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My question is more from applied quantum mechanics. Suppose I have a 2D conductor(or semiconductor). I use eigenstate representation of hamiltonian in transverse direction and real space representation in longitudinal direction (direction of current flow). Now,

1. Hω=Eω , ω being eigenstates and E eigenvalues.

2. To find H we need kinetic energy + U (potential).

3. we can find n = electron density by ωω* . density matrix.

4. once n is found we can calculate U (Hartree potential) by Poissons equation.

1 and 4 are solved self consistently untill U satisfies both equations.

If I have the H matrix after the self consistent loop is over i.e. I have actual value of potential U. Then what is the physical interpretation for Eigenvalues of H, are they the allowed energy levels??

Thanks in advance,

-Arya