Hydrogen molecule in a small box, thought experiment

[edpell]

What would happen if we placed a hydrogen molecule in a box with sides less than the Bohr diameter. That is, a box so small the electrons could not orbit around the protons at a distance of a Bohr radius. What state would the two electrons and two protons take?

 

[Nugatory]

What would happen if we placed a hydrogen molecule in a box with sides less than the Bohr diameter. That is, a box so small the electrons could not orbit around the protons at a distance of a Bohr radius. What state would the two electrons and two protons take?

You can't build a box that size, but I expect that you don't literally mean a "box" and you'd be happy with an infinite square well potential... And we have a perfectly good Hamiltonian based on the interaction between the four charged particles... Just solve the Schrödinger equation for those conditions and see what comes out. It won't be an unexcited hydrogen molecule, but it will be some sensible arrangement of two protons and two electrons.

 

[edpell]

Yes, an infinite square well potential is fine. "Just solve Schrödinger equation" is easier said than done for four interacting particles in a potential well. Does anybody know of a software package that might be able to do this? Thanks.

 

[The_Duck]

You can probably get some qualitative insight into the solution for a very small box size without doing too much calculation. Suppose the box has a width of L. Suppose there were no interactions between the particles; then we just have a 3D infinite square well. The spacings between the energy levels in an infinite square well go like 1/L^2. Now add in the Coulomb interactions. The Coulomb potential is 1/r so we should expect the interaction energy to go like 1/L. So for small enough L (presumably L << the Bohr radius) the interaction energy becomes negligible because 1/L << 1/L^2. The particles essentially act like free particles, because they have too much kinetic energy to really notice the Coulomb force. If you wanted I think you could do time-independent perturbation theory to calculate the leading effects of the Coulomb interaction in this limit. I expect, though that this approach only really works for L << the Bohr radius.