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Solutions of the Schrödinger equation for the ground helium by finite element method

Explore the challenges of solving the Schrödinger equation for ground helium using the finite element method. Learn about boundary conditions, formulation, and validation process. Discover the simplifications and limitations of the FEMlab tool for quantum systems.

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Solutions of the Schrödinger equation for the ground helium by finite element method

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  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 • Boundary conditions When out of the boundary • Formulation Coefficient form of eigenvalue PDE in FEMlab:

  5. Solution E = -2.7285 hartree = -74.22eV (Experimental result: Eexp = -78.98eV The slice scheme of the helium wave function with the lowest eigenvalue

  6. Validation • Energy levels for hydrogen atom • 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!

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