Using Pyslise2D

Ixaru’s problem

As a first example we’ll take a look at the potential:

\[V(x, y) = (1+x^2)(1+y^2)\]

on the domain \([-5.5; 5.5]\times[-5.5; 5.5]\).

Using pyslise this problem can be easily worked on:

from pyslise import Pyslise2D

def V(x, y):
    return (1 + x**2) * (1 + y**2)

problem = Pyslise2D(V, -5.5,5.5, -5.5,5.5, x_tolerance=1e-6, y_count=25)

To find the closest eigenvalue in the neighbourhood of 5 one can use:

problem.eigenvalue(5)
# 5.52674387

pyslise is able to find the closest eigenvalue to a certain guess because the implemented algorithm is able to calculate an error-matrix that expresses of that given guess is an eigenvalue. In turn, this matrix can be used to improve that initial guess. Until a sufficiently accurate estimate is found.

There is method implemented to find all eigenvalues of the two-dimensional Schrödinger equation in a certain interval. But, as of yet, this method isn’t perfect. It is based on a few heuristics to ‘guess’ that all eigenvalues are found. This heuristic is implemented in .eigenvalues(Emin, Emax):

problem.eigenvalues(0,13)

[3.1959180850800877,
 5.526743864002774,
 5.526743877339405,
 7.55780334350954,
 8.031272354757498,
 8.444581365360518,
 9.92806092943532,
 9.928061003007299,
 11.311817072021494,
 11.31181710814099,
 12.103256481915713,
 12.201180767897501]