An electron wavefunction creates a local potential well, and the shape of the potential well changes the wavefunction. I tried to see how wavefunction tunneling affects the shape of the bubble and its energy levels. First, I recreated the results for an electron bubble without tunneling. Then, I modelled the effect of tunneling as a perturbation of the non-tunneling eigenstate and solved the problem using the Finite Difference and the Jacobi method. This solution took too long to run and I tried to find more efficient ways of solving it.
The techniques I learnt include
- General numerical methods like Runge-Kuttah, Gauss-Jordan elimination, Finite Difference method, Jacobi method, etc.
- Doing a complexity analysis of a numerical solution
- reducing problems with Dirichlet boundary conditions PDEs into numerically solvable systems by expanding in the basis of Bessel and Legendre polynomials
- imaginary time evolution of the Schrodinger equation to find the energy eigenstates
[Reference: (Humphrey J. MARIS, 10.1143/JPSJ.77.111008)