Studying the potential [42]: # Do not modify this function import numpy as np # Create a function giving the potential of the atom. # It is defined for x positive only located between 0 and the infinite. The potential barreer is centered at x = 0.5 def potential(x): if (x >= 0.8): return 0 return 0.5*np.e**-(((x-0.5)/0.1)**2) + np.e** (10*(1-x)) / 10000 + np.log(x+0.2) Your professor has created a function giving the potential of an atom as a function of the distance to the center of the atom. The units are arbitrary, as the values of the potential Task 1 Plot the potential and explain with your own words what this potential represents. You will need for this to use the explanations of the lectures, and it is also possible to have some help from the internet. Task 2 Explain how you will solve the following problem: you want to study the wave function evolution of a particle initially trapped into this potential, as a function of the energy of that particle. You will need to define the following: ⚫ which regions of interest you will use to split the problem ⚫ what are the conditions on the energy for each of these regions ⚫ what are the boundaries conditions at the limits of the regions Task 3 Solve mathematically the problem to reach the equations of the coefficients of the wave function. Your professor will not need to have the mathematical solution, but you will need to write down below the equations you obtained

Studying the potential [42]: # Do not modify this function import numpy as np # Create a function giving the potential of the atom. # It is defined for x positive only located between 0 and the infinite. The potential barreer is centered at x = 0.5 def potential(x): if (x >= 0.8): return 0 return 0.5*np.e-(((x-0.5)/0.1)2) + np.e** (10*(1-x)) / 10000 + np.log(x+0.2) Your professor has created a function giving the potential of an atom as a function of the distance to the center of the atom. The units are arbitrary, as the values of the potential Task 1 Plot the potential and explain with your own words what this potential represents. You will need for this to use the explanations of the lectures, and it is also possible to have some help from the internet. Task 2 Explain how you will solve the following problem: you want to study the wave function evolution of a particle initially trapped into this potential, as a function of the energy of that particle. You will need to define the following: ⚫ which regions of interest you will use to split the problem ⚫ what are the conditions on the energy for each of these regions ⚫ what are the boundaries conditions at the limits of the regions Task 3 Solve mathematically the problem to reach the equations of the coefficients of the wave function. Your professor will not need to have the mathematical solution, but you will need to write down below the equations you obtained