Visualizing the 3D Schrödinger Equation: Quantum Eigenstates of a Particle Confined in 3D Wells

What do the quantum eigenstates of 3D wells look like?
In this video, we visualize the solutions of the 3D Schrödinger Equation computed for more than a total of 500 eigenstates of 2, 4, 8, and 12 wells, illustrating what the molecular orbitals for a molecule with that number of atoms look like.

These simulations are made with qmsolve, an open-source python package that we are developing for solving and visualizing quantum physics. You can find the source code here:
https://github.com/quantum-visualizat...

The way this simulator works is by discretizing the Hamiltonian of an arbitrary potential and diagonalizing it for getting the energies and the eigenstates of the system.
The eigenstates of this video are computed with high accuracy (less than 1% of relative error) by diagonalizing a Hamiltonian matrix with a shape of 1,000,000 x 1,000,000.

For a molecule that contains only a single electron, an orbital is exactly the same that its eigenstate.
Therefore in these examples, the eigenstates are equivalent to the orbitals.

In the video, it can be noticed that the first molecular orbitals can be visualized as a first-order approximation as a simple linear combination of the orbitals of a single well. However, as the energy of the eigenstates raises, their wave function starts to take much more complex shapes.

Between each eigenstate is plotted a transition between two eigenstates. This is made by preparing a quantum superposition of the two eigenstates involved.

#QuantumPhysics #MolecularOrbitals #QuantumChemistry