# Computing eigenvalues of Schrodinger equation with spin

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.

• Schrodinger equation does not include spin inherently, can you write down the equations that you want to solve? Oct 17, 2021 at 20:21

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.
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