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I want to solve a 2-dimensional particle in box problem with two electrons in the quantum well.I would like to take into account spin of electrons and Coulomb interactions to compute singlet and triplet eigenstates.

My question is it possible to solve this problem using finite difference method, if yes then please guide me through it.

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    $\begingroup$ Schrodinger equation does not include spin inherently, can you write down the equations that you want to solve? $\endgroup$
    – nicoguaro
    Oct 17, 2021 at 20:21

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When modelling spin in the Schrödinger equation, one has several alternatives which need to be chosen in advance. I'll copy an excerpt from a work of mine to give an overview (it considers atomic orbitals, but the idea is the same for a quantum well):

enter image description here

For each alternative, the sought two-particle wavefunction would look like $$ \Psi(x_1,x_2) = \sum_{ij} c_{ij}(t) \; \hat{\mathcal A} \phi_i(x_1) \phi_j(x_2) $$ where $\hat{\mathcal A}$ is the antisymmetrization operator.

After stating the available alternatives, let's go on by assuming you want to follow the spin-restricted ansatz. This makes things the easiest here, because for two particles, you either have a spin-polarization or (something comparable to) a "closed-shell" here, and by the comment in the excerpt, you then obtain spin-eigenfunctions.

Ok, one large step ahead, here is your TODO: discretize the two-particle Schrödinger equation on a two-dimensional Finite Difference grid, and numerically diagonalize the resulting matrix. The eigenfunctions $\phi_i(x_1,x_2)$ you get can be separated according to their behaviour with respect to the line $x_1=x_2$:

  • Odd functions, where $\phi_i(x_1,x_2)=0$ for $x_1=x_2$ correspond to the spin-polarized triplet state
  • Even functions correspond to the singlet state. You'll also find the groundstate here, which is the eigenfunction corresponding to the lowest eigenenergy
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