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 :)