Computational Physics Lectures: How to write a scientific project, with examples

Algorithm : A finite set of unambiguous instructions that, given some set of initial conditions, can be performed in a prescribed sequence to achieve a certain goal.

Computing means solving scientific problems using computers. It covers numerical as well as symbolic computing. Computing is also about developing an understanding of the scientific process by enhancing algorithmic thinking when solving problems.

And this competence is about:

• derivation, verification, and implementation of algorithms
• understanding what can go wrong with algorithms
• overview of important, known algorithms
• understanding how algorithms are used to solve complicated problems
• reproducible science and ethics
• algorithmic thinking for gaining deeper insights about scientific problems

All these elements (and many more) will hopefully aid you in maturing and gaining a better understanding of the scientific process per se. Writing good reports is a central element in achieving such insights.

The standard situation we meet at an almost daily basis:

• Theory+experiment+simulation is almost the norm in research and industry
• To be able to model complex systems with no simple answers. Solve real problems.
• Emphasis on insight and understanding of fundamental principles and laws in the Sciences.
• Be able to visualize, present, discuss, interpret and come with a critical analysis of the results, and develop a sound ethical attitude to own and other's work.
• Enhance reasoning about the scientific method

Again, a proper presentation of obtained results via good scientic reports, aids in including all the above aspects.

What should it contain? A typical structure.

• An abstract where you give the main summary of your work
• An introduction where you explain the aims and rationale for the physics case and what you have done. At the end of the introduction you should give a brief summary of the structure of the report
• Theoretical models and technicalities. This is the methods section
• Results and discussion
• Conclusions and perspectives
• Appendix with extra material
• Bibliography

Keep always a good log of what you do.

The abstract gives the reader a quick overview of what has been done and the most important results. Here is a typical example taken from a scientific article

We study the collective motion of a suspension of rodlike microswimmers in a two-dimensional film of viscoelastic fluids. We find that the fluid elasticity has a small effect on a suspension of pullers, while it significantly affects the pushers. The attraction and orientational ordering of the pushers are enhanced in viscoelastic fluids. The induced polymer stresses break down the large-scale flow structures and suppress velocity fluctuations. In addition, the energy spectra and induced mixing in the suspension of pushers are greatly modified by fluid elasticity.

What should I focus on? Introduction.

You don't need to answer all questions in a chronological order. When you write the introduction you could focus on the following aspects

• Motivate the reader, the first part of the introduction gives always a motivation and tries to give the overarching ideas
• What I have done
• The structure of the report, how it is organized etc

What should I focus on? Methods sections.

• Describe the methods and algorithms
• You need to explain how you implemented the methods and also say something about the structure of your algorithm and present some parts of your code
• You should plug in some calculations to demonstrate your code, such as selected runs used to validate and verify your results. The latter is extremely important!! A reader needs to understand that your code reproduces selected benchmarks and reproduces previous results, either numerical and/or well-known closed form expressions.

What should I focus on? Results.

• Present your results
• Give a critical discussion of your work and place it in the correct context.
• Relate your work to other calculations/studies
• An eventual reader should be able to reproduce your calculations if she/he wants to do so. All input variables should be properly explained.
• Make sure that figures and tables should contain enough information in their captions, axis labels etc so that an eventual reader can gain a first impression of your work by studying figures and tables only.

What should I focus on? Conclusions.

• State your main findings and interpretations
• Try as far as possible to present perspectives for future work
• Try to discuss the pros and cons of the methods and possible improvements

What should I focus on? additional material.

• Additional calculations used to validate the codes
• Selected calculations, these can be listed with few comments
• Listing of the code if you feel this is necessary

You can consider moving parts of the material from the methods section to the appendix. You can also place additional material on your webpage.

What should I focus on? References.

• Give always references to material you base your work on, either scientific articles/reports or books.
• Refer to articles as: name(s) of author(s), journal, volume (boldfaced), page and year in parenthesis.
• Refer to books as: name(s) of author(s), title of book, publisher, place and year, eventual page numbers

• With a UiO IP number you can access freely all books and scientific journals available at our University library
• For scientific articles, go to for example the journal Physical Review Letters of the American Physical Society
• For project examples and how we give feedback on projects, go to the project folder of the course and click on the ProjectExample link.

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#Program which solves the one-particle Schrodinger equation 

#for a potential specified in function

#potential(). This example is for the harmonic oscillator in 3d

​

from  matplotlib import pyplot as plt

import numpy as np

#Function for initialization of parameters

def initialize():

    RMin = 0.0

    RMax = 10.0

    lOrbital = 0

    Dim = 400

    return RMin, RMax, lOrbital, Dim

# Here we set up the harmonic oscillator potential

def potential(r):

    return r*r

​

#Get the boundary, orbital momentum and number of integration points

RMin, RMax, lOrbital, Dim = initialize()

​

#Initialize constants

Step    = RMax/(Dim+1)

DiagConst = 2.0 / (Step*Step)

NondiagConst =  -1.0 / (Step*Step)

OrbitalFactor = lOrbital * (lOrbital + 1.0)

​

#Calculate array of potential values

v = np.zeros(Dim)

r = np.linspace(RMin,RMax,Dim)

for i in xrange(Dim):

    r[i] = RMin + (i+1) * Step;

    v[i] = potential(r[i]) + OrbitalFactor/(r[i]*r[i]);

​

#Setting up a tridiagonal matrix and finding eigenvectors and eigenvalues

Hamiltonian = np.zeros((Dim,Dim))

Hamiltonian[0,0] = DiagConst + v[0];

Hamiltonian[0,1] = NondiagConst;

for i in xrange(1,Dim-1):

    Hamiltonian[i,i-1]  = NondiagConst;

    Hamiltonian[i,i]    = DiagConst + v[i];

    Hamiltonian[i,i+1]  = NondiagConst;

Hamiltonian[Dim-1,Dim-2] = NondiagConst;

Hamiltonian[Dim-1,Dim-1] = DiagConst + v[Dim-1];

# diagonalize and obtain eigenvalues, not necessarily sorted

EigValues, EigVectors = np.linalg.eig(Hamiltonian)

# sort eigenvectors and eigenvalues

permute = EigValues.argsort()

EigValues = EigValues[permute]

EigVectors = EigVectors[:,permute]

# now plot the results for the three lowest lying eigenstates

for i in xrange(3):

    print EigValues[i]

FirstEigvector = EigVectors[:,0]

SecondEigvector = EigVectors[:,1]

ThirdEigvector = EigVectors[:,2]

plt.plot(r, FirstEigvector**2 ,'b-',r, SecondEigvector**2 ,'g-',r, ThirdEigvector**2 ,'r-')

plt.axis([0,4.6,0.0, 0.025])

plt.xlabel(r'$r$')

plt.ylabel(r'Radial probability $r^2|R(r)|^2$')

plt.title(r'Radial probability distributions for three lowest-lying states')

plt.savefig('eigenvector.pdf')

plt.show()