Python library for solving spherically symmetric quantum-mechanical problems through basis expansion, with a focus on a complex-momentum basis.
This code was part of a Bachelor thesis project. For more information see the thesis Quantum Resonances in a Complex-Momentum Basis. Some functionality has been removed during cleanup since then, so a legacy branch was created for historical purposes.
- Python 2.7.x (not Python 3)
- SciPy 0.11.0 or later (including NumPy 1.6.2 or later)
- Optional: matplotlib 1.1.1 or later (Great for sketches, but we recommend using something like PGFPlots for quality graphics.)
To use the library, choose a problem, a basis to expand the problem in and values for the quantum numbers l and j. Import what you need:
from nhqm.solve import solve
from nhqm.quantum_numbers import QuantumNumbers
from nhqm.bases.momentum import MomentumBasis, gauss_contour
from nhqm.problems import Helium5
Then do something like this:
problem = Helium5()
quantum_numbers = QuantumNumbers(l=1, j=1.5)
contour = gauss_contour([0, 10], points_per_segment=10)
basis = MomentumBasis(contour)
states = solve(problem, quantum_numbers, basis)
The states obtained are solutions to the time-independent Schrödinger equation, i.e. stationary states. These can be used as a basis for a many body problem. The states contain information you can use:
ground_state = states[0]
print ground_state.energy
plot(ground_state.wavefunction) #pseudo-code
The scripts folder contains examples. Be sure to run and look at those.
##Problems
A problem is an object with a potential and a characteristic mass. It has some other parameters as well for convenience.
Two problems are included: the Hydrogen atom and the He5 core.
##Bases
To solve a problem, we expand it in a basis. The basis is an object with methods that can create a Hamiltonian matrix and generate wavefunctions from the eigenvectors of the Hamiltonian.
Two bases are included: The isotropic harmonic oscillator and the momentum basis (free particle basis).
The momentum basis can use complex momenta by integrating along a complex contour, yielding a non-hermitian (but symmetric) hamiltonian with possibly complex eigenvalues. The eigensolutions to such a Hamiltonian form the so-called Berggren basis which can be used as a single-particle basis when working with many-particle systems.
When solving a single-particle problem, the result is a list of SingleParticleStationaryState objects. These objects each represent a solution to the time-independent Schrödinger equation and contain information about problem, basis, quantum numbers, energy and wavefunction. This object representation can be useful when using the states as a single-particle basis for a many-body system.
The Quantum Number object is very simple, only containing l and j.
There are a number of helper functions included, ranging from very important to unimportant. The matrix and quantum modules contain the most important functions.