1. Solutions of the Schrödinger equation for the ground helium by finite element method
Jiahua Guo

2. Introduction Schrödinger equation Problems with traditional methods
Helium atom system
Challenge of solving He with FEM

3. Governing equations
In Cartesian coordinates, the spin-independent, nonrelativistic Schrödinger equation for the two electrons in the helium atom is:
In spherical coordinates:
L is the angular momentum operator and can be written as:
Thus the Hamiltonian operator can be finally written as:

4. Boundary conditions & Formulation
When out of the boundary
Formulation
Coefficient form of eigenvalue PDE in FEMlab:

5. Solution E = -2.7285 hartree = -74.22eV (Experimental result:
Eexp = eV
The slice scheme of the helium wave function with the lowest eigenvalue

6. Validation Energy levels for hydrogen atom Atomic orbits Energy level
Energy value based on Bohr model (hartree)
Energy value calculated by FemLab (hartree)
Error
n=1
0.28%
n=2
0.24%
Atomic orbits 1s 2s
2px
2py
2pz

7. Conclusion
Schrödinger equation can be simplified by decreasing some variables, making it an equation with fewer dimensions.
FEMlab is a good tool when trying to find out the eigenvalues of energy of some three-dimensional systems (e.g. hydrogen and helium atoms). However, it can’t deal with a complicated many-body Schrödinger equation.

8. Thank you!