOR;
6. Weekly emails with summary of activities will be mailed to all participants;

Grading scale: Grades are awarded on a scale from A to F, where A is the best grade and F is a fail. There are two projects which are graded and each project counts 1/2 of the final grade. The total score is thus the average from the two projects.

The final number of points is based on the average of all projects (including eventual additional points) and the grade follows the following table:

• 92-100 points: A
• 77-91 points: B
• 58-76 points: C
• 46-57 points: D
• 40-45 points: E
• 0-39 points: F-failed

Course participants are expected to have their own laptops/PCs. We use Git as version control software and the usage of providers like GitHub, GitLab or similar are strongly recommended.

We will make extensive use of Python and C++ as programming languages.

If you have Python installed and you feel pretty familiar with installing different packages, we recommend that you install the following Python packages via pip as

• pip install numpy scipy matplotlib ipython scikit-learn mglearn sympy pandas pillow

For OSX users we recommend, after having installed Xcode, to install brew. Brew allows for a seamless installation of additional software via for example

• brew install python3

For Linux users, with its variety of distributions like for example the widely popular Ubuntu distribution, you can use pip as well and simply install Python as

• sudo apt-get install python3

If you don't want to perform these operations separately and venture into the hassle of exploring how to set up dependencies and paths, we recommend two widely used distrubutions which set up all relevant dependencies for Python, namely

• Anaconda:https://docs.anaconda.com/,

which is an open source distribution of the Python and R programming languages for large-scale data processing, predictive analytics, and scientific computing, that aims to simplify package management and deployment. Package versions are managed by the package management system conda.

• Enthought canopy:https://www.enthought.com/product/canopy/

is a Python distribution for scientific and analytic computing distribution and analysis environment, available for free and under a commercial license.

Furthermore, Google's Colab:https://colab.research.google.com/notebooks/welcome.ipynb is a free Jupyter notebook environment that requires no setup and runs entirely in the cloud. Try it out!

Here we list several useful Python libraries we strongly recommend (if you use anaconda many of these are already there)

• NumPy:https://www.numpy.org/ is a highly popular library for large, multi-dimensional arrays and matrices, along with a large collection of high-level mathematical functions to operate on these arrays
• The pandas:https://pandas.pydata.org/ library provides high-performance, easy-to-use data structures and data analysis tools
• Xarray:http://xarray.pydata.org/en/stable/ is a Python package that makes working with labelled multi-dimensional arrays simple, efficient, and fun!
• Scipy:https://www.scipy.org/ (pronounced “Sigh Pie”) is a Python-based ecosystem of open-source software for mathematics, science, and engineering.
• Matplotlib:https://matplotlib.org/ is a Python 2D plotting library which produces publication quality figures in a variety of hardcopy formats and interactive environments across platforms.
• Autograd:https://github.com/HIPS/autograd can automatically differentiate native Python and Numpy code. It can handle a large subset of Python's features, including loops, ifs, recursion and closures, and it can even take derivatives of derivatives of derivatives
• SymPy:https://www.sympy.org/en/index.html is a Python library for symbolic mathematics.
• scikit-learn:https://scikit-learn.org/stable/ has simple and efficient tools for machine learning, data mining and data analysis
• TensorFlow:https://www.tensorflow.org/ is a Python library for fast numerical computing created and released by Google
• Keras:https://keras.io/ is a high-level neural networks API, written in Python and capable of running on top of TensorFlow, CNTK, or Theano
• And many more such as pytorch:https://pytorch.org/, Theano:https://pypi.org/project/Theano/ etc

• Armadillo: http://arma.sourceforge.net/ Armadillo is a high quality linear algebra library (matrix maths) for the C++ language, aiming towards a good balance between speed and ease of use
• Eigen: http://eigen.tuxfamily.org/index.php?title=Main_Page Eigen is a C++ template library for linear algebra: matrices, vectors, numerical solvers, and related algorithms.
• Scythe: http://scythe.lsa.umich.edu/ The Scythe Statistical Library is an open source C++ library for statistical computation
• Autodiff: https://autodiff.github.io/ autodiff is a C++17 library that uses modern and advanced programming techniques to enable automatic computation of derivatives in an efficient and easy way.
• Optmilib: https://www.kthohr.com/optimlib.html OptimLib is a lightweight C++ library of numerical optimization methods for nonlinear functions.

Recommended textbooks: Lecture Notes by Morten Hjorth-Jensen and Bernd A. Berg, Markov Chain Monte Carlo Simulations and their Statistical Analysis, World Scientific, 2004, chapters 1, 2

As of now this is not required, but the situation may change. If face covering will be required during the semester, we will fill in more details.

We will be practicing physical distancing in the classroom dedicated to the lab sessions. Thus, everybody should maintain at least one meter distance between themselves and others (excluding those with whom they live). This applies to all aspects of the classroom setting, including seating arrangements, informal conversations, and dialogue between teachers and students.

All participants attending the laboratory sessions must maintain proper hygiene and health practices, including:

• frequently wash with soap and water or, if soap is unavailable, using hand sanitizer with at least 60% alcohol;
• Routinely cleaning and sanitizing living spaces and/or workspace;
• Using the bend of the elbow or shoulder to shield a cough or sneeze;
• Refraining from shaking hands;

Course participants will (a) look for instructional signs posted by UiO or public health authorities, (b) observe instructions from UiO or public health authorities that are emailed to my “uio.no” account, and (c) follow those instructions. The relevant links are https://www.uio.no/om/hms/korona/index.html and https://www.uio.no/om/hms/korona/retningslinjer/veileder-smittevern.html

Students will self-monitor for flu-like symptoms (for example, cough, shortness of breath, difficulty breathing, fever, sore throat or loss of taste or smell). If a student experiences any flu-like symptoms, they will stay home and contact a health care provider to determine what steps should be taken.

If a student is exposed to someone who is ill or has tested positive for the COVID-19 virus, they will stay home, contact a health care provider and follow all public health recommendations. You may also contact the study administration of the department where you are registered as student.

Those who come to UiO facilities must commit to the personal responsibility necessary for us to remain as safe as possible, including following the specific guidelines outlined in this syllabus and provided by UiO more broadly (see links below here).

See https://www.uio.no/om/hms/korona/index.html and https://www.uio.no/om/hms/korona/retningslinjer/veileder-smittevern.html. For English version, click on the relevant link.