blog.HCNOP.net

[whordijk-blog]

MathJax: a JavaScript display engine for mathematics

In a previous post I used mathURL to generate a .png image of a mathematical expression: the image would be hosted at said website so that I could link to it in my post. While doing this I realized this was quite a cumbersome method and there really should be an easier way to have mathematical expressions in my blog posts.

The guys from MathJax must have thought the same thing when they started developing their JavaScript display engine for mathematics: utilizing this engine it's now a breeze to render mathematical expressions using \(\LaTeX\\) code directly in blog posts yielding great looking results:

\[ \oint_{\partial V}\!\mathbf{A}\cdot\mathbf{\hat{n}} \; \mathrm{d}a = \int_{V}\!\left(\nabla\cdot\mathbf{A}\right) \; \mathrm{d}V \]

[whordijk-blog]

Bachelor Thesis finished

Just before Christmas I finished and defended my Bachelor Thesis. It was graded with 8.5 out of 10 points.

This thesis describes the process of deriving the Master Equation for the time evolution of the density operator for a lead coupled spinless Hubbard chain using a Markov approximation. The derivation is done in a direct, conventional way of expanding operators and switching pictures in order to separate the system Hamiltonian in a convenient way and through the Super Operator formalism, yielding a different form of the Master Equation. We will implement both results in software and we will iteratively find the stationary state of the chain from some initial state when the lead is kept at a constant chemical potential. We will show that for each initial state the chain density operator converges to this stationary state that is determined by the chemical potential of the lead, but at a rate that is independent of the chemical potential of the lead and that both the conventional and the Super Operator approach yield the exact same results. We will also show and discuss a small systematic error we find in the converged results yielding non-physical interpretations and how we can improve the numerics regarding the implementation of the model.

[whordijk-blog]

Progress on the research

My previous idea proved to be a little ambitious. I've reset my goals to first successfully set up a model for a weak lead coupled Hubbard chain, solve the Von Neumann equation for the chain density matrix and integrate numerically to calculate said density matrix for all times.

So far, so good: using the second quantization formalism I've found a general expression for the chain creation matrices in the Fock space binary basis:

which utilizes only the upper shift matrix and identity matrices. The annihilation matrices have the same form with the only difference that instead of an upper shift matrix we use a lower shift matrix.

[whordijk-blog]

Functional programming with Haskell

I recently dived into the functional programming language Haskell. I'm considering using it with my bachelor thesis, 'translating' the DMRG algorithm from Fortran: this will be quite a challenge!

[whordijk-blog]

I think I can safely say that nobody understands quantum mechanics.

Richard Feynman - The Character of Physical Law (1965)

[whordijk-blog]

My workstation for the coming semester: I have a nice view on the TU Delft "Mekelpark".