This repository follows the procedure from David Berenstein et al. - Bootstrapping Simple QM Systems. The detail mathematical derivation can be found in bootstrap_derivation.pdf, and the result will be discussed in bootstrap_result.pdf.
Since most of the code is written in Sympy, it is highly recommended to run with Jupyter Notebook or Jupyter Lab. The package we used is in requirements.txt.
python -m pip install -r requirements.txt- Construct a
classwith specific potential, also contains functions to compute recursion relation and the determinant of sub-matrices. - Find the interval for energy eigenvalues such that the Hankel Matrix is positive-semi definite. (Mainly processed by bootstrap_sympy.py)
- Plot the solved energy eigenvalues interval with different size of matrices.
To get a feeling how we do the Bootstrap Method, you can run bootstrap_numpy.ipynb, however the performance is restricted to due to the decimal precision.
Tha full main codes are written with ipynb, and run with sympy, select a specific potential and open the correspoding ipynb file.
-
harmonic_sympy.ipynb : bootstrapping with harmonic potential
$V(x)=kx^2$ -
hydrogen_sympy.ipynb : bootstrapping with Coulomb potential
$V(r)=-\frac{k}{r}$ -
harmonic_sympy.ipynb : bootstrapping with Yukawa potential
$V(r)=-\frac{k}{r}e^{-ar}$ , but approximated to first order
The computed result can be load from the correspoding directories (harmonic, hydrogen, yukawa_order1)