C++ Lab Course - Seminar topics

Computational physics
C++ Lab Course - Seminar topics
Armin Scrinzi

Chapter 1
C++ course project — single particle
TDSE code
We have discussed several important building blocks of a C++ code. During the rest of the course,
we will together build a functioning and extendable code to solve the time-dependent Schrödinger
equation.
1.1 Features
• Bound states solver for general potentials, including truely 2d problems: atomic models, trap
potentials(?), quantum dots,. . .
• Resonance states for general potentials
• Scattering states and cross-sections
• Interaction with dipole fields: Stark effect and ionization rates
1.2 Technical properties
• General coordinate systems
• General (basis set) discretizations: finite elements, gaussian, spherical harmonics, FFT (plane
waves)
• Linear algebra: LAPACK, C++ packages, ARPACK
• “Complex scaling” (for resonances and Stark effect)
• Parallel: MPI, PETSC
1.3 Tasks & grades
We will try to reach this goal together. For that, tasks will be assigned to each team. The teams
will present the results in the course. In these presentations, the team jointly present their solution,
each member of the team should actively participate. The presentation is a prerequisit for getting
the course credited. At the end of the course there will be an exam, where a small programming task
will be assigned to everybody. Grades for the course will be based 50% on the team performance,
50% on the final exam.
Please, from the following list of tasks, pick one for your team. Each task will be formulated
by me in python. (Several solutions already exist in python, others will be presented during the
next 2 weeks.) Your task mostly is to “translate” the python code into C++. Please, supply a
3

list of at least 3 tasks you would like to do, indicating your priorities. Assignement will be on a
first-come-first-serve basis and according to your priorities. Tasks that are not picked, will be taken
by us (Nina and Armin), such that at the end of a course, we have a functioning code that will be
available for all and serve as a model for your future programming.
Presentations start on June 5 (maximal 7 presentations).
1.3.1 List of tasks
1. Interface matrix class to LAPACK routines (for fast algorithms)
2. Tensor class
3. Wave function class
4. Potential module
5. Timer routines
6. Operator class: tensor product operators
7. Operator class: general potentials
8. Time propagator class
9. Read input, I/O
1.3.2 Detailed description of tasks
Extension of Matrix.h
Starting from Matrix.h:
• Build interface to the eigenvalue package into Matrix.h
• Diagonal and banded matrices
• Build interface to linear system solving package for full and banded matrices (a C++ package
will be supplied)
Tensor class
• Generalize the definition of a tensor from the tensor products to any linear map on the product
space. The application for this are interaction operators:
T (⃗a ⊗ ⃗ b) = ∑ ln
T km,ln a l b n
where T may be a tensor product or a 4-index object (like the interaction potential).
• Implement the application of a tensor to a vector:
[(A ⊗ B)⃗c] km = ∑ ln
A kl A mn c ln
• Generalize to 3-fold (or n-fold if you like) tensor products
4

Wave function class
A wave function is the following object
• It has up to three coordinate axes which also defines a discretization by a product basis (the
axis class will be supplied)
• It has a time
• We can form the scalar product of two wave functions.
• We can apply the overlap matrix and the inverse of the overlap matrix to the wave function.
Operator class
Operator representations on product basis sets. This is an important and large class, the task can
be split into smaller parts.
• An operator is constructed given a set of axes and a string that defines the operator. A typical
string could be the kinetic energy in 2-d polar coordinates φ, ρ in the tensor-product form
kin = ’+1/2(|)x(d|d) +1(d|d)x(|1/q^2|)’
or general potentials, e.g the harmonic oscillator:
pot = ’harmonic(kappax,kappay)’
for
pot(x, y) = κ x x 2 + κ y y 2
• An operator can be applied to a wave function, the result is a wave function.
• Operators on finite-element basis sets are stored piecewise for each product of finite elemetns.
• Tensor product operators are stored as the individual factors.
• Potentials are stored by their values on quadrature points and by the transformation matrices
from the basis set to the quadrature points (compare the discussion of DVR).
Timer routines
Functionality like in “mytimer.py”: find C++ class with this functionality or translate the python
code. We want
• Minimal timer overhead — timer operation must be fast!
• Option to switch off the timer (to reduce any possible overhead to the absolute mininum).
• Progress monitor: let the timer print a “.” or “-” if more than n seconds that have elapsed
since it was last called last time.
Time propagator class
A time-propagator takes as an argument and operator, wave function, a new time, and an accuracy
parameter. Advances the wave function from its current time to the new time by solving the TDSE
with the given operator. Controls the accuracy.
5

Read input I/O
Translate the python read input.py into C++ with a few extra features.
• Check input command sanity: on each call of read_input, record the command string. After
you are through all commands, when calling read_input_finish, check all lines of the input
file for command-like strings: if you find one that does not fit any, stop with the message that
there is a misprint. (You may prototype this in Python, to see the logics at work).
• Read vectors from columns of input, supplement empty entries by defaults.
6