Question

Write the Hamiltonian operator for 2 electrons in a 1D box of length a.

I know that the Hamitonian is a sum of the kinetic and potential energies. I was thinking that for 2 electrons in a 1D box it would be:
-2((hbar^2)/2m_e)(d^2/dx^2)) + Coulomb interaction for 2 electrons

I am a bit confused because I don't seem to be using the length of the box anywhere in this.

Answer 1

The box size enters in when writing the potential V(x). It becomes infinite at the ends. There is an interaction potential, but a more important consequence of having two electrons is the Pauli exclusion principle. See the discussion at
http://galileo.phys.virginia.edu/classes/252/symmetry/Symmetry.html

In that discussion, the interaction potential is neglected.