How do you solve degeneracy for 2-D particle in a box?

[PhuongV]

Homework Statement

What is the degeneracy of the energy level E =65 E0 of the two dimensional particle in a box?
Answer

Homework Equations

E=(h_^2/8mL^2)*(nx^2+ny^2)--> I think we use this eq.

The Attempt at a Solution

 

[Ansatz7]

As far as I can tell, it's basically a mathematical question - you need to find the number of combinations of integers n₁ and n₂ such that n₁² + n₂² = 65 * 2 (I think the 2 should be there since for the ground state energy n₁ = n₂ = 1).

 

[gomboc]

Assuming E0 is the zero point energy, you need to determine how many solutions there are to the equation

$$E_0 \left(\frac{8mL^2}{h^2}\right) = n_x^2 + n_y^2 = 2(65)$$

where n_x and n_y are positive integers. For example, one solution would be n_x = 9 and n_y = 7, so obviously n_x = 7 and n_y = 9 is also a solution, so the degeneracy is at LEAST 2. You just need to find all possible solutions, and then count them.

Fortunately the guess and test method works easily for this problem. You could solve it graphically, but the integers are so small I think it's easier to just guess here.

 

[PhuongV]

Thank you everyone! I figured it out some time ago! I was just not thinking at the time :)

 

[paranormal]

To solve for degeneracy in a 2-D particle in a box, we use the equation E=(h^2/8mL^2)*(nx^2+ny^2) where E is the energy level, h is Planck's constant, m is the mass of the particle, and L is the length of the box. The degeneracy of an energy level is the number of different ways a particular energy level can be achieved. In this case, the energy level E=65E0 can be achieved in multiple ways, depending on the values of nx and ny. To determine the degeneracy, we can rearrange the equation to solve for nx and ny:

nx^2+ny^2=(8mL^2E)/(h^2)

We can then use this equation to find all possible combinations of nx and ny that result in the energy level E=65E0. The number of possible combinations is the degeneracy of the energy level.

For example, if we assume that nx and ny can only take on integer values, we can solve for nx and ny using a trial and error method. We can start with nx=1 and solve for ny, then increase nx and repeat the process until we reach the desired energy level. In this case, we would find that nx=8 and ny=1, or nx=1 and ny=8, both result in the energy level E=65E0. Therefore, the degeneracy of the energy level E=65E0 is 2.

In summary, to solve for degeneracy in a 2-D particle in a box, we use the equation E=(h^2/8mL^2)*(nx^2+ny^2) to find all possible combinations of nx and ny that result in the desired energy level. The number of combinations is the degeneracy of the energy level.