GSoC 2014 Application Manish Shukla : Solutions to Quantum Systems - sympy/sympy GitHub Wiki

Personal Details:

I am Manish Kumar Shukla, a 3rd year undergraduate pursuing an Integrated Bachelors in Computer Science and MS in Computational Natural Sciences at IIIT Hyderabad.

Email: [email protected]

IRC: Shukla

Github: gamecoder2012

Science background

I'm pursuing research in Dynamics of quantum systems and its application in quantum computation, for my Masters' Thesis. Thus, I have taken advanced courses in classical and quantum mechanics, wave dynamics and calculus. I have also completed a summer and a semester honours project wherein I looked at time evolution of various quantum systems.

Programming Experience

I am comfortable programming in C,C++, and Python. I have used python for programming in the domain of Optics to implement Kirchhoff-Helmholtz integration for reconstruction of a hologram and I've also implemented its parallel version in pyCUDA. I use Ubuntu 12.04 LTS and my preferred text editor is Sublime Text or Vim. I am new to Git and only understand its basic functionality as yet.

Contribution:

Open : 7308 - Implementation for wave function and energy for particle confined on a ring

Motivation for Project

Not all quantum systems can be perfectly solved, but those simple systems which can be solved are of immense importance in understanding quantum mechanics. Solutions to complex systems can be built upon them. It would be of great help to those in field of quantum mechanics if they can get solutions to these quantum systems. They could then apply various operators on these solutions and also plot them. The quantum module of sympy is still at a basic level and this contribution will add to its functionality.

Current status

Basic solutions for infinite box, particle in a box, hydrogen atom and simple harmonic oscillator with ladder operators been implemented.

Proposal

I want to add solutions to following systems

1. Time Independent Systems

Particle in a box with finite potential
Particle confined to move on ring.
Particle confined on circular plane.
particle on surface of sphere of fixed radius
Particle in two and three dimensional box
Electron in a magnetic field
Two boxes with barrier in between them.
Angular Solutions to hydrogen atom

2. Time Dependent systems

Free Particle
Particle in an expanding box
Particle on expanding ring
Particle in time dependent potential

3. Implementation of operators and expectation values

Position operator
Momentum Operator
Angular Momentum operator

4. Solving complex systems with approximations(Stretch Goal)

Dihydrogen cation
Hydrogen molecule

5. Dynamics of measurements (Future work)

Dynamics of strong and weak measurements on these systems.
Plotting of wave functions.

Timeline(Tentative)

Part 1 Time Independent Solutions

Week 1: Familiarize myself with the work already implemented and make improvements in them, if any.

Week 2: Implement a module to solve particle in 2d and 3d box.

    Deliverable: Wave function and Energy modules for particle in 2d and 3d box.

Week 3: Implement a module to solve particle in finite potential box and two boxes with a barrier in between them.

    Deliverable : Wave functions for outside and inside the box with finite potential and energy levels. Wave function and energy module for two boxes.

Week 4: Implement a module to solve particle confined on circular plane and particle on surface of sphere
```
   Deliverable : Wave function and Energy modules for these two systems.
```

Week 5: Implement a module for angular solutions to hydrogen atom

   Deliverable : Wave function with radial and angular solutions dependent on quantum numbers n,m,l;

Week 6: Documentation and Implementation of electron in a magnetic field.

Part 2 Introducing Time dependency

Week 7: Implement modules for free particle and particle in expanding box.
Week 8: Implement module for Particle on expanding ring
Week 9: Implement module for particle on surface of expanding sphere

Week 10: Implement module for particle in time dependent potential

     Deliverable : Modules for time evolution of wave functions for these system taking time as input to show the wave function at that time.

Part 3: Operators and expectation value calculation

Week 11: Implementing angular momentum operator.

Week 12: Getting expectation value for these operators for systems implemented above( where ever valid)

     Deliverable : Module for applying angular momentum operator on wave functions and calculating expectation value for the operator.

Week 13: Catch up time if needed. Else, continue with stretch goal.
Week 14: Documentation and Evaluation.

Implementation Details:

The major bottleneck in the design lies in solving quantum systems, for which one would require to analyze the Hamiltonian of respective systems and solve them using time dependent and time independent Schrodinger equation. Once the solutions are available in algebraic form I would implement them employing methods that already exist in sympy.

Prototype

Solution of particle confined on ring

     >>> wavefunction(1,x)
     sqrt(2)*exp(i*x)/(2*sqrt(pi))
     >>> wavefunction(2,x)
     sqrt(2)*exp(2*i*x)/(2*sqrt(pi))
     >>> wavefunction(3,x)
     sqrt(2)*exp(3*i*x)/(2*sqrt(pi))
     >>> integrate(wavefunction(2, x)*wavefunction(-2, x), (x, 0, 2*pi))
     1
     >>> integrate(wavefunction(4, x)*wavefunction(-4, x), (x, 0, 2*pi))
     1
     >>> var("m,r")
     (m, r)
     >>> energy(1,m,r)
     hbar**2/(2*m*r**2)
     >>> energy(2,m,r)
     2*hbar**2/(m*r**2)
     >>> energy(-2,2.0,3.0)
     0.111111111111111*hbar**2

Commitments

As I have no other commitments for the summer I will put up 40 hours or more if needed to meet the weekly deadlines. I will maintain a blog to show the progress of the project. I would love to continue on work indicated as future work in proposal after GSOC and would continue to contribute to sympy.

References

1. Books :

Quantum Mechanics by Trilochan Pradhan
Concepts of Modern Physics by Arthur Beiser
Molecular Quantum Mechanics by Peter Atkins