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

In summary: I found.In summary, the conversation is regarding a problem involving a "crystal" consisting of two nuclei and two electrons arranged in a specific configuration. The questions involve finding the potential energy as a function of distance, the minimum kinetic energy of the two electrons, and the value of distance for which the total energy is a minimum. The equations used include the En=n2pi2hbar2/2mL2 and the Schrodinger equation. The conversation also discusses the correct answers for parts a and b and the use of hbar, the reduced Planck constant. A possible solution for part c is also mentioned, with a final answer of d= h^2 / (42ke
  • #1
danmel413
12
0

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


En=n2pi2hbar2/2mL2

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 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)

Thanks!
 
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  • #2
danmel413 said:
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.
Perhaps they are expressing their answer in terms of h rather than hbar. Also, don't forget you have two electrons in the box.
 
  • #3
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)
 
  • #4
ondryice said:
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!
 
  • #5
PeroK said:
You're 8 months too late!
Yeah I figured if someone googles the problem (like I did) then they'll find the whole problem
 

1. What is a 1D box in the context of electrons?

A 1D box refers to a hypothetical space in which two electrons are confined to move in only one dimension. This means that the electrons can only move along a straight line and are restricted from moving in any other direction.

2. How is kinetic energy defined in this scenario?

Kinetic energy in this scenario is the energy associated with the movement of the two electrons in the 1D box. It is calculated using the formula KE = 1/2 * m * v2, where m is the mass of the electron and v is its velocity.

3. Why is it important to find the minimum kinetic energy of two electrons in a 1D box?

Studying the minimum kinetic energy of two electrons in a 1D box helps us understand the behavior of electrons in confined spaces, which is crucial in many fields of physics and chemistry. It also allows us to make predictions about the physical properties of materials that contain confined electrons.

4. How do you calculate the minimum kinetic energy of two electrons in a 1D box?

The minimum kinetic energy of two electrons in a 1D box is calculated using the formula KEmin = (h2 * n2) / (8 * m * L2), where h is Planck's constant, n is the quantum number, m is the mass of the electron, and L is the length of the 1D box.

5. Can the minimum kinetic energy of two electrons in a 1D box be zero?

No, the minimum kinetic energy of two electrons in a 1D box cannot be zero. This is because of the Heisenberg uncertainty principle, which states that it is impossible to know the exact position and momentum of an electron simultaneously. Therefore, even at the lowest possible energy level, there will still be some kinetic energy associated with the movement of the electrons in the 1D box.

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