This software solves the time-independent Schödinger equation for the 1D, 2D and 3D cases employing a finite difference method. For example, to solve the 1D quantum harmonic oscillator one can use the following snippet:
using Brooglie
m = 1
ω = 1
a,b = -10,10 # Limits of the box
# Use atomic units!
V(x) = 0.5*m*ω^2*x^2
# Find the first 5 eigenstates
E, vv = solve(V, a=a, b=b, m=m, nev=5)E will be in Hartree (atomic units), remember that one Hartree is
equivalent to 27.21 eV. As expected, we obtain E≃ω(n+½) noticing that
ℏ=1 in these units:
| Analytical | Numerical |
|---|---|
| 0.5 | 0.500952 |
| 1.5 | 1.50275 |
| 2.5 | 2.50436 |
| 3.5 | 3.50576 |
| 4.5 | 4.50696 |
In the array vv we find 5 vectors (1 dimensional), one for each
eigenstate:
Read the help of solve to see all the available options. It accepts
potentials with arbitrary number of dimensions, like for example
V(x,y,z,k,w), and creates the Hamiltonian accordingly (read the
documentation for the methods used under the doc/ directory),
reshaping the vv vectors into the correct number of dimensions. For
example, if V(x,y) = x+y is used as potential the array vv will
contain matrices.
If the Hamiltonian is needed for a more detailed calculation, it can
be returned skipping the diagonalization with the buildH function.
- Fix all the tests, some will need to tweak
Normaxitersafter the last commits. - Add an hydrogen atom to the tests.