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README: Code instructions

Solving the Schrödinger equation for your own potential.

This code has been developed to be run using Jupyter Notebook. To correctly run the code to solve the Schrödinger equation for an arbitrary potential you choose you should open general_main.ipynb from the notebook. You will see there are some different blocks. You should only modify the initial block to set the potential and other constant values. Code comments are in catalan. The initial block you are to find in the code is the following,

#### Bloc de inputs
# Nota: es necessari tenir instal.lat ffmpeg

### Llibreries
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
from matplotlib import animation
import tridiagonal
import trapezoidal
import wavefunction

### Inputs
## Limits d'integracio
rmin = -5.0
rmax = 5.0
## Trobar valors propis
ninter1 = 5000
error1 = 1e-8
n1 = 1
n2 = 4
## Definim potencial
def potencial(rmin,h,n):
    v = np.zeros(n)
    for i in xrange(1,n+1):
        # Double-well potential
        #v[i-1] = 15*((rmin + h*i)**2 -1)**2
        # Anharmonic potential
        #v[i-1] = (rmin + h*i)**4
        # Square well potential
        if (i<int(n/3)):
            v[i-1] = 10
        elif (i>int(2*n/3)):
            v[i-1] = 10
        # Modified Poschl-Teller
        #v[i-1] = -17/((np.cosh(rmin + i*h))**2) 
    return v
## Wavefunction numerica
ninter2 = 10000
## Simulacio de l'evolucio temporal
# Coeficients del primers eigenstates
c0 = 1
c1 = 1
c2 = 1
c3 = 0
# Limits en el grafic
xlim_e = -3
xlim_d = 3
ylimd = 12
ylimb = 0
# Frequencia de la simulacio
w = 0.05

Integration limits

rmin and rmax correspond to the integration limits of the integration. You may have to change them depending on the potential you are working with.

Defining the potential

You should define the potential you want to use by using the original form but changing x to x = rmin + h·i. Some example potentials (Double-well, Anharmonic, Square well and Modified Poschl-Teller potential) are already defined in a comment on the code.

Eigenvalues

You can determine the error and the number of iterations used to obtain the eigenvalues. As well you can determine which eigenvalues should be found by setting the first eigenvalue n1 you want to obtain and the last one n2.

Simulation

  • c0, c1, c2 and c3 correspond to the coefficients of the different first four eigenstates from which is conformed the state which is simulated.
  • w correspond to the frequency of the simulation.

Once every parameter is set you can run all the blocks and you will obtain the different outputs: Eigenvalues, Eigenstates and a simulation of the probability density temporal evolution.

Please, DO NOT MODIFY other blocks than this first setting block. Modifications in other blocks would imply changing the inner structure of the code and may trigger malfunctions.

Other codes

As well the Github includes a series of codes which include:

  1. The codes which were used to prove the robustness and accuarcy of the numerical methods, using the Poschl-Teller potential (PoschlTeller.ipynb). You can access them freely and run them to obtain the results in question. Please, do not modify this codes.

  2. The codes which were used to solve the hyfrogen atom model and obtain the different components of the probability density function (main_3d.ipynb). In this case, the first block allows to modify the values of the quantic numbers to decide which probability density function is going to be obtained. Please, DO NOT MODIFY other blocks than this first setting block.

  3. The rest of the codes correspond to the different numerical methods used along all the applications developed. All of them have been accurately comented to ensure a good understanding of the codes for an external user in case it was to modify or change any section of it,