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https://github.com/jonocarroll/dirac


https://github.com/jonocarroll/dirac

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# Dirac



## Solve the Dirac Equation for Hydrogen (electron + proton)



### Dr. Jonathan Carroll, PhD; Original ca 2010. Revised 2023.



This Fortran code was written to solve the Dirac equation for _muonic_ hydrogen, but has

been retrofitted to solve for regular hydrogen. Some comments removing additional

functionality remain for context or for re-use.



A (draft?) copy of the research article relevant to this calculation is

[included](https://github.com/jonocarroll/Dirac/blob/main/muH.pdf). This contains the 

relevant equations and a brief explanation of the approach.



## Running the program



1. Download or clone this repository, e.g. `git clone [email protected]:jonocarroll/Dirac.git`

1. Edit the `Makefile` to use whichever Fortran compiler you have. This has been tested with `gfortran`

1. Ensure a clean start with `make clean` which removes all `.o` and `.mod` files as well as the compiled binary

1. Compile the code with `make` (which automatically builds `make Dirac`)

1. Run the program with `./Dirac` which will perform the integration and produce the `Wavefunctions.dat` data file. 

   This will take some time, but provides progress updates to the terminal

1. Use your favourite plotting program to visualise the `F` and `G` components of the spinor



![F and G components of the spinor](Wavefunction.png)



## Editing Configuration



Two solvers are implemented; [Runge-Kutta](https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods)

and [Adams-Bashforth](https://en.wikipedia.org/wiki/Linear_multistep_method#Adams%E2%80%93Bashforth_methods). These

are selected by changing the

[`solving_method` parameter](https://github.com/jonocarroll/Dirac/blob/d94cd1988a0e5ccef8351424acd57a1c5950981f/FindEigenvalue.f90#L31-L32).



The step-size, maximum number of iterations, and other shared parameters are found in the

[shared.f90](https://github.com/jonocarroll/Dirac/blob/main/shared.f90) file. As this code was originally designed for

muonic hydrogen, many equations still refer to that, but I have set `m_muon = m_electron` as a shortcut to

solving for traditional hydrogen.



The Coulomb potential is set to the point distribution in [funcs.f90](https://github.com/jonocarroll/Dirac/blob/main/funcs.f90)

but this can be changed to a finite charge distribution if desired. Additional integration is required in that case, in which

case the additional required files are provided (as-is, without any cleanup) in the

[integration/](https://github.com/jonocarroll/Dirac/tree/main/integration) subdirectory.



The 1s state is solved for by default, but this can be changed by setting the

[`state`](https://github.com/jonocarroll/Dirac/blob/164fbcf5581fc054ed3b6c71de123a17e699e6db/FindEigenvalue.f90#L68)

variable. Three options are available; the 1s, 2s, and 2p states. Coding for additional states requires

adjusting many parameters including the known number of nodes, etc...



## Questions



I'm happy to take questions, but keep in mind I haven't really worked on this for more than a dozen years, so

my memory of how all the pieces fit together is limited. You can contact me either via 

[Issues here](https://github.com/jonocarroll/Dirac/issues/new/choose) or my website, 

[jcarroll.com.au](https://jcarroll.com.au).