Problem: Consider a "crystal" consisting of two nuclei and two electrons arranged like this:
q1 q2 q1 q2
with a distance d betweem each. (q1=e, q2=-e)
a) Find the potential energy as a function of d.
b) Assuming the electrons to be restricted to a one-dimensional box of length 3d, find the minimum kinetic energy of the two electrons.
c) Find the value of d for which the total energy is a minimum.
And the Schrodinger equation
The Attempt at a Solution
The Potential energy I found to be (-7/3)(ke2/d) which is correct. (k=coulomb constant).
I assumed the minimum Kinetic energy would be the lowest allowed energy (basically E and n=1) because Potential energy should be zero inside the box. I got as a result pi2hbar2/18md2, but the correct answer is hbar2/36md2.
I have a factor of pi2 that I don't know how to get rid of
I'm missing a factor of 1/2 - is that because there are two electrons and it is thus 2m instead of m?
For c, d is supposed to equal hbar2/42mke2 and I assume it comes from the fact that Eelectric=kq2/r, but I'm not sure how to continue there. I'm guessing it has something to do with the kinetic energy I can't find.
hbar is the reduced planck constant (h/2pi)