Numerov Method

 

Numerov

A python script that solves the one dimensional time-independent Schrodinger equation for bound states. The script uses a Numerov method to solve the differential equation and displays the desired energy levels and a figure with an approximate wave function for each of these energy levels.

Running

To run this code simply clone this repository and run the Numerov.py script with python (the numpy and matplotlib modules are required):

$ git clone https://github.com/FelixDesrochers/Numerov/
$ cd Numerov
$ python Numerov.py

Then the program will ask you to enter the number of energy levels you want to display and the desired potential (make sure that the potential is approximately centered at x=0):

$ >> Which first energy levels do you want (enter an integer) : 4
$ >> Potential (as a fonction of x): 3*(x^4)-2*(x^3)-6*(x^2)+x+5

Note: The programm may sometimes display less energy levels than what has been asked. To solve this problem modify the values of x_V_min and x_V_max in the parameters section of the Numerov.py script.

Examples

Harmonic Oscillator

For instance, if we want the energy levels for the quantum harmonic oscillator we would run the following commands:

$ python Numerov.py
$ >> Which first energy levels do you want (enter an integer) : 8
$ >> Potential (as a fonction of x): x**2

The program then displays the following figure:

And the following energies (please note that these are obtained using atomic units, see https://en.wikipedia.org/wiki/Atomic_units for more details):

Energy level 0 : 0.707658207399
Energy level 1 : 2.12132034435
Energy level 2 : 3.5355339059
Energy level 3 : 4.94974746826
Energy level 4 : 6.36396103051
Energy level 5 : 7.77817459266
Energy level 6 : 9.19238815459
Energy level 7 : 10.6066017163

Other Examples

This program can also solve the Schrödinger equation for all sorts of unorthodox problems such as the double-well potential or the absolute value potential.

Double-Well Potential

For a "double-well potential" with the following input,

$ python Numerov.py
$ >> Which first energy levels do you want (enter an integer) : 7
$ >> Potential (as a fonction of x): (x^4)-6*(x^2)+9

we get the following results:

Energy level 0 : 2.35727297545
Energy level 1 : 2.35937176485
Energy level 2 : 6.54881394689
Energy level 3 : 6.68442950475
Energy level 4 : 9.41561275062
Energy level 5 : 10.6784965326
Energy level 6 : 12.8773418447

Absolute value potential

And for the absolute value potential (V(x)=|x|), we get:

Energy level 0 : 0.81639999999
Energy level 1 : 1.85575743448
Energy level 2 : 2.57809582976
Energy level 3 : 3.24460762395
Energy level 4 : 3.82571482969
Energy level 5 : 4.38167123906
Energy level 6 : 4.89181971232
Energy level 7 : 5.38661378006
Energy level 8 : 5.8513002713
Energy level 9 : 6.30526300457

Algorithm

All informations about the used algorithm are described in the explain_algorithm.pdf file (incomplete).

Contributing

I am open to any improvement suggestion or contribution. If you wish to contribute to this repository just follow these simple steps:

1. Fork it (https://github.com/yourname/yourproject/fork)
2. Create your feature branch (git checkout -b feature/fooBar)
3. Commit your changes (git commit -am 'Add some fooBar')
4. Push to the branch (git push origin feature/fooBar)
5. Create a new Pull Request

License

MIT - http://alco.mit-license.org

(See the LICENSE.md for more informations)