Find the minimum kinetic energy of two electrons in a 1D box

[danmel413]

Homework Statement

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.

Homework Equations

E_n=n²pi²h_bar²/2mL²

And the Schrodinger equation

The Attempt at a Solution

[/B]
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 pi²h_bar²/18md², but the correct answer is h_bar²/36md².

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 h_bar²/42mke² and I assume it comes from the fact that E_electric=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.

h_bar is the reduced Planck constant (h/2pi)

Thanks!

 

[TSny]

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 pi²h_bar²/18md², but the correct answer is h_bar²/36md².

Perhaps they are expressing their answer in terms of h rather than hbar. Also, don't forget you have two electrons in the box.

 

[ondryice]

For part c, you should add the total potential energy you found in (a) to the minimum kinetic energy of the two electrons (answer to b), then differentiate the sum and solve for zero. When I solved, I got an answer of d= h^2 / (42ke^2m)

 

[PeroK]

For part c, you should add the total potential energy you found in (a) to the minimum kinetic energy of the two electrons (answer to b), then differentiate the sum and solve for zero. When I solved, I got an answer of d= h^2 / (42ke^2m)

You're 8 months too late!

 

[ondryice]

You're 8 months too late!

Yeah I figured if someone googles the problem (like I did) then they'll find the whole problem